Symmetry-protected entangling boundary zero modes in crystalline topological insulators

2.1.1. Hamiltonian.

We consider a lattice $\Lambda \subset {{\mathbb{Z}}^{d}}$ whose N sites are labeled by the vector r:= (r₁, ··· , r_N)^T. Each site r is also associated with N_orb orbital (flavors) labeled by the Greek letter α = 1, ··· , N_orb. We define the non-interacting second-quantized Hamiltonian

$$\begin{eqnarray}&&\hat{H} :=\underset{r,r\prime =1}{\overset{N}{\sum}} \underset{\alpha ,\alpha \prime =1}{\overset{{{N}_{\text{orb}}}}{\sum}} \hat{\psi}_{\alpha ,r}^{\dagger}{{\mathcal{H}}_{\alpha ,r;\alpha \prime ,r\prime}}{{\hat{\psi}}_{\alpha \prime ,r\prime}}.\end{eqnarray} \tag{ 2.1a }$$

The pair of creation $\left(\hat{\psi}_{\alpha ,r}^{\dagger}\right)$ and annihilation operators $\left({{\hat{\psi}}_{\alpha \prime ,r\prime}}\right)$ obey the fermion algebra

$$\begin{eqnarray}&&\begin{array}{l} \left\{\hat{\psi}_{\alpha ,r}^{\dagger},{{{\hat{\psi}}}_{\alpha \prime ,r\prime}}\right\}={{\delta}_{\alpha ,\alpha \prime}} {{\delta}_{r,r\prime}}, \\ \left\{\hat{\psi}_{\alpha ,r}^{\dagger},\hat{\psi}_{\alpha \prime ,r\prime}^{\dagger}\right\}=\left\{{{{\hat{\psi}}}_{\alpha ,r}},{{{\hat{\psi}}}_{\alpha \prime ,r\prime}}\right\}=0. \\ \end{array}\end{eqnarray} \tag{ 2.1b }$$

Hermiticity $\hat{H}={{\hat{H}}^{\dagger}}$ is imposed by demanding that the single-particle matrix elements obey

$$\begin{eqnarray}&&{{\mathcal{H}}_{\alpha ,r;\alpha \prime ,r\prime}}=\mathcal{H}_{\alpha \prime ,r\prime ;\alpha ,r}^{*}.\end{eqnarray} \tag{ 2.1c }$$

Locality, in the orbital-lattice basis, is imposed by demanding that

$$\begin{eqnarray}&&\mathop {{\rm{lim}}}\limits_{|r - r\prime | \to \infty } |{{\cal H}_{\alpha ,r;\alpha \prime ,r\prime }}| \leqslant \mathop {{\rm{lim}}}\limits_{|r - r\prime | \to \infty } c \times {{\rm{e}}^{ - |r - r\prime |/\ell }}\end{eqnarray} \tag{ 2.1d }$$

for some positive constant c and for some characteristic length scale ℓ independent of the lattice sites r, r' = 1, ···, N and of the orbital indices α, α' = 1, ··· , N_orb. If we collect the orbital (α) and lattice (r) indices into a single collective index I ≡ (α, r), we may introduce the short-hand notation

$$\begin{eqnarray}&&\hat{H}\equiv \underset{I,{{I}^{\prime}}\in \Omega}{\sum} \hat{\psi}_{I}^{\dagger}{{\mathcal{H}}_{I{{I}^{\prime}}}}{{\hat{\psi}}_{{{I}^{\prime}}}}\equiv {{\hat{\psi}}^{\dagger}}\mathcal{H}\hat{\psi},\end{eqnarray} \tag{ 2.1e }$$

where $\mathcal{H}$ is a Hermitian N_tot × N_tot matrix with N_tot = N_orb × N and Ω is the set

$$\begin{eqnarray}&&\Omega :=\left\{1,\cdots ,{{N}_{\text{orb}}}\right\}\times \Lambda \end{eqnarray} \tag{ 2.1f }$$

obtained by taking the Cartesian product of the set of orbitals with the set of lattice points.

The Hermitian matrix $\mathcal{H}$ has N_tot pairs of eigenvalues and eigenvectors (_I, υ_I) that are defined by demanding that

$$\begin{eqnarray}&&\mathcal{H}{{\upsilon}_{I}}={{\varepsilon}_{I}}{{\upsilon}_{I}},\quad \upsilon _{I}^{\dagger}{{\upsilon}_{I{{\prime}^{{}}}}}={{\delta}_{I,I{{\prime}^{}}}},\end{eqnarray} \tag{ 2.2a }$$

for I, I' = 1, ···, N_tot. With the help of the unitary matrix

$$\begin{eqnarray}&&\mathcal{U}=\left({{\upsilon}_{1}},\ldots ,{{\upsilon}_{{{N}_{\text{tot}}}}}\right),\end{eqnarray} \tag{ 2.2b }$$

we may represent the single-particle Hamiltonian $\mathcal{H}$ as the diagonal matrix

$$\begin{eqnarray}&&{{\mathcal{U}}^{\dagger}}\mathcal{H}\mathcal{U}=\text{diag}~\left({{\varepsilon}_{1}},\ldots ,{{\varepsilon}_{{{N}_{\text{tot}}}}}\right).\end{eqnarray} \tag{ 2.2c }$$

The canonical transformation

$$\begin{eqnarray}&&{{\hat{\psi}}_{I}}=:\underset{J=1}{\overset{{{N}_{\text{tot}}}}{\sum}} {{\mathcal{U}}_{IJ}}~{{\hat{\chi}}_{J}},\end{eqnarray} \tag{ 2.2d }$$

gives the representation

$$\begin{eqnarray}&&\hat{H}=\underset{I=1}{\overset{{{N}_{\text{tot}}}}{\sum}} \hat{\chi}_{I}^{\dagger}~{{\varepsilon}_{I}}~{{\hat{\chi}}_{I}}.\end{eqnarray} \tag{ 2.2e }$$

The ground state of N_f fermions is then the Fermi sea

$$\begin{eqnarray}&&|{{\Psi}_{\text{FS}}}\rangle : =\underset{I=1}{\overset{{{N}_{\text{f}}}}{\prod}} \hat{\chi}_{I}^{\dagger}|0\rangle ,\quad {{\hat{\chi}}_{I}}|0\rangle =0,\end{eqnarray} \tag{ 2.3a }$$

whereby we have assumed that

$$\begin{eqnarray}&&I<I{{\prime}^{{}}}\Rightarrow {{\varepsilon}_{I}}\leqslant {{\varepsilon}_{I{{\prime}^{}}}}.\end{eqnarray} \tag{ 2.3b }$$

2.1.2. Entanglement spectrum of the equal-time one-point correlation matrix.

We seek to partition the Fock space $\mathfrak{F}$, on which the non-interacting Hamiltonian $\hat{H}$ defined by equation (2.2) acts, into two Fock subspace, which we denote by $\mathfrak{F}$_A and $\mathfrak{F}$_B, according to the tensorial decomposition

$$\begin{eqnarray}&&\mathfrak{F}={{\mathfrak{F}}_{A}}\otimes {{\mathfrak{F}}_{B}}.\end{eqnarray} \tag{ 2.4 }$$

Two ingredients are necessary to define the subspaces $\mathfrak{F}$_A and $\mathfrak{F}$_B. We need a state from $\mathfrak{F}$ that is a single Slater determinant. It is for this quality that we choose the ground state (2.3). We need a basis, that we choose to be the orbital-lattice basis defined by the representation (2.2).

Following [29], we start from the equal-time one-point correlation function (matrix)

$$\begin{eqnarray}&&{{\mathcal{C}}_{IJ}} :=\langle {{\Psi}_{\text{FS}}}|\hat{\psi}_{I}^{\dagger}{{\hat{\psi}}_{J}}|{{\Psi}_{\text{FS}}}\rangle ,\quad I,J=1,\cdots ,{{N}_{\text{tot}}}.\end{eqnarray} \tag{ 2.5 }$$

Insertion of equations (2.3) and (2.2d) delivers

$$\begin{eqnarray}&&{{\mathcal{C}}_{IJ}}=\underset{I{{\prime}^{{}}}=1}{\overset{{{N}_{\text{f}}}}{\sum}} \mathcal{U}_{II\prime}^{*}{{\mathcal{U}}_{JI\prime}}\equiv \underset{I\prime =1}{\overset{{{N}_{\text{f}}}}{\sum}} \left\langle {{\upsilon}_{I\prime}}|I\right\rangle \left\langle J|{{\upsilon}_{I\prime}}\right\rangle ,\end{eqnarray} \tag{ 2.6 }$$

where I, J = 1, ···, N_tot. We define the N_tot × N_tot correlation matrix $\mathcal{C}$ by its matrix elements (2.6). One then verifies that

$$\begin{eqnarray}&&{{\mathcal{C}}^{\dagger}}=\mathcal{C},\quad {{\mathcal{C}}^{2}}=\mathcal{C},\end{eqnarray} \tag{ 2.7 }$$

i.e., the correlation matrix $\mathcal{C}$ is a Hermitian projector. The last equality of equation (2.6) introduces the bra and ket notation of Dirac for the single-particle eigenstates of $\mathcal{H}$ defined by equation (2.2) to emphasize that this projector is nothing but the sum over all the single-particle eigenstates of $\mathcal{H}$ that are occupied in the ground state. Thus, all N_tot eigenvalues of the correlation matrix $\mathcal{C}$ are either the numbers 0 or 1. When it is convenient to shift the eigenvalues of the correlation matrix from the numbers 0 or 1 to the numbers ± 1, this is achieved through the linear transformation

$$\begin{eqnarray}&&{{\mathcal{Q}}_{IJ}} :=\mathbb{I}-2{{\mathcal{C}}_{IJ}},\quad I,J=1,\cdots ,{{N}_{\text{tot}}}.\end{eqnarray} \tag{ 2.8 }$$

The occupied single-particle eigenstates of $\mathcal{H}$ in the ground state of $\hat{H}$ are all eigenstates of $\mathcal{Q}$ with eigenvalue −1. The unoccupied single-particle eigenstates of $\mathcal{H}$ in the ground state of $\hat{H}$ are all eigenstates of $\mathcal{Q}$ with eigenvalue +1. In other words, $\mathcal{Q}$ is the difference between the projector onto the unoccupied single-particle eigenstates of $\mathcal{H}$ in the ground state of $\hat{H}$ and the projector onto the occupied single-particle eigenstates of $\mathcal{H}$ in the ground state of $\hat{H}$. As such $\mathcal{Q}$ possesses all the symmetries of $\mathcal{H}$ and all the spectral symmetries of $\mathcal{H}$.

We denote the N_tot-dimensional single-particle Hilbert space on which the correlation function $\mathcal{C}$ acts by $\mathfrak{H}$. The labels A and B for the partition are introduced through the direct sum decomposition

$$\begin{eqnarray}&&\mathfrak{H}=~{{\mathfrak{H}}_{A}}\oplus ~{{\mathfrak{H}}_{B}},\end{eqnarray} \tag{ 2.9a }$$

whereby A and B are two non-intersecting subsets of the set Ω defined by equation (2.1f) such that

$$\begin{eqnarray}&&\Omega =A{\cup}^{}B,\quad A{\cap}^{}B=\varnothing ,\end{eqnarray} \tag{ 2.9b }$$

and

$$\begin{eqnarray}\mathcal{C}=\left(\begin{array}{c} {{C}_{A}} & {{C}_{AB}} \\ C_{AB}^{\dagger} & {{C}_{B}} \\ \end{array}\right),\quad \mathcal{Q}=\left(\begin{array}{c} \mathbb{I}-2{{C}_{A}} & -2{{C}_{AB}} \\ -2C_{AB}^{\dagger} & \mathbb{I}-2{{C}_{B}} \\ \end{array}\right).\end{eqnarray} \tag{ 2.9c }$$

By construction, the N_A × N_A block C_A and the N_B × N_B block C_B are Hermitian matrices. These blocks inherit the property that their eigenvalues are real numbers bounded between the numbers 0 and 1 from $\mathcal{C}$ being a Hermitian projector. We call the set

$$\begin{eqnarray}&&\sigma \left({{C}_{A}}\right):=\left\{{{\zeta}_{\iota}}|\exists {{v}_{\iota}}\in {{\mathbb{C}}^{{{N}_{\text{A}}}}},~{{C}_{A}}{{v}_{\iota}}={{\zeta}_{\iota}}{{v}_{\iota}}, \iota =1,\cdot \cdot \cdot ,{{N}_{A}}\right\}\end{eqnarray} \tag{ 2.10a }$$

of single-particle eigenvalues of the block C_A the entanglement spectrum of the correlation matrix $\mathcal{C}$. Any eigenvalue from σ(C_A) obeys

$$\begin{eqnarray}&&0\leqslant {{\zeta}_{\iota}}\leqslant 1,\quad \iota =1,\cdot \cdot \cdot ,{{N}_{A}}.\end{eqnarray} \tag{ 2.10b }$$

The single-particle eigenvalues of the N_A × N_A Hermitian matrix C_A can be shifted from their support in the interval [0, 1] to the interval [−1, +1] through the linear transformation

$$\begin{eqnarray}&&{{Q}_{A}} :=\mathbb{I}-2{{C}_{A}}.\end{eqnarray} \tag{ 2.11 }$$

We refer to the set σ(Q_A) of single-particle eigenvalues of the N_A × N_A Hermitian matrix Q_A as the entanglement spectrum of the correlation matrix $\mathcal{Q}$, which we shall abbreviate as the entanglement spectrum.

It is shown in [29] that there exists a N_A × N_A block Hermitian matrix (entanglement Hamiltonian) ${{H}^{E}}\equiv {{\left(H_{K,L}^{E}\right)}_{K,L\in A}}$ with the positive definite operator

$$\begin{eqnarray}&&{{\hat{\rho}}_{A}} :=\frac{{{\text{e}}^{-{{{\sum}^{}}_{K{{\prime}^{{}}},L{{\prime}^{{}}}\in A}}\hat{\psi}_{K{{\prime}^{{}}}}^{\dagger} H_{K{{\prime}^{{}}}L{{\prime}^{}}}^{E}{{{\hat{\psi}}}_{L{{\prime}^{{}}}}}}}}{\text{t}{{\text{r}}_{{{\mathfrak{F}}_{A}}}}{{\text{e}}^{-{{{\sum}^{}}_{K{{\prime}^{{}}},L{{\prime}^{{}}}\in A}}\hat{\psi}_{K{{\prime}^{{}}}}^{\dagger} H_{K{{\prime}^{{}}}L{{\prime}^{{}}}}^{E}{{{\hat{\psi}}}_{L{{\prime}^{{}}}}}}}}\end{eqnarray} \tag{ 2.12a }$$

whose domain of definition defines the Fock space $\mathfrak{F}$_A such that the block C_A from the correlation matrix (2.9c) is

$$\begin{eqnarray}&&{{C}_{A}}={{\left(\text{t}{{\text{r}}_{{{\mathfrak{F}}_{A}}}}~{{\hat{\rho}}_{A}}~\hat{\psi}_{K}^{\dagger}~{{\hat{\psi}}_{L}}\right)}_{K,L\in A}}.\end{eqnarray} \tag{ 2.12b }$$

The positive definite matrix ${{\hat{\rho}}_{A}}$ is the reduced density matrix acting on the Fock space $\mathfrak{F}$_A obtained by tracing the degrees of freedom from the Fock space $\mathfrak{F}$_B in the density matrix

$$\begin{eqnarray}&&{{\hat{\rho}}_{\text{FS}}} :=|{{\Psi}_{\text{FS}}}\rangle \langle {{\Psi}_{\text{FS}}}|\end{eqnarray} \tag{ 2.12c }$$

whose domain of definition is the Fock space $\mathfrak{F}$.

It is also shown in [29] that the single-particle spectrum σ(C_A) of the N_A × N_A Hermitian matrix C_A is related to the single-particle spectrum σ(H) of the N_A × N_A Hermitian matrix H^E by

$$\begin{eqnarray}&&{{\zeta}_{\iota}}=\frac{1}{{{\text{e}}^{{{\varpi}_{\iota}}}}+1},\quad \iota =1,\cdots ,{{N}_{A}}.\end{eqnarray} \tag{ 2.13 }$$

Equation (2.13) states that the dependence of the eigenvalue ζ_ι of C_A on the eigenvalue ϖ_ι of H is the same as that of the Fermi–Dirac function on the single-particle energy ϖ_ι when the inverse temperature is unity and the chemical potential is vanishing in units for which the Boltzmann constant is unity. Equation (2.13) is a one-to-one mapping between σ(C_A) and σ(H).

Equation (2.13) allows to express the entanglement entropy

$$\begin{eqnarray}&&S_{A}^{\text{ee}} :=-\text{t}{{\text{r}}_{{{\mathfrak{F}}_{A}}}}~\left({{\hat{\rho}}_{A}} \ln {{\hat{\rho}}_{A}}\right)\end{eqnarray} \tag{ 2.14 }$$

of the reduced density matrix ${{\hat{\rho}}_{A}}$ in terms of σ(C_A) through

$$\begin{eqnarray}&&S_{A}^{\text{ee}}=-\underset{\iota =1}{\overset{{{N}_{A}}}{\sum}} \left[{{\zeta}_{\iota}} \ln {{\zeta}_{\iota}}+\left(1-{{\zeta}_{\iota}}\right) \ln \left(1-{{\zeta}_{\iota}}\right)\right].\end{eqnarray} \tag{ 2.15 }$$

Zero modes of C_A are eigenstates of C_A with vanishing eigenvalues. The vector space spanned by the zero modes of C_A is denoted by ker (C_A) in linear algebra. We conclude that the zero modes of C_A do not contribute to the entanglement entropy.

2.1.3. Equal-time one-point correlation matrix and SUSY QM.

It was observed in [12]and [13] that the 2 × 2 block structure (2.9c) on the Hermitian correlation matrix $\mathcal{C}$ defined by its matrix elements (2.5) is compatible with the condition (2.7) that the correlation matrix is a projector if and only if the four conditions

$$\begin{eqnarray}&&C_{A}^{2}-{{C}_{A}}=-{{C}_{AB}}~C_{AB}^{\dagger},\end{eqnarray} \tag{ 2.16a }$$

$$\begin{eqnarray}&&{{Q}_{A}}{{C}_{AB}}=-{{C}_{AB}}~{{Q}_{B}},\end{eqnarray} \tag{ 2.16b }$$

$$\begin{eqnarray}&&C_{AB}^{\dagger}{{Q}_{A}}=-{{Q}_{B}}C_{AB}^{\dagger},\end{eqnarray} \tag{ 2.16c }$$

$$\begin{eqnarray}&&C_{B}^{2}-{{C}_{B}}=-C_{AB}^{\dagger}{{C}_{AB}},\end{eqnarray} \tag{ 2.16d }$$

hold. Here, the N_A × N_A matrix Q_A was defined in equation (2.11) and we have introduced the N_B × N_B matrix ${{Q}_{B}} :=\mathbb{I}-2{{C}_{B}}$.

The family of 2N_susy + 1 operators

$$\begin{eqnarray}&&\left(\hat{\mathcal{Q}}_{1}^{\dagger},{{\hat{\mathcal{Q}}}_{1}},\cdots ,\hat{\mathcal{Q}}_{{{N}_{\text{susy}}}}^{\dagger},{{\hat{\mathcal{Q}}}_{{{N}_{\text{susy}}}}},\hat{\mathcal{H}}\right)\end{eqnarray} \tag{ 2.17a }$$

acting on a common Hilbert space realizes the graded Lie algebra of supersymmetric quantum mechanics (SUSY QM) if and only if

$$\begin{eqnarray}&&\left[\hat{\mathcal{Q}}_{i}^{\dagger},\hat{\mathcal{H}}\right]=\left[{{\hat{\mathcal{Q}}}_{i}},\hat{\mathcal{H}}\right]=0,\end{eqnarray} \tag{ 2.17b }$$

$$\begin{eqnarray}&&\left\{{{\hat{\mathcal{Q}}}_{i}},\hat{\mathcal{Q}}_{j}^{\dagger}\right\}={{\delta}_{i,j}}\hat{\mathcal{H}}\end{eqnarray} \tag{ 2.17c }$$

holds for i, j = 1, ··· , N_susy. SUSY QM is realized when $\hat{\mathcal{H}}$, which is Hermitian by construction because of equation (2.17c), can be identified with a Hamiltonian. Conversely, a Hamiltonian in quantum mechanics is supersymmetric if there exists a factorization of the form (2.17). It is then convention to call the operators $\hat{\mathcal{Q}}_{i}^{\dagger}$ and ${{\hat{\mathcal{Q}}}_{i}}$ with i = 1, ··· , N_susy supercharges owing to equation (2.17b). We are going to show that the algebra (2.16) is an example of SUSY QM with N_susy = 1 in disguise.

To this end, we define the N_A × N_A, N_B × N_B, N_A × N_B, and N_B × N_A block matrices

$$\begin{eqnarray}&&{{S}_{A}} :=\mathbb{I}-Q_{A}^{2},\quad {{S}_{B}} :=\mathbb{I}-Q_{B}^{2},\end{eqnarray} \tag{ 2.18a }$$

$$\begin{eqnarray}&&{{M}^{+}}\equiv {{M}^{\dagger}} :=2{{C}_{AB}},\quad {{M}^{-}}\equiv M :=2C_{AB}^{\dagger},\end{eqnarray} \tag{ 2.18b }$$

respectively. By construction, the spectrum of the Hermitian N_A × N_A matrix S_A and that of the Hermitian N_B × N_B matrix S_B belong to the interval [0, 1]. One verifies with the help of equation (2.16) that

$$\begin{eqnarray}&&{{M}^{+}}{{S}_{B}}-{{S}_{A}}{{M}^{+}}=0,\quad {{M}^{-}}{{S}_{A}}-{{S}_{B}}{{M}^{-}}=0,\end{eqnarray} \tag{ 2.19a }$$

$$\begin{eqnarray}&&{{M}^{+}}{{M}^{-}}={{S}_{A}},\quad {{M}^{-}}{{M}^{+}}={{S}_{B}}.\end{eqnarray} \tag{ 2.19b }$$

We cannot close the graded Lie algebra (2.17) with the four block matrices (2.18). However, we still have the possibility to define the triplet of N_tot × N_tot matrices

$$\begin{eqnarray}{{\mathcal{S}}_{\text{susy}}}=\left(\begin{array}{c} {{S}_{A}} & 0 \\ 0 & {{S}_{B}} \\ \end{array}\right),\quad {{\mathcal{Q}}_{\text{susy}}} :=\left(\begin{array}{c} 0 & 0 \\ M & 0 \\ \end{array}\right),\end{eqnarray} \tag{ 2.20 }$$

and ${{\mathcal{Q}}^{\dagger}}$. One verifies that they satisfy the graded Lie algebra

$$\begin{eqnarray}&&\left[{{\mathcal{Q}}_{\text{susy}}},{{\mathcal{S}}_{\text{susy}}}\right]=\left[\mathcal{Q}_{\text{susy}}^{\dagger},{{\mathcal{S}}_{\text{susy}}}\right]=0,\end{eqnarray} \tag{ 2.21a }$$

$$\begin{eqnarray}&&\left\{{{\mathcal{Q}}_{\text{susy}}},{{\mathcal{Q}}_{\text{susy}}}\right\}=\left\{\mathcal{Q}_{\text{susy}}^{\dagger},\mathcal{Q}_{\text{susy}}^{\dagger}\right\}=0,\end{eqnarray} \tag{ 2.21b }$$

$$\begin{eqnarray}&&\left\{{{\mathcal{Q}}_{\text{susy}}},\mathcal{Q}_{\text{susy}}^{\dagger}\right\}={{\mathcal{S}}_{\text{susy}}},\end{eqnarray} \tag{ 2.21c }$$

i.e., the pair of supercharge matrices $\mathcal{Q}_{\text{susy}}^{\dagger}$ and ${{\mathcal{Q}}_{\text{susy}}}$ and the Hamiltonian matrix ${{\mathcal{S}}_{\text{susy}}}$ realize SUSY QM with N_tot = 1.

The grading labeled by A and B in the definition of the matrices in equation (2.21) originates from the decomposition of the single-particle Hilbert space $\mathcal{H}$ into the direct sum (2.9a). The matrix ${{\mathcal{S}}_{\text{susy}}}$ is block diagonal, i.e., it does not mix the subspaces $\mathfrak{H}$_A and $\mathfrak{H}$_B. The pair of matrices $\mathcal{Q}_{\text{susy}}^{\dagger}$ and ${{\mathcal{Q}}_{\text{susy}}}$ are off-diagonal with respect to the labels A and B, i.e., they mix the subspaces $\mathfrak{H}$_A and $\mathfrak{H}$_B. The matrix M⁻ ≡ M maps $\mathfrak{H}$_A into $\mathfrak{H}$_B. Its adjoint M⁺ ≡ M^† maps $\mathfrak{H}$_B into $\mathfrak{H}$_A. In the context of SUSY QM, the pair M⁻ and M⁺ are called intertwiner.

We are now going to show that the equations (2.18), (2.20), and (2.21) imply that the number of linearly independent eigenstates of ${{\mathcal{S}}_{\text{susy}}}$ with vanishing eigenvalues, i.e., the number of linearly independent zero modes is larger or equal to

$$\begin{eqnarray}&&|\text{dim}\text{}{{\mathfrak{H}}_{A}}-\text{dim}\text{}{{\mathfrak{H}}_{B}}|\equiv |{{N}_{A}}-{{N}_{B}}|.\end{eqnarray} \tag{ 2.22 }$$

To see this, we assume without loss of generality that N_A > N_B. When the dimension N_A of $\mathfrak{H}$_A is larger than the dimension N_B of $\mathfrak{H}$_B, M⁺ is a rectangular matrix with more rows than columns while M⁻ is a rectangular matrix with more columns than rows. There follows two consequences. On the one hand, the condition

$$\begin{eqnarray}&&{{M}^{-}}{{\mathfrak{H}}_{A}}=0\end{eqnarray} \tag{ 2.23 }$$

that defines the null space of M⁻ necessarily admits N_A − N_B > 0 linearly independent solutions in $\mathfrak{H}$_A, for we must solve N_B equations for N_A unknowns. On the other hand, the condition

$$\begin{eqnarray}&&{{M}^{+}}{{\mathfrak{H}}_{B}}=0\end{eqnarray} \tag{ 2.24 }$$

that defines the null space of M⁺ is overdetermined, for we must solve N_A equations for N_B unknowns. The condition

$$\begin{eqnarray}&&{{\mathcal{S}}_{\text{susy}}}~\mathfrak{H}=0\end{eqnarray} \tag{ 2.25 }$$

that defines the null space of ${{\mathcal{S}}_{\text{susy}}}$ delivers at least N_A − N_B > 0 linearly independent solutions of the form

$$\begin{eqnarray}&&\upsilon =\left(\begin{array}{c} {{\upsilon}_{A}} \\ 0 \\ \end{array}\right),\end{eqnarray} \tag{ 2.26 }$$

where

$$\begin{eqnarray}&&{{M}^{-}}{{\upsilon}_{A}}=0.\end{eqnarray} \tag{ 2.27 }$$

As observed by Witten in [30], the number of zero modes plays an important role in SUSY QM. The number of zero modes of ${{\mathcal{S}}_{\text{susy}}}$ is given by the Witten index

$$\begin{eqnarray}&&{{\Delta}_{\text{w}}} :=|\text{dim ker}\left({{M}^{-}}\right)-\text{dim ker}\left({{M}^{+}}\right)|\geqslant |{{N}_{A}}-{{N}_{B}}|.\end{eqnarray} \tag{ 2.28 }$$

The relevance of equation (2.28) to the entanglement spectrum of non-interacting fermions is the main result of section 2.1.3.

2.1.4. Chiral symmetry of the entanglement spectrum σ(Q_A).

We have seen that the existence of zero modes in the spectrum $\sigma \left({{\mathcal{S}}_{\text{susy}}}\right)$ of ${{\mathcal{S}}_{\text{susy}}}$ defined in equation (2.20) is guaranteed when the dimensionalities N_A and N_B of the single-particle Hilbert spaces $\mathfrak{H}$_A and $\mathfrak{H}$_B, respectively, are unequal. We have shown that this property is a consequence of a hidden supersymmetry. When N_A = N_B, we cannot rely on SUSY QM to decide if $0\in \sigma \left({{\mathcal{S}}_{\text{susy}}}\right)$.

The non-interacting Hamiltonian $\hat{H}$ defined by equation (2.1a) in the orbital-lattice basis acts on the Fock space $\mathfrak{F}$. Let $\mathscr{O}$ denote an operation, i.e., an invertible mapping of time, of the lattice, of the orbital degrees of freedom, or of compositions thereof. We represent this operation by either a unitary or an anti-unitary transformation on the Fock space $\mathfrak{F}$. In turn, it suffices to specify how the creation and annihilation operators defined by their algebra (2.1b) transform under the operation $\mathscr{O}$ to represent $\mathscr{O}$ on the Fock space $\mathfrak{F}$.

For example, the transformation law

$$\begin{eqnarray}&&{{\hat{\psi}}_{I}}\to \hat{O} {{\hat{\psi}}_{I}} {{\hat{O}}^{\dagger}}\equiv \underset{{{I}^{\prime}}=1}{\overset{{{N}_{\text{tot}}}}{\sum}} {{\mathcal{O}}_{{{I}^{\prime}}I}} {{\hat{\psi}}_{{{I}^{\prime}}}},\quad I=1,\cdots ,{{N}_{\text{tot}}},\end{eqnarray} \tag{ 2.29 }$$

where $\mathcal{O}=\left({{\mathcal{O}}_{IJ}}\right)\in U\left({{N}_{\text{tot}}}\right)$ is a unitary matrix, realizes in a unitary fashion the operation $\mathscr{O}$. The non-interacting Hamiltonian $\hat{H}$ defined in equation (2.1) and the correlation matrix $\mathcal{C}$ defined in equation (2.5) obey the transformation laws

$$\begin{eqnarray}&&\hat{H}\to \hat{O} \hat{H} {{\hat{O}}^{\dagger}}\Rightarrow {{\mathcal{H}}^{\text{T}}}\to {{\mathcal{O}}^{\dagger}} {{\mathcal{H}}^{\text{T}}} \mathcal{O},\end{eqnarray} \tag{ 2.30 }$$

and

$$\begin{eqnarray}&&\mathcal{C}\to {{\mathcal{O}}^{\dagger}} \mathcal{C} \mathcal{O},\end{eqnarray} \tag{ 2.31 }$$

respectively.

For comparison, the operations of time reversal $\left({{\mathscr{O}}_{\text{tr}}},{{\hat{O}}_{\text{tr}}},{{\mathcal{O}}_{\text{tr}}}\right)$ and charge conjugation (that exchanges particle and holes) $\left({{\mathscr{O}}_{\text{ph}}},{{\hat{O}}_{\text{ph}}},{{\mathcal{O}}_{\text{ph}}}\right)$ are anti-unitary transformations for which

$$\begin{eqnarray}&&\mathcal{C}\to \mathcal{O}_{\text{tr}}^{\dagger}\mathcal{C} {{\mathcal{O}}_{\text{tr}}},\quad {{\mathcal{O}}_{\text{tr}}}\equiv {{\mathcal{T}}_{\text{tr}}} K,\quad \mathcal{T}_{\text{tr}}^{-1}=\mathcal{T}_{\text{tr}}^{\dagger},\end{eqnarray} \tag{ 2.32 }$$

and

$$\begin{eqnarray}&&\mathcal{C}\to \mathcal{O}_{\text{ph}}^{\dagger}\mathcal{C} {{\mathcal{O}}_{\text{ph}}},\quad {{\mathcal{O}}_{\text{ph}}}\equiv {{\mathcal{T}}_{\text{ph}}} K,\quad \mathcal{T}_{\text{ph}}^{-1}=\mathcal{T}_{\text{ph}}^{\dagger},\end{eqnarray} \tag{ 2.33 }$$

respectively. Here, K is the anti-linear operation of complex conjugation.

The non-interacting Hamiltonian $\hat{H}$ has $\mathscr{O}$ as a symmetry if and only if

$$\begin{eqnarray}&&\hat{O} \hat{H} {{\hat{O}}^{\dagger}}=\hat{H}\Leftrightarrow {{\mathcal{H}}^{\text{T}}}={{\mathcal{O}}^{\dagger}} {{\mathcal{H}}^{\text{T}}} \mathcal{O}.\end{eqnarray} \tag{ 2.34 }$$

Moreover, if we assume the transformation law

$$\begin{eqnarray}&&|{{\Psi}_{\text{FS}}}\rangle \to {{\text{e}}^{\text{i}\Theta}}|{{\Psi}_{\text{FS}}}\rangle ,\quad 0\leqslant \Theta <2\pi ,\end{eqnarray} \tag{ 2.35 }$$

for the ground state (2.3), i.e., if we assume that the ground state does not break spontaneously the symmetry (2.34), then

$$\begin{eqnarray}&&\mathcal{C}={{\mathcal{O}}^{\dagger}} \mathcal{C} \mathcal{O}\Leftrightarrow \mathcal{Q}={{\mathcal{O}}^{\dagger}} \mathcal{Q} \mathcal{O}.\end{eqnarray} \tag{ 2.36a }$$

We want to derive what effect condition (2.36a) has on the entanglement spectrum σ(Q_A), i.e., when the direct sum decompositions (2.9) and

$$\begin{eqnarray}\mathcal{O}=\left(\begin{array}{c} {{O}_{A}} & {{O}_{AB}} \\ {{O}_{BA}} & {{O}_{B}} \\ \end{array}\right)\in U\left({{N}_{\text{tot}}}\right).\end{eqnarray} \tag{ 2.36b }$$

hold for two special cases.

First, we assume that

$$\begin{eqnarray}\mathcal{O}=\left(\begin{array}{c} {{O}_{A}} & 0 \\ 0 & {{O}_{B}} \\ \end{array}\right),\quad {{O}_{A}}\in U\left({{N}_{A}}\right),\quad {{O}_{B}}\in U\left({{N}_{B}}\right).\end{eqnarray} \tag{ 2.37a }$$

This situation arises when the operation $\mathscr{O}$ is compatible with the partitioning encoded by the two subsets A and B of Ω in the sense that

$$\begin{eqnarray}&&A=\mathscr{O}A,\quad B=\mathscr{O}B.\end{eqnarray} \tag{ 2.37b }$$

If so, condition (2.36) simplifies to

$$\begin{eqnarray}&&{{Q}_{A}}=O_{A}^{\dagger} {{Q}_{A}} {{O}_{A}},\end{eqnarray} \tag{ 2.38a }$$

$$\begin{eqnarray}&&{{C}_{AB}}=O_{A}^{\dagger} {{C}_{AB}} {{O}_{B}},\end{eqnarray} \tag{ 2.38b }$$

$$\begin{eqnarray}&&{{Q}_{B}}=O_{B}^{\dagger} {{Q}_{B}} {{O}_{B}}.\end{eqnarray} \tag{ 2.38c }$$

Thus, the N_A × N_A Hermitian block matrices C_A and ${{Q}_{A}}=\mathbb{I}-2{{C}_{A}}$ inherit the symmetry obeyed by the N_tot × N_tot Hermitian matrix $\mathcal{H}$ when equation (2.37) holds.

Second, we assume that

$$\begin{eqnarray}&&{{N}_{A}}={{N}_{B}}={{N}_{\text{tot}}}/2\end{eqnarray} \tag{ 2.39a }$$

and

$$\begin{eqnarray}\mathcal{O}=\left(\begin{array}{c} 0 & {{O}_{AB}} \\ {{O}_{BA}} & 0 \\ \end{array}\right),\quad {{O}_{AB}},{{O}_{BA}}\in U\left({{N}_{\text{tot}}}/2\right).\end{eqnarray} \tag{ 2.39b }$$

This situation arises when the operation $\mathcal{O}$ interchange the two subsets A and B of Ω,

$$\begin{eqnarray}&&A=\mathscr{O}B,\quad B=\mathscr{O}A.\end{eqnarray} \tag{ 2.39c }$$

If so, condition (2.36) simplifies to

$$\begin{eqnarray}&&{{Q}_{A}}=O_{BA}^{\dagger} {{Q}_{B}} {{O}_{BA}},\end{eqnarray} \tag{ 2.40a }$$

$$\begin{eqnarray}&&{{C}_{AB}}=O_{BA}^{\dagger} {{C}_{BA}} {{O}_{AB}},\end{eqnarray} \tag{ 2.40b }$$

$$\begin{eqnarray}&&{{Q}_{B}}=O_{AB}^{\dagger} {{Q}_{A}} {{O}_{AB}}.\end{eqnarray} \tag{ 2.40c }$$

We are going to combine equation (2.40) with equation (2.16). Multiplication of equation (2.40a) from the right by equation (2.40b) gives the relation

$$\begin{eqnarray}&&{{Q}_{A}} {{C}_{AB}}=O_{BA}^{\dagger} {{Q}_{B}} {{C}_{BA}} {{O}_{AB}},\end{eqnarray} \tag{ 2.41 }$$

while multiplication of equation (2.40b) from the right by equation (2.40c) gives the relation

$$\begin{eqnarray}&&{{C}_{AB}} {{Q}_{B}}=O_{BA}^{\dagger} {{C}_{BA}} {{Q}_{A}} {{O}_{AB}}.\end{eqnarray} \tag{ 2.42 }$$

We may then use equation (2.16b) to infer that

$$\begin{eqnarray}&&{{Q}_{A}} {{C}_{AB}}=-{{C}_{AB}} {{Q}_{B}}.\end{eqnarray} \tag{ 2.43 }$$

A second use of equation (2.40c) delivers

$$\begin{eqnarray}&&{{Q}_{A}} {{C}_{AB}}=-{{C}_{AB}} O_{AB}^{\dagger} {{Q}_{A}} {{O}_{AB}}.\end{eqnarray} \tag{ 2.44 }$$

We introduce the auxiliary (N_tot/2) ×(N_tot/2) matrix

$$\begin{eqnarray}&&{{\Gamma}_{\mathscr{O} A}}:={{C}_{AB}} O_{AB}^{\dagger},\end{eqnarray} \tag{ 2.45a }$$

whose domain and co-domain are the single-particle Hilbert space $\mathfrak{H}$_A. We may then rewrite equation (2.44) as the vanishing anti-commutator

$$\begin{eqnarray}&&\left\{{{Q}_{A}}, {{\Gamma}_{\mathscr{O} A}}\right\}=0.\end{eqnarray} \tag{ 2.45b }$$

The same manipulations with the substitution A ↔ B deliver

$$\begin{eqnarray}&&\left\{{{Q}_{B}}, {{\Gamma}_{\mathscr{O}B}}\right\}=0,\end{eqnarray} \tag{ 2.45c }$$

where

$$\begin{eqnarray}&&{{\Gamma}_{\mathscr{O}B}}:={{C}_{BA}} O_{BA}^{\dagger}\end{eqnarray} \tag{ 2.45d }$$

Equations (2.45b) and (2.45c) are the main result of section 2.1.4.

An important consequence of equation (2.45b) is that the spectra σ(Q_A) and σ(Q_B), are endowed with a symmetry not necessarily present in the spectrum $\sigma \left(\mathcal{H}\right)$ when equations (2.34), (2.36), and (2.39) hold. This spectral symmetry is reminiscent of the so-called chiral symmetry, the property that a single-particle Hamiltonian anti-commutes with a unitary operator, although, here, the operator ${{\Gamma}_{\mathscr{O}}}$ is not necessarily unitary. If we assume that C_AB is an invertible (N_tot/2) ×(N_tot/2) matrix, to any pair (1 − 2 ζ_ι ≠ 0, υ_ι) that belongs to σ(Q_A), the image pair $\left(-1+2 {{\zeta}_{\iota}}\ne 0, {{\Gamma}_{\mathscr{O}}} {{\upsilon}_{\iota}}\right)$ also belongs to σ(Q_A), and conversely. We shall say that the entanglement spectrum σ(Q_A) is chiral symmetric, when equations (2.34), (2.36), (2.39) hold with ${{\Gamma}_{\mathscr{O}}}$ defined in equation (2.45d) invertible.

In this paper, we shall consider a regular d-dimensional lattice Λ, say a Bravais lattice. We denote with ${{P}_{\parallel}}\subset {{\mathbb{R}}^{d}}$ the plane with the coordinates

$$\begin{eqnarray}&&\left(\begin{array}{c} {{r}_{\parallel}} \\ \mathbf{0} \\ \end{array}\right),\quad {{r}_{\parallel}}\in {{\mathbb{R}}^{n}},\quad n=1,\cdots ,d-1.\end{eqnarray} \tag{ 2.46a }$$

We denote with ${{P}_{\bot}}\subset {{\mathbb{R}}^{d}}$ the plane with the coordinates

$$\begin{eqnarray}&&\left(\begin{array}{c} \mathbf{0} \\ {{r}_{\bot}} \\ \end{array}\right),\quad {{r}_{\bot}}\in {{\mathbb{R}}^{d-n}},\quad n=1,\cdots ,d-1.\end{eqnarray} \tag{ 2.46b }$$

Any lattice point r from Λ can be written as

$$\begin{eqnarray}&&r\equiv \left(\begin{array}{c} {{r}_{\parallel}} \\ {{r}_{\bot}} \\ \end{array}\right)\in {{\mathbb{R}}^{d}}.\end{eqnarray} \tag{ 2.46c }$$

We define the operation of reflection $\mathcal{R}$ about the plane ${{P}_{\bot}}\subset {{\mathbb{R}}^{d}}$ by

$$\begin{eqnarray}&&r\equiv \left(\begin{array}{c} {{r}_{\parallel}} \\ {{r}_{\bot}} \\ \end{array}\right)\to \mathcal{R} r :=\left(\begin{array}{c} -{{r}_{\parallel}} \\ +{{r}_{\bot}} \\ \end{array}\right),\end{eqnarray} \tag{ 2.47a }$$

$$\begin{eqnarray}&&{{\hat{\psi}}_{\alpha ,r}}\to \hat{R} {{\hat{\psi}}_{\alpha ,r}} {{\hat{R}}^{\dagger}}\equiv \underset{\beta =1}{\overset{{{N}_{\text{orb}}}}{\sum}} {{\text{R}}_{\alpha \beta}} {{\hat{\psi}}_{\beta ,\mathcal{R}r}},\end{eqnarray} \tag{ 2.47b }$$

for any α = 1, ··· , N_orb and r ∈ Λ. We define the operation of parity (inversion) $\mathscr{P}$ by

$$\begin{eqnarray}&&r\to \mathscr{P} r :=-r,\end{eqnarray} \tag{ 2.48a }$$

$$\begin{eqnarray}&&{{\hat{\psi}}_{\alpha ,r}}\to \hat{P} {{\hat{\psi}}_{\alpha ,r}} {{\hat{P}}^{\dagger}}\equiv \underset{\beta =1}{\overset{{{N}_{\text{orb}}}}{\sum}} {{\text{P}}_{\alpha \beta}} {{\hat{\psi}}_{\beta ,\mathscr{P} r}},\end{eqnarray} \tag{ 2.48b }$$

for any α = 1, ···, N_orb and r ∈ Λ.

Even though we may impose on the non-interacting Hamiltonian (2.1) the conditions of translation and point-group symmetries when the lattice Λ is regular, a partitioning (2.9) might break these symmetries, for it involves lattice degrees of freedom. We seek a partitioning that preserves translation invariance within the 'plane' (2.46b) that is normal to the 'plane' (2.46a). With this partition in mind, we first perform the Fourier transformations

$$\begin{eqnarray}&&{{\mathcal{H}}_{I,{{I}^{\prime}}}}\left({{k}_{\bot}}\right) :=\frac{1}{{{N}_{\bot}}}\underset{{{r}_{\bot}},r_{\bot}^{\prime}}{\sum} \text{}\text{}{{\text{e}}^{\text{i}{{k}_{\bot}}\cdot \left({{r}_{\bot}}-r_{\bot}^{\prime}\right)}}{{\mathcal{H}}_{I,{{r}_{\bot}};{{I}^{\prime}},r_{\bot}^{\prime}}},\end{eqnarray} \tag{ 2.49a }$$

and

$$\begin{eqnarray}&&{{\mathcal{C}}_{I,{{I}^{\prime}}}}\left({{k}_{\bot}}\right) :=\frac{1}{{{N}_{\bot}}}\underset{{{r}_{\bot}},r_{\bot}^{\prime}}{\sum} \text{}\text{}{{\text{e}}^{\text{i}{{k}_{\bot}}\cdot \left({{r}_{\bot}}-r_{\bot}^{\prime}\right)}}{{\mathcal{C}}_{I,{{r}_{\bot}};{{I}^{\prime}},r_{\bot}^{\prime}}}\text{,}\end{eqnarray} \tag{ 2.49b }$$

on the single-particle Hamiltonian (2.1c) and equal-time one-point correlation matrix (2.5), respectively, where

$$\begin{eqnarray}&&I\equiv \left(\alpha ,{{r}_{\parallel}}\right),\quad {{I}^{\prime}}\equiv \left({{\alpha}^{\prime}},r_{\parallel}^{\prime}\right),\end{eqnarray} \tag{ 2.49c }$$

and N_⊥ is the number of lattice sites in Λ holding a suitably chosen r_|| fixed. For some suitably chosen r_⊥ held fixed, we may then define the non-intersecting partitioning of the set

$$\begin{eqnarray}&&\Omega :=\left\{\left(\alpha ,{{r}_{\parallel}}\right)|\alpha =1,\cdots ,{{N}_{\text{orb}}}, \left(\begin{array}{c} {{r}_{\parallel}} \\ {{r}_{\bot}} \\ \end{array}\right)\in \Lambda \right\}\end{eqnarray} \tag{ 2.49d }$$

into

$$\begin{eqnarray}&&A=\mathcal{R}B=\mathscr{P} B\end{eqnarray} \tag{ 2.49e }$$

and

$$\begin{eqnarray}&&B=\mathcal{R}A=\mathscr{P} A.\end{eqnarray} \tag{ 2.49f }$$

The discussion in section 2.1.3 is now applicable for each k_⊥ separately, i.e., equation (2.16) becomes

$$\begin{eqnarray}&&C_{A}^{2}\left({{k}_{\bot}}\right)-{{C}_{A}}\left({{k}_{\bot}}\right)=-{{C}_{AB}}\left({{k}_{\bot}}\right) C_{AB}^{\dagger}\left({{k}_{\bot}}\right),\end{eqnarray} \tag{ 2.50a }$$

$$\begin{eqnarray}&&{{Q}_{A}}\left({{k}_{\bot}}\right) {{C}_{AB}}\left({{k}_{\bot}}\right)=-{{C}_{AB}}\left({{k}_{\bot}}\right) {{Q}_{B}}\left({{k}_{\bot}}\right),\end{eqnarray} \tag{ 2.50b }$$

$$\begin{eqnarray}&&C_{AB}^{\dagger}\left({{k}_{\bot}}\right) {{Q}_{A}}\left({{k}_{\bot}}\right)=-{{Q}_{B}}\left({{k}_{\bot}}\right) C_{AB}^{\dagger}\left({{k}_{\bot}}\right),\end{eqnarray} \tag{ 2.50c }$$

$$\begin{eqnarray}&&C_{B}^{2}\left({{k}_{\bot}}\right)-{{C}_{B}}\left({{k}_{\bot}}\right)=-C_{AB}^{\dagger}\left({{k}_{\bot}}\right) {{C}_{AB}}\left({{k}_{\bot}}\right).\end{eqnarray} \tag{ 2.50d }$$

We impose the symmetry conditions

$$\begin{eqnarray}&&{{\mathcal{R}}^{\dagger}} C\left({{k}_{\bot}}\right)\mathcal{R}=C\left(+{{k}_{\bot}}\right),\end{eqnarray} \tag{ 2.51a }$$

$$\begin{eqnarray}&&{{\mathcal{P}}^{\dagger}} C\left({{k}_{\bot}}\right)\mathcal{P}=C\left(-{{k}_{\bot}}\right),\end{eqnarray} \tag{ 2.51b }$$

where $\mathcal{R}$ and $\mathcal{P}$ are unitary representations of the actions of the operations of reflection $\mathcal{R}$ and parity (inversion) $\mathscr{P}$ on the labels (2.49c) for each k_⊥ separately, respectively. We assume that N_A = N_B, for which it is necessary that the dimensionality d of the lattice is even with the pair of orthogonal planes spanned by r_|| and r_⊥ each (d/2)-dimensional, respectively. We then conclude section 2.1.4 with the identities

$$\begin{eqnarray}&&\left\{{{Q}_{A}}\left({{k}_{\bot}}\right),{{\Gamma}_{\mathcal{R}}}\left({{k}_{\bot}}\right)\right\}=0,\end{eqnarray} \tag{ 2.52a }$$

$$\begin{eqnarray}&&{{Q}_{A}}\left({{k}_{\bot}}\right) {{\Gamma}_{\mathscr{P}}}\left({{k}_{\bot}}\right)+{{\Gamma}_{\mathscr{P}}}\left({{k}_{\bot}}\right) {{Q}_{A}}\left(-{{k}_{\bot}}\right)=0,\end{eqnarray} \tag{ 2.52b }$$

where ${{\Gamma}_{\mathcal{R}}}\left({{k}_{\bot}}\right)$ and ${{\Gamma}_{\mathscr{P}}}\left({{k}_{\bot}}\right)$ are not assumed invertible for each k_⊥ separately.

2.1.5. Equal-time one-point correlation matrix and $\mathscr{P}\,\mathscr{CT}$ symmetry.

Any local quantum field theory with a Hermitian Hamiltonian for which Lorentz invariance is neither explicitly nor spontaneously broken must preserve the composition $\left(\mathscr{P}\ \mathscr{CT} \right)$ of parity $\left(\mathscr{P}\right)$, charge conjugation $\left(\mathscr{C}\right)$, and time-reversal $\left(\mathscr{T}\right)$, even though neither $\mathscr{P}$, nor $\mathscr{C}$, nor $\mathscr{T}$ need to be separately conserved [31]. The $\mathscr{P}\ \mathscr{CT}$ theorem implies the existence of antiparticles. It also implies that any composition of two out of the triplet of transformations $\mathscr{P}$, $\mathscr{C}$, and $\mathscr{T}$ is equivalent to the third. (Both the $\mathscr{P}$ and the $\mathscr{CP}$ symmetries are violated by the weak interactions [32–34]. Consequently, the $\mathscr{P}\ \mathscr{CT}$ theorem predicts that the $\mathscr{T}$ symmetry is violated by the weak interactions. A direct observation for the violation of the $\mathscr{T}$ symmetry by the weak interaction has been reported in [35].)

The $\mathscr{PCT}$ theorem does not hold anymore after relaxing the condition of Lorentz invariance. The effective Hamiltonians used in condensed matter physics generically break Lorentz invariance. However, the extent to which $\mathscr{P}$, $\mathscr{C}$, and $\mathscr{T}$ are separately conserved is not to be decided as a matter of principle but depends on the material and the relevant energy scales in condensed matter physics. For example, the effects of the Earth magnetic field or those of cosmic radiation are for most practical purposes irrelevant to the properties of materials in condensed matter physics. A non-relativistic counterpart to charge conjugation symmetry holds in a mean-field treatment of superconductivity. Inversion symmetry, the non-relativistic counterpart to symmetry under parity, is common to many crystalline states of matter. The effective Hamiltonian of electrons in magnetically inert materials preserves time-reversal symmetry in the absence of an external magnetic field. Conversely, the effective Hamiltonian of electrons in the presence of magnetic impurities in metals or if the crystalline host is magnetic break explicitly time-reversal symmetry.

It has become clear in the last eight years that the notion of symmetry protected phases of matter is useful both theoretically and experimentally in condensed matter physics. It has first lead to a classification for the possible insulating phases of electrons that are adiabatically connected to Slater determinants [36–41]. Some of the predicted insulating phases have then been observed in suitable materials. For each insulating phase, there exists a non-interacting many-body Hamiltonian $\hat{H}$ that is smoothly connected to all Hamiltonians describing this insulating phase by local unitary transformations [9]. This classification depends on the dimensionality of space and on the presence or absence of the following three discrete (i.e., involutive) symmetries. There is the symmetry of $\hat{H}$ under time reversal ${{\hat{O}}_{\text{tr}}}$,

$$\begin{eqnarray}&&\left[\hat{H},{{\hat{O}}_{\text{tr}}}\right]=0,\quad \hat{O}_{\text{tr}}^{2}=\pm 1.\end{eqnarray} \tag{ 2.53 }$$

There is the spectral symmetry of $\hat{H}$ under an anti-unitary transformation (particle–hole) ${{\hat{O}}_{\text{ph}}}$,

$$\begin{eqnarray}&&\left\{\hat{H},{{\hat{O}}_{\text{ph}}}\right\}=0,\quad \hat{O}_{\text{ph}}^{2}=\pm 1.\end{eqnarray} \tag{ 2.54 }$$

There is the spectral symmetry of $\hat{H}$ under a unitary transformation (chiral) ${{\hat{O}}_{\text{ch}}}$,

$$\begin{eqnarray}&&\left\{\hat{H},{{\hat{O}}_{\text{ch}}}\right\}=0.\end{eqnarray} \tag{ 2.55 }$$

There are two distinct insulating phases characterized by the presence or absence of the chiral spectral symmetry when both time reversal and particle–hole symmetry are broken. There are other eight distinct insulating phases when both time reversal and particle–hole symmetry are satisfied. For a fixed dimension of space, five of the ten insulating phases support gapless extended boundary states in geometries with open boundaries. Moreover, these three discrete symmetries can be augmented by discrete (involutive) symmetries that enforce reflection or mirror symmetries resulting in a rearrangement of which of the five insulating phases supporting gapless extended boundary states.

The single-particle matrix $\mathcal{Q}$ defined in equation (2.8) inherits from the single-particle matrix $\mathcal{H}$ defined by equation (2.2) any one of the three symmetries

$$\begin{eqnarray}&&\left[\mathcal{Q},{{\mathcal{O}}_{\text{tr}}}\right]=0,\quad \mathcal{O}_{\text{tr}}^{2}=\pm 1,\end{eqnarray} \tag{ 2.56a }$$

$$\begin{eqnarray}&&\left\{\mathcal{Q},{{\mathcal{O}}_{\text{ph}}}\right\}=0,\quad \mathcal{O}_{\text{ph}}^{2}=\pm 1,\end{eqnarray} \tag{ 2.56b }$$

$$\begin{eqnarray}&&\left\{\mathcal{Q},{{\mathcal{O}}_{\text{ch}}}\right\}=0,\end{eqnarray} \tag{ 2.56c }$$

with ${{\mathscr{O}}_{\text{ch}}}$ implementing the chiral transformation and ${{\mathcal{O}}_{\text{ch}}}$ its single-particle representation, ${{\mathscr{O}}_{\text{ph}}}$ implementing the particle–hole transformation and ${{\mathcal{O}}_{\text{ph}}}$ its single-particle representation, and ${{\mathscr{O}}_{\text{tr}}}$ implementing reversal of time and ${{\mathcal{O}}_{\text{tr}}}$ its single-particle representation, respectively. Here, we are assuming that there exists a partition

$$\begin{eqnarray}&&\Omega =A{\cup}^{}B,\quad A{\cap}^{}B=\varnothing ,\end{eqnarray} \tag{ 2.56d }$$

that obeys

$$\begin{eqnarray}&&A=\mathscr{O}A,\quad B=\mathscr{O}B,\end{eqnarray} \tag{ 2.56e }$$

with $\mathscr{O}={{\mathscr{O}}_{\text{tr}}},{{\mathscr{O}}_{\text{ph}}},{{\mathscr{O}}_{\text{ch}}}$, when the time-reversal symmetry or the particle–hole or the chiral spectral symmetries of $\mathcal{Q}$ are inherited from those of $\mathcal{H}$.

Conservation of $\mathscr{P}\mathscr{CT}$, as occurs in a relativistic local quantum-field theory, implies that the operations of parity $\mathscr{P}$, charge conjugation $\mathscr{C}$, and time-reversal $\mathscr{T}$ are not independent. The composition of any two of them is equivalent to the third one.

Even though $\mathscr{P}\,\mathscr{CT}$ may not be conserved for the non-interacting Hamiltonian $\mathcal{H}$ defined in equation (2.2), its correlation matrix $\mathcal{Q}$ defined in equation (2.8) or its entanglement matrix Q_A defined in equation (2.11) may obey a weaker form of a $\mathscr{P}\,\mathscr{CT}$-like relation as we now illustrate by way of two examples.

Case when $\mathscr{P}\,\mathscr{C}$ is equivalent to $\mathscr{T}$. When the parity (inversion) transformation $\mathscr{P}$ is a symmetry of $\mathcal{H}$, there exists a ${{\Gamma}_{\mathscr{P} A}}$ such that, by equation (2.45b), it anti-commutes with Q_A, i.e.,

$$\begin{eqnarray}&&\left\{{{Q}_{A}},{{\Gamma}_{\mathscr{P}A}}\right\}=0.\end{eqnarray} \tag{ 2.57 }$$

If we also assume that there exists a particle–hole transformation O_ph (an anti-unitary transformation) that anti-commutes with Q_A (e.g., if $\mathcal{H}$ is the Bogoliubov–de-Gennes Hamiltonian of a superconductor), i.e.,

$$\begin{eqnarray}&&\left\{{{Q}_{A}},{{O}_{\text{ph}}}\right\}=0,\end{eqnarray} \tag{ 2.58 }$$

it then follows that the composition

$$\begin{eqnarray}&&{{O}_{\text{tr}}} :={{O}_{\text{ph}}} {{\Gamma}_{\mathscr{P} A}}\end{eqnarray} \tag{ 2.59 }$$

is an anti-linear transformation that commutes with Q_A, i.e.,

$$\begin{eqnarray}&&\left[{{Q}_{A}},{{O}_{\text{tr}}}\right]=0.\end{eqnarray} \tag{ 2.60 }$$

The symmetry of $\mathcal{H}$ under parity and the spectral symmetry of $\mathcal{H}$ under particle–hole transformation have delivered an effective anti-linear symmetry of Q_A.

Case when $\mathscr{CT}$ is equivalent to $\mathscr{P}$. We assume that $\mathcal{H}$ defined in equation (2.2) anti-commutes with the conjugation of charge $\mathscr{C}$ represented by ${{\mathcal{O}}_{\text{ph}}}$, while it commutes with reversal of time $\mathscr{T}$ represented by ${{\mathcal{O}}_{\text{tr}}}$. We also assume that the partition (2.56d) obeys equation (2.56e) when $\mathscr{O}$ is ${{\mathcal{O}}_{\text{ph}}}$ or ${{\mathscr{O}}_{\text{tr}}}$, i.e.,

$$\begin{eqnarray}{{\mathcal{O}}_{\text{ph}}}=\left(\begin{array}{c} {{O}_{\text{ph}A}} & 0 \\ 0 & {{O}_{\text{ph}B}} \\ \end{array}\right),\quad {{\mathcal{O}}_{\text{tr}}}=\left(\begin{array}{c} {{O}_{\text{tr}A}} & 0 \\ 0 & {{O}_{\text{tr}B}} \\ \end{array}\right).\end{eqnarray} \tag{ 2.61 }$$

We define

$$\begin{eqnarray}{{\mathcal{O}}_{\text{ch}}} :={{\mathcal{O}}_{\text{ph}}} {{\mathcal{O}}_{\text{tr}}},\quad {{\mathcal{O}}_{\text{ch}}}=\left(\begin{array}{c} {{O}_{\text{ch}A}} & 0 \\ 0 & {{O}_{\text{ch}B}} \\ \end{array}\right).\end{eqnarray} \tag{ 2.62 }$$

It follows that ${{\mathcal{O}}_{\text{ch}}}$ is unitary, obeys equation (2.56e), and anti-commutes with $\mathcal{H}$. Hence, we may interpret ${{\mathcal{O}}_{\text{ch}}}$ as an effective chiral transformation. The symmetries and spectral symmetries of $\mathcal{H}$ are passed on to $\mathcal{Q}$ defined in equation (2.8). In particular,

$$\begin{eqnarray}&&{{\mathcal{O}}_{\text{ch}}} \mathcal{Q}=-\mathcal{Q} {{\mathcal{O}}_{\text{ch}}}.\end{eqnarray} \tag{ 2.63 }$$

We make the additional assumption that $\mathfrak{h}$_A and $\mathfrak{h}$_B are isomorphic and that the action of O_chA on $\mathfrak{h}$_A is isomorphic to that of O_chB on $\mathfrak{h}$_B. If so, we write

$$\begin{eqnarray}{{O}_{\text{ch}A}}\cong {{O}_{\text{ch}B}}\cong {{O}_{\text{ch}}},\quad {{\mathcal{O}}_{\text{ch}}}=\left(\begin{array}{c} {{O}_{\text{ch}}} & 0 \\ 0 & {{O}_{\text{ch}}} \\ \end{array}\right).\end{eqnarray} \tag{ 2.64 }$$

Equation (2.63) then gives four conditions,

$$\begin{eqnarray}&&{{O}_{\text{ch}}}\ {{Q}_{A}}=-{{Q}_{A}}\ {{O}_{\text{ch}}},\end{eqnarray} \tag{ 2.65a }$$

$$\begin{eqnarray}&&{{O}_{\text{ch}}}\ {{C}_{AB}}=-{{C}_{AB}}\ {{O}_{\text{ch}}},\end{eqnarray} \tag{ 2.65b }$$

$$\begin{eqnarray}&&{{O}_{\text{ch}}}\ {{C}_{BA}}=-{{C}_{BA}}\ {{O}_{\text{ch}}},\end{eqnarray} \tag{ 2.65c }$$

$$\begin{eqnarray}&&{{O}_{\text{ch}}}\ {{Q}_{B}}=-{{Q}_{B}}\ {{O}_{\text{ch}}}.\end{eqnarray} \tag{ 2.65d }$$

With the help of equation (2.16), one verifies that

$$\begin{eqnarray}&&\left({{O}_{\text{ch}}}\ {{C}_{BA}}\right)\ {{Q}_{A}}=-{{O}_{\text{ch}}}\ {{Q}_{B}}\ {{C}_{BA}}={{Q}_{B}}\ \left({{O}_{\text{ch}}}\ {{C}_{BA}}\right),\end{eqnarray} \tag{ 2.66a }$$

$$\begin{eqnarray}&&\left({{O}_{\text{ch}}}\ {{C}_{BA}}\right)\ {{C}_{AB}}={{C}_{BA}}\ \left({{C}_{AB}}\ {{O}_{\text{ch}}}\right),\end{eqnarray} \tag{ 2.66b }$$

$$\begin{eqnarray}&&\left({{C}_{AB}}\ {{O}_{\text{ch}}}\right)\ {{C}_{BA}}={{C}_{AB}}\ \left({{O}_{\text{ch}}}\ {{C}_{BA}}\right),\end{eqnarray} \tag{ 2.66c }$$

$$\begin{eqnarray}&&\left({{C}_{AB}}\ {{O}_{\text{ch}}}\right)\ {{Q}_{B}}=-{{C}_{AB}}\ {{Q}_{B}}\ {{O}_{\text{ch}}}={{Q}_{A}}\ \left({{C}_{AB}}\ {{O}_{\text{ch}}}\right).\end{eqnarray} \tag{ 2.66d }$$

This should be compared with equation (2.40). We conclude with the observation that there exists the transformation

$$\begin{eqnarray}{{\mathcal{O}}_{\text{eff}}} :=\left(\begin{array}{c} 0 & {{C}_{AB}}\ {{O}_{\text{ch}}} \\ {{O}_{\text{ch}}}\ {{C}_{BA}} & 0 \\ \end{array}\right)\end{eqnarray} \tag{ 2.67 }$$

that commutes with $\mathcal{Q}$,

$$\begin{eqnarray}&&\left[\mathcal{Q},{{\mathcal{O}}_{\text{eff}}}\right]=0.\end{eqnarray} \tag{ 2.68 }$$

The spectral symmetry of $\mathcal{H}$ under a particle–hole transformation and the symmetry of $\mathcal{H}$ under time reversal have conspired to provide $\mathcal{Q}$ with a symmetry under the transformation ${{\mathcal{O}}_{\text{eff}}}$.

We consider a many-body Hamiltonian acting on the Fock space $\mathfrak{F}$ introduced in equation (2.2) that describes N_f interacting fermions. Its normalized ground state is

$$\begin{eqnarray}&&|\Psi \rangle :=\underset{n=1}{\overset{{{2}^{{{N}_{\text{tot}}}}}}{\sum}} \underset{n_{1}^{\left(n\right)}=0,1}{\sum} \cdots \underset{n_{{{N}_{\text{tot}}}}^{\left(n\right)}=0,1}{\sum} {{\delta}_{n_{1}^{\left(n\right)}+\cdots +n_{{{N}_{\text{tot}}}}^{\left(n\right)},{{N}_{\text{f}}}}}\times c_{n_{1}^{\left(n\right)},\cdots ,n_{{{N}_{\text{tot}}}}^{\left(n\right)}}^{\left(\Psi \right)}|n_{1}^{\left(n\right)},\cdots ,n_{{{N}_{\text{tot}}}}^{\left(n\right)}\rangle .\end{eqnarray} \tag{ 2.69a }$$

The Slater determinant

$$\begin{eqnarray}&&\langle n_{1}^{\left(n\right)},\cdots ,n_{{{N}_{\text{tot}}}}^{\left(n\right)}| :=\underset{I=1}{\overset{{{N}_{\text{tot}}}}{\prod}} \langle 0|{{\left({{\hat{\chi}}_{I}}\right)}^{n_{I}^{\left(n\right)}}},\end{eqnarray} \tag{ 2.69b }$$

has $c_{n_{1}^{\left(n\right)},\cdots ,n_{{{N}_{\text{tot}}}}^{\left(n\right)}}^{\left(\Psi \right)}$ as its overlap with the ground state |Ψ〉.

In the presence of interactions, the equal-time one-point correlation function (matrix)

$$\begin{eqnarray}&&{{\mathcal{C}}_{IJ}} :=\langle \Psi |\hat{\psi}_{I}^{\dagger} {{\hat{\psi}}_{J}}|\Psi \rangle ,\quad I,J=1,\cdots ,{{N}_{\text{tot}}},\end{eqnarray} \tag{ 2.70 }$$

does not convey anymore the same information as the density matrix

$$\begin{eqnarray}&&{{\hat{\rho}}_{\Psi}} :=|\Psi \rangle \langle \Psi |,\quad \langle \Psi |\Psi \rangle =1.\end{eqnarray} \tag{ 2.71a }$$

In particular, equation (2.70) does not encode the full information contained in the reduced density matrices

$$\begin{eqnarray}&&\hat{\rho}_{\Psi}^{\left(A\right)} :=\text{t}{{\text{r}}_{{{\mathfrak{F}}_{B}}}} {{\hat{\rho}}_{\Psi}}\end{eqnarray} \tag{ 2.71b }$$

and

$$\begin{eqnarray}&&\hat{\rho}_{\Psi}^{\left(B\right)} :=\text{t}{{\text{r}}_{{{\mathfrak{F}}_{A}}}} {{\hat{\rho}}_{\Psi}},\end{eqnarray} \tag{ 2.71c }$$

where we have defined the partition in equations (2.4) and (2.9).

We seek a useful representation of the reduced density matrices (2.71b) and (2.71c) and how they might be related by symmetries of the interacting Hamiltonian. To this end, we assign the labels μ_A = 1, ··· , dim $\mathfrak{F}$_A and μ_B = 1, ··· , dim $\mathfrak{F}$_B to any orthonormal basis $\{|\Psi _{{{\mu}_{A}}}^{\left(A\right)}\rangle \}$ and $\{|\Psi _{{{\mu}_{B}}}^{\left(B\right)}\rangle \}$ that span the Fock spaces $\mathfrak{F}$_A and $\mathfrak{F}$_B, respectively. Without loss of generality, we assume dim $\mathfrak{F}$_A ⩾̸ dim $\mathfrak{F}$_B. We may write the expansion

$$\begin{eqnarray}&&|\Psi \rangle =\underset{{{\mu}_{A}}=1}{\overset{\text{dim}\text{}{{\mathfrak{F}}_{A}}}{\sum}} \underset{{{\mu}_{B}}=1}{\overset{\text{dim}\text{}{{\mathfrak{F}}_{B}}}{\sum}} {{D}_{{{\mu}_{A}} {{\mu}_{B}}}}|\Psi _{{{\mu}_{A}}}^{\left(A\right)}\rangle \otimes |\Psi _{{{\mu}_{B}}}^{\left(B\right)}\rangle \end{eqnarray} \tag{ 2.72 }$$

with the overlaps ${{D}_{{{\mu}_{A}} {{\mu}_{B}}}}\in \mathbb{C}$ defining the matrix elements of the dim $\mathfrak{F}$_A × dim $\mathfrak{F}$_B matrix D.

At this stage, we make use of the singular value decomposition

$$\begin{eqnarray}&&D=U \Sigma {{V}^{\dagger}}\end{eqnarray} \tag{ 2.73a }$$

with

$$\begin{eqnarray}&&U=\left({{U}_{{{\mu}_{A}}\mu _{A}^{\prime}}}\right)\end{eqnarray} \tag{ 2.73b }$$

a dim $\mathfrak{F}$_A × dim $\mathfrak{F}$_A unitary matrix,

$$\begin{eqnarray}\Sigma =\left(\begin{array}{c} {{\sigma}_{1}} & 0 & \cdots & {} & {} & 0 \\ 0 & \ddots & \ddots & {} & {} & \vdots \\ \vdots & \ddots & {{\sigma}_{\Re}} & {} & {} & {} \\ {} & {} & {} & 0 & \ddots & {} \\ {} & {} & {} & \ddots & \ddots & 0 \\ 0 & {} & {} & {} & 0 & 0 \\ \vdots & {} & {} & {} & \vdots & \vdots \\ \end{array}\right),\quad 0<{{\sigma}_{1}}\leqslant \cdots \leqslant {{\sigma}_{\Re}},\end{eqnarray} \tag{ 2.73c }$$

a dim $\mathfrak{F}$_A × dim $\mathfrak{F}$_B rectangular diagonal matrix of rank $\Re$ ⩽̸ dim $\mathfrak{F}$_B, and

$$\begin{eqnarray}&&V=\left({{V}_{{{\mu}_{B}}\mu _{B}^{\prime}}}\right)\end{eqnarray} \tag{ 2.73d }$$

a dim $\mathfrak{F}$_B × dim $\mathfrak{F}$_B unitary matrix.

With the help of equation (2.73), we have the Schmidt decomposition:

$$\begin{eqnarray}&&|\Psi \rangle =\underset{\nu =1}{\overset{\Re}{\sum}} {{\sigma}_{\nu}}|\tilde{\Psi}_{\nu}^{\left(A\right)}\rangle \otimes |\tilde{\Psi}_{\nu}^{\left(B\right)}\rangle ,\end{eqnarray} \tag{ 2.74a }$$

where

$$\begin{eqnarray}&&|\tilde{\Psi}_{\nu}^{(A)}\rangle :=\underset{{{\mu}_{A}}=1}{\overset{\text{dim}\text{}{{\mathfrak{F}}_{A}}}{\sum}} {{U}_{{{\mu}_{A}}\nu}}|\Psi _{{{\mu}_{A}}}^{\left(A\right)}\rangle \end{eqnarray} \tag{ 2.74b }$$

and

$$\begin{eqnarray}&&|\tilde{\Psi}_{\nu}^{\left(B\right)}\rangle :=\underset{{{\mu}_{B}}=1}{\overset{\text{dim}\text{}{{\mathfrak{F}}_{B}}}{\sum}} V_{{{\mu}_{B}}\nu}^{*}|\Psi _{{{\mu}_{B}}}^{\left(B\right)}\rangle .\end{eqnarray} \tag{ 2.74c }$$

The singular values {σ_ν} ≡ σ (Σ) of the rectangular matrix Σ are non-negative and obey the normalization condition

$$\begin{eqnarray}&&\underset{\nu =1}{\overset{\Re}{\sum}} \sigma _{\nu}^{2}=1\end{eqnarray} \tag{ 2.74d }$$

owing to the facts that the ground state (2.69) is normalized to one and that the basis of $\mathfrak{F}$_A and $\mathfrak{F}$_B are chosen orthonormal.

In the basis (2.74),

$$\begin{eqnarray}&&{{\hat{\rho}}_{\Psi}} :=\underset{\nu =1}{\overset{\Re}{\sum}} \underset{{{\nu}^{\prime}}=1}{\overset{\Re}{\sum}} {{\sigma}_{\nu}}{{\sigma}_{{{\nu}^{\prime}}}}|\tilde{\Psi}_{\nu}^{\left(A\right)}\rangle \langle \tilde{\Psi}_{{{\nu}^{\prime}}}^{\left(A\right)}|\otimes |\tilde{\Psi}_{\nu}^{\left(B\right)}\rangle \langle \tilde{\Psi}_{{{\nu}^{\prime}}}^{\left(B\right)}|\end{eqnarray} \tag{ 2.75a }$$

and the reduced density matrices (2.71b) and (2.71c) become the spectral decompositions

$$\begin{eqnarray}&&\hat{\rho}_{\Psi}^{\left(A\right)}=\underset{\nu =1}{\overset{\Re}{\sum}} \sigma _{\nu}^{2}|\tilde{\Psi}_{\nu}^{\left(A\right)}\rangle \langle \tilde{\Psi}_{\nu}^{\left(A\right)}|\end{eqnarray} \tag{ 2.75b }$$

and

$$\begin{eqnarray}&&\hat{\rho}_{\Psi}^{\left(B\right)} :=\underset{\nu =1}{\overset{\Re}{\sum}} \sigma _{\nu}^{2}|\tilde{\Psi}_{\nu}^{\left(B\right)}\rangle \langle \tilde{\Psi}_{\nu}^{\left(B\right)}|,\end{eqnarray} \tag{ 2.75c }$$

respectively. The reduced density matrices $\hat{\rho}_{\Psi}^{\left(A\right)}$ and $\hat{\rho}_{\Psi}^{\left(B\right)}$ are explicitly positive semi-definite. They share the same non-vanishing eigenvalues

$$\begin{eqnarray}&&0<\sigma _{\nu}^{2}\equiv {{\text{e}}^{-{{\omega}_{\nu}}}},\quad \text{0}\text{}\leqslant {{\omega}_{\nu}}<\infty ,\quad \nu \text{= 1},\cdots ,\Re ,\end{eqnarray} \tag{ 2.75d }$$

each of which can be interpreted as the probability for the ground state (2.69) to be in the orthonormal basis state $|\tilde{\Psi}_{\nu}^{\left(A\right)}\rangle \otimes |\tilde{\Psi}_{\nu}^{\left(B\right)}\rangle$.

As we did with equation (2.39a), we assume that

$$\begin{eqnarray}&&\text{dim}\ {{\mathfrak{F}}_{A}}=\text{dim}\ {{\mathfrak{F}}_{B}}=\mathfrak{D}.\end{eqnarray} \tag{ 2.76a }$$

Hence, dim $\mathfrak{F}$_A and dim $\mathfrak{F}$_B are isomorphic. This means that ${{\hat{\Gamma}}_{AB}} :{{\mathfrak{F}}_{B}} \mapsto {{\mathfrak{F}}_{A}}$ defined by

$$\begin{eqnarray}&&{{\hat{\Gamma}}_{AB}} :=\underset{\nu =1}{\overset{\mathfrak{D}}{\sum}} |\tilde{\Psi}_{\nu}^{\left(A\right)}\rangle \langle \tilde{\Psi}_{\nu}^{\left(B\right)}|\end{eqnarray} \tag{ 2.76b }$$

and ${{\hat{\Gamma}}_{BA}} :{{\mathfrak{F}}_{A}} \mapsto {{\mathfrak{F}}_{B}}$ defined by

$$\begin{eqnarray}&&{{\hat{\Gamma}}_{BA}} :=\underset{\nu =1}{\overset{\mathfrak{D}}{\sum}} |\tilde{\Psi}_{\nu}^{\left(B\right)}\rangle \langle \tilde{\Psi}_{\nu}^{\left(A\right)}|\end{eqnarray} \tag{ 2.76c }$$

are linear one-to-one maps. One verifies that

$$\begin{eqnarray}&&\hat{\Gamma}_{AB}^{-1}={{\hat{\Gamma}}_{BA}}=\hat{\Gamma}_{AB}^{\dagger},\end{eqnarray} \tag{ 2.77a }$$

$$\begin{eqnarray}&&\hat{\rho}_{\Psi}^{\left(A\right)}={{\hat{\Gamma}}_{AB}}\ \hat{\rho}_{\Psi}^{\left(B\right)}\ \hat{\Gamma}_{AB}^{\dagger},\quad \hat{\rho}_{\Psi}^{\left(B\right)}=\hat{\Gamma}_{AB}^{\dagger}\ \hat{\rho}_{\Psi}^{\left(A\right)}\ {{\hat{\Gamma}}_{AB}}.\end{eqnarray} \tag{ 2.77b }$$

We conclude section 2.2 with a counterpart for interacting fermions of the spectral symmetry (2.45). We assume that the operation $\mathscr{O}$ under which the partition is interchanged,

$$\begin{eqnarray}&&A=\mathscr{O}B,\quad B=\mathscr{O}A,\end{eqnarray} \tag{ 2.78 }$$

is represented by the unitary map $\mathcal{O} :{{\mathfrak{F}}_{A}} \mapsto {{\mathfrak{F}}_{B}}$ defined either by

$$\begin{eqnarray}&&|\tilde{\Psi}_{\nu}^{\left(B\right)}{{\rangle}_{\mathscr{O}}}:=\underset{\nu {{\prime}^{{}}}=1}{\overset{\Re}{\sum}} \mathcal{O}_{\nu \nu {{\prime}^{{}}}}^{*}|\tilde{\Psi}_{\nu {{\prime}^{}}}^{\left(A\right)}\rangle ,\end{eqnarray} \tag{ 2.79a }$$

or by its inverse ${{\mathcal{O}}^{\dagger}} :{{\mathfrak{F}}_{B}} \mapsto {{\mathfrak{F}}_{A}}$ defined by

$$\begin{eqnarray}&&|\tilde{\Psi}_{\nu {{\prime}^{}}}^{\left(A\right)}\rangle =:\underset{\nu {{\prime}^{{}}}=1}{\overset{\Re}{\sum}} {{\mathcal{O}}_{\nu {{\prime}^{{}}}\nu {{\prime}^{{}}}}}|\tilde{\Psi}_{\nu {{\prime}^{{}}}}^{\left(B\right)}{{\rangle}_{\mathscr{O}}},\end{eqnarray} \tag{ 2.79b }$$

whereby

$$\begin{eqnarray}&&\underset{\nu {{\prime}^{}}=1}{\overset{\Re}{\sum}} \mathcal{O}_{\nu \nu {{\prime}^{}}}^{*}{{\mathcal{O}}_{\nu {{\prime}^{}}\nu {{\prime}^{}}}}={{\delta}_{\nu ,\nu {{\prime}^{}}}}\end{eqnarray} \tag{ 2.79c }$$

with ν, ν ', ν '' = 1, ···, $\Re$.

By equation (2.75b),

$$\begin{eqnarray}&&{{\left(\hat{\rho}_{\Psi}^{\left(A\right)}\right)}_{\mathcal{O}}}:=\underset{\nu ,\nu {{\prime}^{}},\nu {{\prime}^{}}=1}{\overset{\Re}{\sum}} \sigma _{\nu}^{2}\mathcal{O}_{\nu {{\prime}^{}}\nu}^{*}{{\mathcal{O}}_{\nu {{\prime}^{}}\nu}}|\tilde{\Psi}_{\nu {{\prime}^{}}}^{\left(B\right)}{{\rangle}_{\mathcal{O}~\mathcal{O}}}\langle \tilde{\Psi}_{\nu {{\prime}^{}}}^{\left(B\right)}|.\end{eqnarray} \tag{ 2.80 }$$

If we impose the symmetry constraint

$$\begin{eqnarray}\underset{\nu =1}{\overset{\Re}{\sum}} \sigma _{\nu}^{2}\mathcal{O}_{\nu {{\prime}^{{}}}\nu}^{*}{{\mathcal{O}}_{\nu {{\prime}^{{}}}\nu}}=\left\{\begin{array}{c} \sigma _{\nu {{\prime}^{{}}}}^{2}{{\delta}_{\nu {{\prime}^{{}}},{{\nu}^{{}}}}}, & \text{if}~\nu {{\prime}^{{}}}=1,\cdots ,\Re , \\ 0, & \text{otherwise,} \\ \end{array}\right.\end{eqnarray} \tag{ 2.81a }$$

(i.e., ${{\mathcal{O}}^{\dagger}}\Sigma \mathcal{O}=\Sigma )$, we conclude that

$$\begin{eqnarray}&&{{\left(\hat{\rho}_{\Psi}^{\left(A\right)}\right)}_{\mathscr{O}}}=\underset{\nu =1}{\overset{\Re}{\sum}} \sigma _{\nu}^{2}|\tilde{\Psi}_{\nu}^{\left(B\right)}{{\rangle}_{\mathscr{O} \mathscr{O}}}\langle \tilde{\Psi}_{\nu}^{\left(B\right)}|\equiv \hat{\rho}_{\mathscr{O}\Psi}^{\left(B\right)}\end{eqnarray} \tag{ 2.81b }$$

We can combine equations (2.77) and (2.81) as follows. Given are the interacting Hamiltonian $\hat{H}$ acting on the Fock space $\mathfrak{F}$ defined in the orbital-lattice basis, the density matrix ${{\hat{\rho}}_{\Psi}}$ for the ground state |Ψ〉 of $\hat{H}$, the partition $\mathfrak{F}$ = $\mathfrak{F}$_A ⊗ $\mathfrak{F}$_B, and the reduced density matrix $\hat{\rho}_{\Psi}^{\left(A\right)}$ and $\hat{\rho}_{\Psi}^{\left(B\right)}$ acting on the Fock spaces $\mathfrak{F}$_A and $\mathfrak{F}$_B with dim $\mathfrak{F}$_A = dim $\mathfrak{F}$_B, respectively. Let $\hat{\Sigma}$ represent the action by conjugation on the space of operators of the matrix (2.73c). Let $\hat{O}$ represent the action by conjugation on the space of operators of the matrix $\mathcal{O}$ with the matrix elements (2.79). Let

$$\begin{eqnarray}&&\hat{\Xi} :={{\hat{\Gamma}}_{BA}}\ \hat{O}\end{eqnarray} \tag{ 2.82 }$$

and assume the symmetry

$$\begin{eqnarray}&&{{\hat{O}}^{\dagger}} \hat{\Sigma} \hat{O}=\hat{\Sigma}.\end{eqnarray} \tag{ 2.83 }$$

The symmetry

$$\begin{eqnarray}&&\hat{\rho}_{\Psi}^{\left(A\right)}={{\hat{\Xi}}^{\dagger}} \hat{\rho}_{\Psi}^{\left(A\right)} \hat{\Xi}\end{eqnarray} \tag{ 2.84 }$$

obeyed by the reduced density matrix then follows.

In this context, the following observations are crucial. On the one hand, none of the results of section 2 are sensitive to the presence or absence of a spectral gap Δ separating the Slater determinant entering the definition (2.5) of the equal-time one-point correlation matrix $\mathcal{C}$ from all excited states when periodic boundary conditions are imposed. On the other hand, the stability analysis from sections 4–7 only makes sense if Δ > 0, since the exponential decay of the boundary states away from the boundaries is controlled by the characteristic length ξ ∝ 1/Δ. Consequently, the overlap of two boundary states localized on disconnected boundaries a distance ∝ N apart vanishes exponentially fast as the thermodynamic limit N → ∞ is taken, i.e., the matrix elements ${{\mathcal{C}}_{IJ}}$ in equation (2.5) also decay exponentially fast if the pair of repeat unit cells in the collective labels I and J are a distance larger than ξ apart. In particular, the matrix elements entering the upper-right block C_AB in equation (2.9) and those entering the transformation (2.45a) are suppressed exponentially in magnitude if they correspond to two repeat unit cells a distance larger than ξ apart. It is only because of the presence of a spectral gap above the ground state of the Hamiltonian $\mathcal{H}$ defined in equation (2.1) that $\mathcal{C}$ defined in equation (2.9) inherits from $\mathcal{H}$ its locality in the orbital basis.

In the remaining of this paper, we shall be concerned with exploring the effects of the spectral symmetry (2.45a) on the entanglement spectra when a single-particle bulk spectral gap Δ guarantees that the equal-time one-point correlation matrices (2.5) and (2.8) are local for bulk-like separations, i.e., their matrix elements in the position basis can be bounded from above by the exponential factor a exp(−b |r| Δ) where a and b are some numerical factor of order unity and |r| ⩾̸ 1/Δ (in units with the Planck constant set to ℏ = 1 and the speed of light set to c = 1) is the bulk-like distance in space of the lattice degrees of freedom entering the bra and ket. In other words, we seek to understand how the spectral properties of the block Q_A of the equal-time one-point correlation matrix defined in equation (2.8) are affected by the assumption that a symmetry operation $\mathscr{O}$ conspires with the partition of the degrees of freedom of the single-particle Hilbert space so as to obey equation (2.39), i.e., the map

$$\begin{eqnarray}&&{{\Gamma}_{\mathscr{O}A}} :~{{\mathfrak{H}}_{A}} \mapsto ~{{\mathfrak{H}}_{A}}\end{eqnarray} \tag{ 3.1 }$$

defined by the matrix multiplication ${{\Gamma}_{\mathscr{O} A}}:={{C}_{AB}}\ O_{AB}^{\dagger}$ anti-commutes with Q_A.

Typically, the symmetry operation $\mathscr{O}$ is a point-group symmetry, say an inversion or reflection symmetry. Hence, we shall assume that the symmetry operation $\mathscr{O}$ is not local. The spectral symmetry of Q_A generated by ${{\Gamma}_{\mathscr{O}A}}$ allows to draw definitive conclusions on the existence of protected zero modes in the entanglement spectrum σ(Q_A) only if ${{\Gamma}_{\mathscr{O}A}}$ is a local operation for bulk-like separations. For this reason, we devote section (3.2) to studying the conditions under which ${{\Gamma}_{\mathscr{O}A}}$ is local for bulk-like separations.

We shall consider two geometries for simplicity. Either we impose on d-dimensional space the geometry of a cylinder by imposing periodic boundary conditions in all (d − 1) directions in space while imposing open boundary conditions for the last direction. Or we impose on d-dimensional space the geometry of a torus by imposing periodic boundary conditions to all d directions in space. When space is one dimensional, the cases of cylindrical and torus geometry are illustrated in figures 1(a) and(b), respectively.

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For a cylindrical geometry, as illustrated in figure 1(a), there are two disconnected compact physical boundaries at x = ±∞ and one compact entangling boundary at x = 0, where x is the non-compact coordinate along the cylinder axis. We choose the symmetry operation $\mathscr{O}$ to be the mirror operation x ↦ −x that leaves the compact entangling boundary at x = 0 point-wise invariant, while it exchanges the two disconnected compact physical boundaries at x = ±∞. The matrix elements of ${{\Gamma}_{\mathscr{O} A}}$ involving bra and kets with lattice degrees of freedom a distance |r| ≫ 1/Δ away from the entangling boundary at x = 0 are exponentially suppressed by the factor exp(−b|r|Δ) originating from $C_{AB}^{\dagger}$. Hence, ${{\Gamma}_{\mathscr{O} A}}$ is a local operator for bulk-like separations that generates a spectral symmetry on Q_A without mixing boundary states localized on disconnected boundaries (whether physical or entangling).

For a torus geometry, as illustrated in figure 1(a), there are two disconnected compact entangling boundaries at θ = 0 and θ = π. On the one hand, we may choose the symmetry operation $\mathscr{O}$ to be the mirror operation θ ↦ −θ that leaves the compact entangling boundary at θ = 0 and θ = π point-wise invariant. The matrix elements of ${{\Gamma}_{\mathscr{O} A}}$ involving bra and kets with lattice degrees of freedom a distance |r| ≫ 1/Δ away from the entangling boundaries at θ = 0 and θ = π are exponentially suppressed by the factor exp(−b|r|Δ) originating from $C_{AB}^{\dagger}$. Hence, ${{\Gamma}_{\mathscr{O} A}}$ is again a local operator for bulk-like separations that generates a spectral symmetry on Q_A without mixing boundary states localized on disconnected entangling boundaries. On the other hand, we may choose the symmetry operation $\mathscr{O}$ to be the mirror operation θ ↦ π + θ that exchanges the compact entangling boundary at θ = 0 and θ = π. Hence, ${{\Gamma}_{\mathscr{O} A}}$ is not a local operator for it mixes with an amplitude of order unity boundary states localized on disconnected entangling boundaries separated by an arbitrary distance. Hence, even though ${{\Gamma}_{\mathscr{O} A}}$ generates a spectral symmetry of the entanglement spectrum of Q_A, it cannot be used to deduce any stability properties of the zero modes localized on the boundaries.

Sections 4–7 are devoted to the stability analysis of entangling boundary states by way of examples in one and two dimensions. This stability analysis requires distinguishing the nature of the boundaries in the partition (2.9), for whether these boundaries are physical or entangling depends on the choice of the boundary conditions imposed along the d dimensions of space, as we have illustrated for the case of d = 1 in figure 2. This stability analysis also requires determining if the spectral symmetry ${{\Gamma}_{\mathscr{O} A}}$ is local or not, as we have illustrated for the case of d = 1 in figure 1.

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Our first example is defined by choosing d = 1 and N_orb = 4 in equation (2.1). The lattice Λ is one-dimensional with the lattice spacing 2$\mathfrak{a}$. It is labeled by the integers r = 1, ··· , N. To represent the single-particle Hamiltonian with the matrix elements (2.1d), we introduce two sets of Pauli matrices. We associate to the unit 2 × 2 matrix σ₀ and the three Pauli matrices σ₁, σ₂, and σ₃ two geometrical shapes, a square or a circle, corresponding to the eigenvalues of σ₃. We associate to the unit 2 × 2 matrix τ₀ and the three Pauli matrices τ₁, τ₂, and τ₃ two colors, black or white, corresponding to the eigenvalues of τ₃. We choose the representation

$$\begin{eqnarray}\hat{\psi}_{r}^{\dagger}\equiv \left(\begin{array}{c} \hat{\psi}_{\blacksquare;r}^{\dagger} & \hat{\psi}_{\square ;r}^{\dagger} & \hat{\psi}_{\bullet ;r}^{\dagger} & \hat{\psi}_{{}^\bigcirc ;r}^{\dagger}. \\ \end{array}\right)\end{eqnarray} \tag{ 4.1a }$$

and impose periodic boundary conditions

$$\begin{eqnarray}&&\hat{\psi}_{r+N}^{\dagger}=\hat{\psi}_{r}^{\dagger},\quad r=1,\cdots ,N.\end{eqnarray} \tag{ 4.1b }$$

The non-interacting Hamiltonian is then defined by

$$\begin{eqnarray}\begin{array}{c} {\hat{H}} & := & \left(t+\delta t\right)\underset{r=1}{\overset{N}{\sum}} \left(\hat{\psi}_{\blacksquare;r+1}^{\dagger} {{{\hat{\psi}}}_{\square ;r}}+\hat{\psi}_{{}^\bigcirc ;r+1}^{\dagger} {{{\hat{\psi}}}_{\bullet ;r}}+\hat{\psi}_{\square ;r}^{\dagger} {{{\hat{\psi}}}_{\blacksquare;r+1}}+{{{\hat{\psi}}}_{\bullet ;r}}\hat{\psi}_{{}^\bigcirc ;r+1}^{\dagger}\right) \\ {} & {} & +\left(t-\delta t\right)\underset{r=1}{\overset{N}{\sum}} \left(\hat{\psi}_{\square ;r+1}^{\dagger} {{{\hat{\psi}}}_{\blacksquare;r}}+\hat{\psi}_{\bullet ;r+1}^{\dagger} {{{\hat{\psi}}}_{{}^\bigcirc ;r}}+{{{\hat{\psi}}}_{\blacksquare;r}}\hat{\psi}_{\square ;r+1}^{\dagger}+\hat{\psi}_{{}^\bigcirc ;r}^{\dagger} {{{\hat{\psi}}}_{\bullet ;r+1}}\right) \\ \end{array}\end{eqnarray} \tag{ 4.1c }$$

in second quantization. It describes the nearest-neighbor hops of fermions with hopping amplitudes $t\pm \delta t\in \mathbb{R}$ as is illustrated in figure 2. The corresponding single-particle Hamiltonian can be written [τ_±:= (τ₁ ± iτ₂)/2]

$$\begin{eqnarray}&&{{\mathcal{H}}_{r,{{r}^{\prime}}}} ~\left(t,\delta t\right):={{\delta}_{{{r}^{\prime}},r-1}}\left\{\left(t+\delta t\right)\left[\frac{{{\sigma}_{0}}+{{\sigma}_{3}}}{2}\otimes {{\tau}_{+}}+\frac{{{\sigma}_{0}}-{{\sigma}_{3}}}{2}\otimes {{\tau}_{-}}\right]\right.\nonumber\\ &&\left.+\left(t-\delta t\right)\left[\frac{{{\sigma}_{0}}+{{\sigma}_{3}}}{2}\otimes {{\tau}_{-}}+\frac{{{\sigma}_{0}}-{{\sigma}_{3}}}{2}\otimes {{\tau}_{+}}\right]\right\}\end{eqnarray} \tag{ 4.1d }$$

$$\begin{eqnarray}&&+{{\delta}_{{{r}^{\prime}},r+1}}\left\{\left(t+\delta t\right)\left[\frac{{{\sigma}_{0}}+{{\sigma}_{3}}}{2}\otimes {{\tau}_{-}}+\frac{{{\sigma}_{0}}-{{\sigma}_{3}}}{2}\otimes {{\tau}_{+}}\right]\right.\nonumber\\ &&\left.+\left(t-\delta t\right)\left[\frac{{{\sigma}_{0}}+{{\sigma}_{3}}}{2}\otimes {{\tau}_{+}}+\frac{{{\sigma}_{0}}-{{\sigma}_{3}}}{2}\otimes {{\tau}_{-}}\right]\right\}.\end{eqnarray} \tag{ 4.1e }$$

In the Bloch basis, the single-particle Hamiltonian (4.1e) becomes

$$\begin{eqnarray}&&{{\mathcal{H}}_{k,{{k}^{\prime}}}}={{\mathcal{H}}_{k}} {{\delta}_{k,{{k}^{\prime}}}},\quad {{\mathcal{H}}_{k}}=2\ t\ \text{cos}\ k\ {{\sigma}_{0}}\ \otimes\ {{\tau}_{1}}-2\delta\ t\ \text{sin}\ k\ {{\sigma}_{3}}\otimes {{\tau}_{2}},\quad k,{{k}^{\prime}}=-\pi +\frac{2\pi}{N},\cdots ,\pi ,\end{eqnarray} \tag{ 4.2a }$$

with the single-particle spectrum consisting of the two-fold degenerate pair of bands

$$\begin{eqnarray}&&{{\varepsilon}_{\pm ;k}}=\pm 2|t|\sqrt{ \text{co}{{\text{s}}^{2}} k+{{\left(\delta t/t\right)}^{2}} \text{si}{{\text{n}}^{2}}\ k}.\end{eqnarray} \tag{ 4.2b }$$

Consequently, the band gap

$$\begin{eqnarray}&&\Delta =4|\delta t|\end{eqnarray} \tag{ 4.2c }$$

opens at the boundary of the first BZ for any non-vanishing δt.

The symmetries of the single-particle Hamiltonian defined by equation (4.1e) are the following, for any pair of sites r, r ' = 1, ··· , N.

• 1  
  Translation symmetry holds,

  $$\begin{eqnarray}&&{{\mathcal{H}}_{r,{{r}^{\prime}}}} ~\left(t,\delta t\right)={{\mathcal{H}}_{r+n,{{r}^{\prime}}+n}} \left(t,\delta t\right),\quad \forall\ n\in \mathbb{Z}.\end{eqnarray} \tag{ 4.3 }$$
• 2  
  When the dimerization vanishes

  $$\begin{eqnarray}&&{{\mathcal{H}}_{r,{{r}^{\prime}}}} ~\left(t,\delta t=0\right)=t \left({{\delta}_{{{r}^{\prime}},r-1}}+{{\delta}_{{{r}^{\prime}},r+1}}\right) {{\sigma}_{0}}\otimes {{\tau}_{1}},\end{eqnarray} \tag{ 4.4 }$$

  the discrete symmetries

  $$\begin{eqnarray}&&0=\left[{{\mathcal{H}}_{r,{{r}^{\prime}}}} ~\left(t,\delta t=0\right),{{\sigma}_{\mu}}\otimes {{\tau}_{\nu}}\right]\end{eqnarray} \tag{ 4.5a }$$

  hold for μ = 0, 1, 2, 3 and ν = 0, 1, whereas the discrete spectral symmetries

  $$\begin{eqnarray}&&0=\left\{{{\mathcal{H}}_{r,{{r}^{\prime}}}} ~\left(t,\delta t=0\right),{{\sigma}_{\mu}}\otimes {{\tau}_{\mu}}\right\}\end{eqnarray} \tag{ 4.5b }$$

  hold for μ = 0, 1, 2, 3 and ν = 2, 3.
• 3  
  For any dimerization, the transformation laws

  $$\begin{eqnarray}&&{{\mathcal{H}}_{r,{{r}^{\prime}}}} ~\left(t,+\delta t\right)={{\sigma}_{\mu}}\otimes {{\tau}_{1}}{{\mathcal{H}}_{r,{{r}^{\prime}}}} ~\left(t,-\delta t\right) {{\sigma}_{\mu}}\otimes {{\tau}_{1}}\end{eqnarray} \tag{ 4.6a }$$

  with μ = 0, 3 implement the symmetry of figure 2 under the composition of the interchange of the full and dashed lines with the interchange of the black and white filling colors with a reflection about the horizontal dash-one-dot (red) line RH, whereas the transformation laws

  $$\begin{eqnarray}&&{{\mathcal{H}}_{r,{{r}^{\prime}}}} ~\left(t,+\delta t\right)={{\sigma}_{\mu}}\otimes {{\tau}_{1}}{{\mathcal{H}}_{r,{{r}^{\prime}}}} ~\left(t,+\delta t\right) {{\sigma}_{\mu}}\otimes {{\tau}_{1}}\end{eqnarray} \tag{ 4.6b }$$

  hold for μ = 1, 2 otherwise.
• 4  
  Let ${{\mathscr{O}}_{\text{R}}}$ be the operation that interchanges site r₁ with site r_N, site r₂ with site r_N − 1, and so on, i.e., a reflection about the vertical dash-two-dots (blue) line RVO if N is odd or the vertical dash-three-dots (green) line RVE if N is even. For any dimerization, the transformation laws

  $$\begin{eqnarray}&&{{\mathcal{H}}_{r,{{r}^{\prime}}}} ~\left(t,\delta t\right)=+{{\sigma}_{\mu}}\otimes {{\tau}_{0}}{{\mathcal{H}}_{{{\mathcal{O}}_{\text{R}}}r,{{\mathcal{O}}_{\text{R}}}{{r}^{\prime}}}} ~\left(t,\delta t\right) {{\sigma}_{\mu}}\otimes {{\tau}_{0}}\end{eqnarray} \tag{ 4.7 }$$

  with μ = 1, 2 are unitary symmetries, whereas the transformation laws

  $$\begin{eqnarray}&&{{\mathcal{H}}_{r,{{r}^{\prime}}}} ~\left(t,\delta t\right)=-{{\sigma}_{\mu}}\otimes {{\tau}_{3}}{{\mathcal{H}}_{{{\mathcal{O}}_{\text{R}}}r,{{\mathcal{O}}_{\text{R}}}{{r}^{\prime}}}} ~\left(t,\delta t\right) {{\sigma}_{\mu}}\otimes {{\tau}_{3}}\end{eqnarray} \tag{ 4.8 }$$

  with μ = 1, 2 are unitary spectral symmetries.

It is also instructive to derive the symmetries and spectral symmetries of the single-particle Hamiltonian (4.2) for any k = π/N, ··· , π from the first BZ.

To this end, it is convenient to introduce the more compact notation

$$\begin{eqnarray}&&{{\mathcal{X}}_{\mu \nu}}:={{\sigma}_{\mu}}\otimes {{\tau}_{\nu}},\quad \mu ,\nu =0,1,2,3,\end{eqnarray} \tag{ 4.9a }$$

for the sixteen linearly independent 4 × 4 Hermitian matrices that generate the unitary group U (4). In the Bloch basis (4.2a),

$$\begin{eqnarray}&&{{\mathcal{H}}_{k,{{k}^{\prime}}}}={{\mathcal{H}}_{k}} {{\delta}_{k,{{k}^{\prime}}}},\quad {{\mathcal{H}}_{k}}=2 t\ \text{cos}\ k\ {{\mathcal{X}}_{01}}-2\ \delta t\ \text{sin}\ k{{\mathcal{X}}_{32}},\end{eqnarray} \tag{ 4.9b }$$

for k, k ' = π/N , ··· , π. We have taken advantage of the fact that ${{\mathcal{X}}_{01}}$ and ${{\mathcal{X}}_{32}}$ anti-commute to derive the band dispersions (4.2b).

There are eight matrices ${{\mathcal{X}}_{\mu \nu}}$ with μ = 0, 1, 2, 3 and ν = 0, 1 that commute with ${{\mathcal{X}}_{01}}$, there are eight matrices ${{\mathcal{X}}_{\mu \nu}}$ with μ = 1, 2 and ν = 1, 3 or μ = 0, 3 and ν = 0, 2 that commute with ${{\mathcal{X}}_{32}}$. This leaves the four matrices ${{\mathcal{X}}_{00}}$, ${{\mathcal{X}}_{30}}$, ${{\mathcal{X}}_{11}}$, and ${{\mathcal{X}}_{21}}$ that commute with ${{\mathcal{H}}_{k}}$ for all k in the BZ.

There are eight matrices ${{\mathcal{X}}_{\mu \nu}}$ with μ = 0, 1, 2, 3 and ν = 2, 3 that anti-commute with ${{\mathcal{X}}_{01}}$, there are eight matrices ${{\mathcal{X}}_{\mu \nu}}$ with μ = 1, 2 and ν = 0, 2 or μ = 0, 3 and ν = 1, 3 that anti-commute with ${{\mathcal{X}}_{32}}$. This leaves the four matrices ${{\mathcal{X}}_{12}}$, ${{\mathcal{X}}_{22}}$, ${{\mathcal{X}}_{03}}$, and ${{\mathcal{X}}_{33}}$ that anti-commute with ${{\mathcal{H}}_{k}}$ for all k in the BZ.

The symmetries

$$\begin{eqnarray}&&\mathcal{O}_{\mathscr{P}}^{\dagger} {{\mathcal{H}}_{-k}}{{\mathcal{O}}_{\mathscr{P}}}={{\mathcal{H}}_{+k}},\end{eqnarray} \tag{ 4.10a }$$

$$\begin{eqnarray}&&\mathcal{O}_{\mathscr{T}}^{\dagger}\mathcal{H}_{-k}^{*}{{\mathcal{O}}_{\mathscr{T}}}={{\mathcal{H}}_{+k}},\end{eqnarray} \tag{ 4.10b }$$

with

$$\begin{eqnarray}&&{{\mathcal{O}}_{\mathscr{P}}}\in \left\{{{\mathcal{X}}_{01}}, {{\mathcal{X}}_{10}}, {{\mathcal{X}}_{20}}, {{\mathcal{X}}_{31}}\right\},\end{eqnarray} \tag{ 4.10c }$$

$$\begin{eqnarray}&&{{\mathcal{O}}_{\mathscr{T}}}\in \left\{{{\mathcal{X}}_{00}}, {{\mathcal{X}}_{11}}, {{\mathcal{X}}_{21}}, {{\mathcal{X}}_{30}}\right\},\end{eqnarray} \tag{ 4.10d }$$

whereas the spectral symmetries

$$\begin{eqnarray}&&\mathcal{O}_{\mathscr{C}}^{\dagger}\mathcal{H}_{-k}^{\text{T}}{{\mathcal{O}}_{\mathscr{C}}}=-{{\mathcal{H}}_{k}},\end{eqnarray} \tag{ 4.11a }$$

$$\begin{eqnarray}&&\mathcal{O}_{\mathscr{S}}^{\dagger} {{\mathcal{H}}_{k}}{{\mathcal{O}}_{\mathscr{S}}}=-{{\mathcal{H}}_{k}},\end{eqnarray} \tag{ 4.11b }$$

with

$$\begin{eqnarray}&&{{\mathcal{O}}_{\mathscr{C}}}\in \left\{{{\mathcal{X}}_{03}}, {{\mathcal{X}}_{12}}, {{\mathcal{X}}_{22}}, {{\mathcal{X}}_{33}}\right\},\end{eqnarray} \tag{ 4.11c }$$

$$\begin{eqnarray}&&{{\mathcal{O}}_{\mathscr{S}}}\in \left\{{{\mathcal{X}}_{03}}, {{\mathcal{X}}_{12}}, {{\mathcal{X}}_{22}}, {{\mathcal{X}}_{33}}\right\},\end{eqnarray} \tag{ 4.11d }$$

follow. As anticipated, the set of chiral-like spectral symmetries is identical the set of particle–hole-like spectral symmetries in view of the presence of four time-reversal-like symmetries. We shall use the notation K for the anti-linear operation of complex conjugation. We shall also introduce the notation

$$\begin{eqnarray}&&\text{Parity}:{{\mathcal{P}}_{10}}:={{\sigma}_{1}}\otimes {{\tau}_{0}}=+\mathcal{P}_{10}^{\text{T}},\end{eqnarray} \tag{ 4.12a }$$

$$\begin{eqnarray}&&\text{Charge}\text{ Conjugation:}{{\mathcal{C}}_{03}}:={{\sigma}_{0}}\otimes {{\tau}_{3}} K=+\mathcal{C}_{03}^{\text{T}},\end{eqnarray} \tag{ 4.12b }$$

$$\begin{eqnarray}&&\text{Time}\text{ Reversal:}{{\mathcal{T}}_{21}}:={{\sigma}_{2}}\otimes {{\tau}_{1}}~K=-\mathcal{T}_{21}^{\text{T}},\end{eqnarray} \tag{ 4.12c }$$

$$\begin{eqnarray}&&\text{Chirality:}{{\mathcal{S}}_{22}}:={{\sigma}_{2}}\otimes {{\tau}_{2}}=+\mathcal{S}_{22}^{\text{T}},\end{eqnarray} \tag{ 4.12d }$$

say, to distinguish the operations of parity (reflection or inversion), charge conjugation (particle–hole interchange), time reversal, and chirality, respectively. The symmetry under parity of $\mathcal{H}$ can be realized in four inequivalent ways. Correspondingly, we define the 4 × 4 matrices

$$\begin{eqnarray}&&\text{Parity}\mathscr{P}:\quad {{\mathcal{P}}_{\text{01}}}, {{\mathcal{P}}_{\text{10}}}, {{\mathcal{P}}_{\text{20}}}, {{\mathcal{P}}_{\text{31}}},\end{eqnarray} \tag{ 4.13a }$$

that realize the algebra of the unit 2 × 2 matrix ρ₀ and of the three Pauli matrices ρ₁, ρ₂, and ρ₃. The symmetry under time reversal of $\mathcal{H}$ can be realized in four inequivalent ways. Correspondingly, we define the 4 × 4 matrices

$$\begin{eqnarray}&&\text{Time}\ \text{Reversal} \mathscr{T}:\quad {{\mathcal{T}}_{00}}, {{\mathcal{T}}_{11}}, {{\mathcal{T}}_{21}}, {{\mathcal{T}}_{30}},\end{eqnarray} \tag{ 4.13b }$$

that realize the algebra of ρ₀ K, ρ₁ K, ρ₂ K, and ρ₃ K. The symmetry under charge conjugation of $\mathcal{H}$ can be realized in four inequivalent ways. Correspondingly, we define the 4 × 4 matrices

$$\begin{eqnarray}&&\text{Charge Conjuction}~\mathcal{C}:\quad {{\mathcal{C}}_{03}}, {{\mathcal{C}}_{12}}, {{\mathcal{C}}_{22}}, {{\mathcal{C}}_{33}},\end{eqnarray} \tag{ 4.13c }$$

that realize the algebra of ρ₀ K, ρ₁ K, ρ₂ K, and ρ₃ K. All the possible compositions of the operations for charge conjugation and time reversal give four realizations for the chiral symmetry of $\mathcal{H}$. Correspondingly, we define the 4 × 4 matrices

$$\begin{eqnarray}&&\text{Chiral}~\mathscr{S}:\quad {{\mathcal{S}}_{03}}, {{\mathcal{S}}_{12}}, {{\mathcal{S}}_{22}}, {{\mathcal{S}}_{33}},\end{eqnarray} \tag{ 4.13d }$$

that realize the algebra of ρ₀, ρ₁, ρ₂, and ρ₃. There are sixteen columns in table 1, each of which correspond to one of these matrix operations. We shall then select 12 triplets $\left({{\mathcal{P}}_{\mu \nu}},{{\mathcal{T}}_{\mu \nu}},{{\mathcal{C}}_{\mu \nu}}\right)$ from equations (4.13) to build the rows of table 2.

Table 1. The spectrum $\sigma \left({{\tilde{\mathcal{H}}}_{\mu \nu}}\right)$ of the single-particle Hamiltonian ${{\tilde{\mathcal{H}}}_{\mu \nu}}$ defined by equation (2.2) with open boundary conditions. Hamiltonian ${{\tilde{\mathcal{H}}}_{\mu \nu}}$ is nothing but Hamiltonian $\mathcal{H}$ (defined in equation (87e)) perturbed additively by the term ${{\delta}_{r,r\prime}}\left(t/10\right){{}_{\mathcal{X}\mu \nu}}$ with ${{\mathcal{X}}_{\mu \nu}}\equiv {{\sigma}_{\mu}}\otimes {{\tau}_{\nu}}$. The choices for ${{\mathcal{X}}_{\mu \nu}}$ made in the first eight rows enumerate all perturbations ${{\mathcal{V}}_{\mu \nu}}$ defined by equation (2.2b) that enter equation (4.22), i.e. that preserve the symmetry under the parity of $\mathcal{H}$ generated by ${{\mathcal{P}}_{10}}$. The last two rows are two examples of a perturbation ${{\mathcal{V}}_{\mu \nu}}$ that breaks parity. The entanglement spectrum $\sigma \left({{\tilde{Q}}_{\mu \nu A}}\right)$ defined by equation (12) for the single-particle Hamiltonian ${{\tilde{\mathcal{H}}}_{\mu \nu}}$ obeying periodic boundary conditions. The entry ○ or × denotes the presence or the absence, respectively, of the symmetries under parity $\mathscr{P}$, charge conjugation $\mathscr{C}$, and time reversal $\mathscr{T}$ of the perturbation ${{\delta}_{r,r\prime}}\left(t/10\right){{}_{\mathcal{X}\mu \nu}}$ for the first sixteen columns. In the last two columns, the entry ○ or × denotes the presence or the absence, respectively, of zero modes (mid-gap) states in the spectra $\sigma \left({{\tilde{\mathcal{H}}}_{\mu \nu}}\right)$ and $\sigma \left({{\tilde{\mathcal{Q}}}_{\mu \nu}}\right)$ as determined by extrapolation to the thermodynamic limit of exact diagonalization with N = 12.

${{\cal X}_{\mu \nu }} \equiv {\sigma _\mu } \otimes {\tau _\upsilon }$	Symmetry under $\mathscr{P}$				Symmetry under $\mathscr{T}$				Symmetry under $\mathscr{C}$				Symmetry under $\mathscr{S}$				$\sigma \left({{\tilde{\mathcal{H}}}_{\mu \nu}}\right)$	$\sigma \left({{\tilde{Q}}_{\mu \nu A}}\right)$
	${{\mathcal{P}}_{01}}$	${{\mathcal{P}}_{10}}$	${{\mathcal{P}}_{20}}$	${{\mathcal{P}}_{31}}$	${{\mathcal{T}}_{00}}$	${{\mathcal{T}}_{11}}$	${{\mathcal{T}}_{21}}$	${{\mathcal{T}}_{30}}$	${{\mathcal{C}}_{03}}$	${{\mathcal{C}}_{12}}$	${{\mathcal{C}}_{22}}$	${{\mathcal{C}}_{33}}$	${{\mathcal{S}}_{03}}$	${{\mathcal{S}}_{12}}$	${{\mathcal{S}}_{22}}$	${{\mathcal{S}}_{33}}$
${{\mathcal{X}}_{00}}$	○	○	○	○	○	○	○	○	×	×	×	×	×	×	×	×	×	○
${{\mathcal{X}}_{01}}$	○	○	○	○	○	○	○	○	○	○	○	○	○	○	○	○	○	○
${{\mathcal{X}}_{02}}$	×	○	○	×	×	○	○	×	×	○	○	×	○	×	×	○	○	○
${{\mathcal{X}}_{03}}$	×	○	○	×	○	×	×	○	×	○	○	×	×	○	○	×	×	×
${{\mathcal{X}}_{10}}$	○	○	×	×	○	○	×	×	×	×	○	○	×	×	○	○	○	○
${{\mathcal{X}}_{11}}$	○	○	×	×	○	○	×	×	○	○	×	×	○	○	×	×	×	○
${{\mathcal{X}}_{12}}$	×	○	×	○	×	○	×	○	×	○	×	○	○	×	○	×	×	×
${{\mathcal{X}}_{13}}$	×	○	×	○	○	×	○	×	×	○	×	○	×	○	×	○	○	○
${{\mathcal{X}}_{20}}$	○	×	○	×	×	○	×	○	○	×	○	×	×	○	×	○	○	○
${{\mathcal{X}}_{21}}$	○	×	○	×	×	○	×	○	×	○	×	○	○	×	○	×	×	○
${{\mathcal{X}}_{22}}$	×	×	○	○	○	○	×	×	○	○	×	×	○	○	×	×	×	×
${{\mathcal{X}}_{23}}$	×	×	○	○	×	×	○	○	○	○	×	×	×	×	○	○	○	○
${{\mathcal{X}}_{30}}$	○	×	×	○	○	×	×	○	×	○	○	×	×	○	○	×	×	○
${{\mathcal{X}}_{31}}$	○	×	×	○	○	×	×	○	○	×	×	○	○	×	×	○	○	○
${{\mathcal{X}}_{32}}$	×	×	×	×	×	×	×	×	×	×	×	×	○	○	○	○	○	○
${{\mathcal{X}}_{33}}$	×	×	×	×	○	○	○	○	×	×	×	×	×	×	×	×	×	×

Table 2. The first column gives all possible combinations for the triplet of symmetries ${{\mathcal{P}}_{\mu \nu}}$, ${{\mathcal{T}}_{\mu \nu}}$, and ${{\mathcal{C}}_{\mu \nu}}$, from table 1 that are compatible with the Cartan symmetry class CI defined by the conditions $\mathcal{T}_{\mu \nu}^{2}=+1$ and $\mathcal{C}_{\mu \nu}^{2}=-1$. The second column gives for each row the most general perturbation ${{\mathcal{V}}_{\mathscr{P};\mathscr{T};\mathscr{C}}}$ that obeys the triplet of symmetries $\left(\mathcal{P},\mathcal{T},\mathcal{C}\right)$ on any given row. The third column gives the doublet $\left({{\eta}_{\mathscr{T}}},{{\eta}_{\mathscr{C}}}\right)\in \left\{-,+\right\}\times \left\{-,+\right\}$ where the sign ${{\eta}_{\mathscr{T}}}$ is defined by $\mathcal{P}\mathcal{T}\mathcal{P}={{\eta}_{\mathscr{T}}}\mathcal{T}$ and similarly for ${{\eta}_{\mathscr{C}}}$. The fourth column is an application of the classification for the symmetry-protected topological band insulators in one-dimensional space derived in [15 and 16] (table VI from [16] was particularly useful). The topological index $\mathbb{Z}$ and 0 correspond to topologically non-trivial and trivial bulk phases, respectively. The entry ○ or × in the last column denotes the presence or absence of zero modes in the spectrum $\sigma \left({{\tilde{Q}}_{\mu \nu A}}\right)$ as is explained in section 4.4 and verified by numerics.

Triplet $\left(\mathscr{P},\mathscr{T},\mathscr{C}\right)$ of symmetries in class CI	Generic perturbation ${{\mathcal{V}}_{\mathscr{P};\mathscr{T};\mathscr{C}}}$	$\left({{\eta}_{\mathscr{T}}},{{\eta}_{\mathscr{C}}}\right)$	Topological index	$\sigma \left({{Q}_{\mathscr{P};\mathscr{T};\mathscr{C} A}}\right)$
${{\mathcal{P}}_{01}}$, ${{\mathcal{T}}_{00}}$, ${{\mathcal{C}}_{12}}$	${{v}_{01}} {{\mathcal{X}}_{01}}+{{v}_{11}} {{\mathcal{X}}_{11}}+{{v}_{30}} {{\mathcal{X}}_{30}}$	(+, −)	$\mathbb{Z}$	○
${{\mathcal{P}}_{01}}$, ${{\mathcal{T}}_{11}}$, ${{\mathcal{C}}_{12}}$	${{v}_{01}} {{\mathcal{X}}_{01}}+{{v}_{11}} {{\mathcal{X}}_{11}}+{{v}_{21}} {{\mathcal{X}}_{21}}$	(+, −)	$\mathbb{Z}$	○
${{\mathcal{P}}_{01}}$, ${{\mathcal{T}}_{30}}$, ${{\mathcal{C}}_{12}}$	${{v}_{01}} {{\mathcal{X}}_{01}}+{{v}_{21}} {{\mathcal{X}}_{21}}+{{v}_{30}} {{\mathcal{X}}_{30}}$	(+, −)	$\mathbb{Z}$	○
${{\mathcal{P}}_{10}}$, ${{\mathcal{T}}_{00}}$, ${{\mathcal{C}}_{12}}$	${{v}_{01}} {{\mathcal{X}}_{01}}+{{v}_{03}} {{\mathcal{X}}_{03}}+{{v}_{11}} {{\mathcal{X}}_{11}}+{{v}_{13}} {{\mathcal{X}}_{13}}$	(+, +)	0	×
${{\mathcal{P}}_{10}}$, ${{\mathcal{T}}_{11}}$, ${{\mathcal{C}}_{12}}$	${{v}_{01}} {{\mathcal{X}}_{01}}+{{v}_{02}} {{\mathcal{X}}_{02}}+{{v}_{11}} {{\mathcal{X}}_{11}}+{{v}_{12}} {{\mathcal{X}}_{12}}$	(+, +)	0	×
${{\mathcal{P}}_{10}}$, ${{\mathcal{T}}_{30}}$, ${{\mathcal{C}}_{12}}$	${{v}_{01}} {{\mathcal{X}}_{01}}+{{v}_{03}} {{\mathcal{X}}_{03}}+{{v}_{12}} {{\mathcal{X}}_{12}}$	(−, +)	0	×
${{\mathcal{P}}_{20}}$, ${{\mathcal{T}}_{00}}$, ${{\mathcal{C}}_{12}}$	${{v}_{01}} {{\mathcal{X}}_{01}}+{{v}_{03}} {{\mathcal{X}}_{03}}+{{v}_{22}} {{\mathcal{X}}_{22}}$	(−, +)	0	×
${{\mathcal{P}}_{20}}$, ${{\mathcal{T}}_{11}}$, ${{\mathcal{C}}_{12}}$	${{v}_{01}} {{\mathcal{X}}_{01}}+{{v}_{02}} {{\mathcal{X}}_{02}}+{{v}_{21}} {{\mathcal{X}}_{21}}+{{v}_{22}} {{\mathcal{X}}_{22}}$	(+, +)	0	×
${{\mathcal{P}}_{20}}$, ${{\mathcal{T}}_{30}}$, ${{\mathcal{C}}_{12}}$	${{v}_{01}} {{\mathcal{X}}_{01}}+{{v}_{03}} {{\mathcal{X}}_{03}}+{{v}_{21}} {{\mathcal{X}}_{21}}+{{v}_{23}} {{\mathcal{X}}_{23}}$	(+, +)	0	×
${{\mathcal{P}}_{31}}$, ${{\mathcal{T}}_{00}}$, ${{\mathcal{C}}_{12}}$	${{v}_{01}} {{\mathcal{X}}_{01}}+{{v}_{13}} {{\mathcal{X}}_{13}}+{{v}_{22}} {{\mathcal{X}}_{22}}+{{v}_{30}} {{\mathcal{X}}_{30}}$	(+, +)	0	×
${{\mathcal{P}}_{31}}$, ${{\mathcal{T}}_{11}}$, ${{\mathcal{C}}_{12}}$	${{v}_{01}} {{\mathcal{X}}_{01}}+{{v}_{12}} {{\mathcal{X}}_{12}}+{{v}_{22}} {{\mathcal{X}}_{22}}$	(−, +)	0	×
${{\mathcal{P}}_{31}}$, ${{\mathcal{T}}_{30}}$, ${{\mathcal{C}}_{12}}$	${{v}_{01}} {{\mathcal{X}}_{01}}+{{v}_{12}} {{\mathcal{X}}_{12}}+{{v}_{23}} {{\mathcal{X}}_{23}}+{{v}_{30}} {{\mathcal{X}}_{30}}$	(+, +)	0	×

The single-particle Hamiltonian $\mathcal{H}$ is extremely sparse. This is reflected by it obeying the symmetries $\left({{\mathcal{P}}_{\mu}},{{\mathcal{T}}_{\mu}},{{\mathcal{C}}_{\mu}}\right)$ with the pair μ and ν fixed by the columns from table 1. In particular, $\mathcal{H}$ cannot be assigned in a unique way the symmetry under time-reversal and the spectral symmetry under charge conjugation without additional information of microscopic origin. Identifying the symmetric space generated by $\mathcal{H}$ is thus ambiguous. Example 1, $\mathcal{H}$ can be thought of as representative of the Cartan symmetry class CI if the choice ${{\mathcal{T}}_{\mu \nu}}$ obeying $\mathcal{T}_{\mu \nu}^{2}=+\mathbb{I}$ and ${{\mathcal{C}}_{\mu \nu}}$ obeying $\mathcal{C}_{\mu \nu}^{2}=-\mathbb{I}$ for the symmetry under time-reversal and the spectral symmetry under charge-conjugation is dictated by a microscopic derivation of $\mathcal{H}$. Example 2, $\mathcal{H}$ can be thought of as representative of the symmetry class DIII if the choice ${{\mathcal{T}}_{\mu \nu}}$ obeying $\mathcal{T}_{\mu \nu}^{2}=-\mathbb{I}$ and ${{\mathcal{C}}_{\mu \nu}}$ obeying $\mathcal{C}_{\mu \nu}^{2}=+\mathbb{I}$ for the symmetry under time-reversal and the spectral symmetry under charge-conjugation is dictated by a microscopic derivation of $\mathcal{H}$. Example 3, $\mathcal{H}$ can be thought of as representative of the Cartan symmetry class BDI if the choice ${{\mathcal{T}}_{\mu \nu}}$ obeying $\mathcal{T}_{\mu \nu}^{2}=+\mathbb{I}$ and ${{\mathcal{C}}_{\mu \nu}}$ obeying $\mathcal{C}_{\mu \nu}^{2}=+\mathbb{I}$ for the symmetry under time-reversal and the spectral symmetry under charge-conjugation is dictated by a microscopic derivation of $\mathcal{H}$. In fact, all complex Cartan symmetry classes AI, BDI, D, DIII, AII, CII, C, and CI are obtained from perturbing $\mathcal{H}$ under the condition that either a symmetry under time reversal or a spectral symmetry under charge conjugation is imposed by a microscopic derivation of $\mathcal{H}$. Finally, if a microscopic derivation of $\mathcal{H}$ does not prevent perturbations that break all the symmetries under time reversal and all the spectral symmetries under charge conjugations from table 4.2, then the remaining two real Cartan symmetry classes A and AIII are realized.

Table 3. The spectrum $\sigma \left({{\tilde{\mathcal{H}}}_{{{k}_{i}} \mu \nu}}\right)$ of the single-particle Hamiltonian ${{\tilde{\mathcal{H}}}_{{{k}_{i}} \mu \nu}}$ defined by equation (2.2) and obeying periodic boundary conditions along the i = 1, 2 direction and open boundary conditions along the i + 1 (modulo 2) direction. The entanglement spectrum $\sigma \left({{\tilde{Q}}_{{{k}_{i}} \mu \nu {{A}_{i+1}}}}\right)$ defined by equation (12) for the single-particle Hamiltonian ${{\tilde{\mathcal{H}}}_{{{k}_{i}}\ \mu \nu}}$ obeying periodic boundary conditions along both the i = 1, 2 and the i + 1 (modulo 2) direction. The entry ○ or × in the second to sixth columns denotes the presence or the absence, respectively, of the symmetries under reflections about the directions 1 $\left({{\mathcal{R}}_{1}}\right)$ and 2 $\left({{\mathcal{R}}_{2}}\right)$, charge conjugation $\mathscr{C}$, time reversal $\mathscr{T}$, and chiral $\mathscr{S}$ of the perturbation ${{\delta}_{{{r}_{i+1}},r_{i+1}^{\prime}}}0.3\ t\ {{\mathcal{X}}_{\mu \nu}}$ for the sixteen rows. In the last two columns, the entries ○ and × denote the presence and the absence, respectively, of non-propagating zero modes in the spectra $\sigma \left({{\tilde{\mathcal{H}}}_{{{k}_{i}}\ \mu \nu}}\right)$ and $\sigma \left({{\tilde{\mathcal{Q}}}_{{{k}_{i}}\ \mu \nu {{A}_{i+1}}}}\right)$ as determined by extrapolation to the thermodynamic limit of exact diagonalization with the open direction running over 32 repeat unit cells and the momentum along the compactified direction running over 128 values. The entry ⊗ in the last two columns denotes the existence of crossings between the mid-gap branches, whereby the crossings are away from vanishing energy (entanglement eigenvalue) and vanishing momentum in the spectra $\sigma \left({{\tilde{\mathcal{H}}}_{{{k}_{i}}\ \mu \nu}}\right)$ and $\sigma \left({{\tilde{\mathcal{Q}}}_{{{k}_{i}}\ \mu \nu {{A}_{i+1}}}}\right)$.

${{\cal X}_{\mu \nu }} \equiv {\sigma _\mu } \otimes {\tau _\upsilon }$	${{\mathcal{R}}_{1}}$		${{\mathcal{R}}_{2}}$		$\mathscr{P}$		$\mathscr{C}$		$\mathscr{S}$
	${{\mathcal{R}}_{13}}$	${{\mathcal{R}}_{23}}$	${{\mathcal{R}}_{10}}$	${{\mathcal{R}}_{20}}$	${{\mathcal{T}}_{10}}$	${{\mathcal{T}}_{20}}$	${{\mathcal{C}}_{01}}$	${{\mathcal{C}}_{31}}$	${{\mathcal{S}}_{11}}$	${{\mathcal{S}}_{21}}$	$\sigma \left({{\tilde{\mathcal{H}}}_{{{k}_{1}}\mu \nu}}\right)$	$\sigma \left({{\tilde{Q}}_{{{k}_{1}}\mu \nu {{A}_{2}}}}\right)$	$\sigma \left({{\tilde{\mathcal{H}}}_{{{k}_{2}}\mu \nu}}\right)$	$\sigma \left({{\tilde{Q}}_{{{k}_{2}}\mu \nu {{A}_{1}}}}\right)$
${{\mathcal{X}}_{00}}$	○	○	○	○	○	○	×	×	×	×	⊗	○	⊗	○
${{\mathcal{X}}_{01}}$	×	×	○	○	○	○	×	×	×	×	⊗	⊗	○	○
${{\mathcal{X}}_{02}}$	×	×	○	○	×	×	×	×	○	○	○	○	⊗	⊗
${{\mathcal{X}}_{03}}$	○	○	○	○	○	○	○	○	○	○	○	○	○	○
${{\mathcal{X}}_{10}}$	○	×	○	×	○	×	×	○	×	○	×	○	○	○
${{\mathcal{X}}_{11}}$	×	○	○	×	○	×	×	○	×	○	×	×	×	×
${{\mathcal{X}}_{12}}$	×	○	○	×	×	○	×	○	○	×	○	○	○	○
${{\mathcal{X}}_{13}}$	○	×	○	×	○	×	○	×	○	×	○	○	×	○
${{\mathcal{X}}_{20}}$	×	○	×	○	○	×	○	×	○	×	×	○	○	○
${{\mathcal{X}}_{21}}$	○	×	×	○	○	×	○	×	○	×	×	×	×	×
${{\mathcal{X}}_{22}}$	○	×	×	○	×	○	○	×	×	○	○	○	○	○
${{\mathcal{X}}_{23}}$	×	○	×	○	○	×	×	○	×	○	○	○	×	○
${{\mathcal{X}}_{30}}$	×	×	×	×	×	×	×	×	○	○	⊗	○	⊗	○
${{\mathcal{X}}_{31}}$	○	○	×	×	×	×	×	×	○	○	⊗	⊗	○	○
${{\mathcal{X}}_{32}}$	○	○	×	×	○	○	×	×	×	×	○	○	⊗	⊗
${{\mathcal{X}}_{33}}$	×	×	×	×	×	×	○	○	×	×	○	○	○	○

It is time to turn our attention to the topological properties of the single-particle Hamiltonian $\mathcal{H}$ defined by its matrix elements(4.1) or (4.2) in the orbital basis or in the Bloch basis, respectively.

The representation (4.2) demonstrates that the single-particle Hamiltonian $\mathcal{H}$ is reducible for all k in the BZ,

$$\begin{eqnarray}&&\mathcal{H}=\underset{N}{\overset{n=1}{{\oplus}}} {{\mathcal{H}}_{\pi n/N}},\end{eqnarray} \tag{ 4.14a }$$

$$\begin{eqnarray}&&{{\mathcal{H}}_{k}}=\mathcal{H}_{k}^{\left(+\right)}\oplus \mathcal{H}_{k}^{\left(-\right)},\end{eqnarray} \tag{ 4.14b }$$

where the 4 × 4 Hermitian matrices $\mathcal{H}_{k}^{\left(+\right)}$ and $\mathcal{H}_{k}^{\left(-\right)}$ are isomorphic to the 2 × 2 Hermitian matrices

$$\begin{eqnarray}&&H_{k}^{\left(+\right)}:=2\ t\ \text{cos}\ k\ {{\tau}_{1}}-2\ \delta\ t\ \text{sin}\ k\ {{\tau}_{2}}\end{eqnarray} \tag{ 4.14c }$$

and

$$\begin{eqnarray}&&H_{k}^{\left(-\right)}:=2\ t\ \text{cos}\ k\ {{\tau}_{1}}+2\ \delta\ t\ \text{sin}\ k\ {{\tau}_{2}},\end{eqnarray} \tag{ 4.14d }$$

respectively.

The reducibility (4.14) defines the partition (2.9) for the one-dimensional example (4.2). For simplicity, we take the thermodynamic limit N → ∞ with N = 2M so that we may define

$$\begin{eqnarray}&&{{\mathfrak{H}}_{A}}:=\underset{n=1}{\overset{M}{{\oplus}}} \underset{\mu ,\nu =\pm}{{\oplus}} |\chi _{\nu ;\pi n/2M}^{\left(\mu \right)}\rangle \langle \chi _{\nu ;\pi n/2M}^{\left(\mu \right)}|\end{eqnarray} \tag{ 4.15a }$$

and

$$\begin{eqnarray}&&{{\mathfrak{H}}_{B}}:=\underset{n=M+1}{\overset{2M}{{\oplus}}} \underset{\mu ,\nu =\pm}{{\oplus}} |\chi _{\nu ;\pi n/2M}^{\left(\mu \right)}\rangle \langle \chi _{\nu ;\pi n/2M}^{\left(\mu \right)}|.\end{eqnarray} \tag{ 4.15b }$$

Here, $\chi _{-;k}^{\left(+\right)}$ and $\chi _{+;k}^{\left(+\right)}$ are the pair of eigenstates with eigenvalues $\varepsilon _{-;k}^{\left(+\right)}\leqslant\varepsilon _{+;k}^{\left(+\right)}$ of $H_{k}^{\left(+\right)}$. Similarly, $\chi _{-,k}^{\left(-\right)}$, $\chi _{+,k}^{\left(-\right)}$, and $\varepsilon _{-;k}^{\left(-\right)}\leqslant\varepsilon _{+;k}^{\left(-\right)}$ denote the eigenstates and their eigenenergies from the lower and upper bands of $H_{k}^{\left(-\right)}$. This partition satisfies (recall equation (4.12))

$$\begin{eqnarray}&&\mathscr{P}A=B,\quad \mathscr{P} B=A,\end{eqnarray} \tag{ 4.16a }$$

$$\begin{eqnarray}&&\mathscr{C} A=A,\quad \mathscr{C} B=B,\end{eqnarray} \tag{ 4.16b }$$

$$\begin{eqnarray}&&\mathscr{T} A=A,\quad \mathscr{T} B=B,\end{eqnarray} \tag{ 4.16c }$$

$$\begin{eqnarray}&&\mathscr{S} A=A,\quad \mathscr{S} B=B.\end{eqnarray} \tag{ 4.16d }$$

Hamiltonian (4.14c) describes a single particle that hops between sites labeled by ▪ and □ in figure 2 with the uniform hopping amplitude t and the staggered hopping amplitude δt. Hamiltonian (4.14d) describes a single particle that hops between sites labeled by • and ○ in figure 4 with the uniform hopping amplitude t and the staggered hopping amplitude δt.

If we take the thermodynamic limit

$$\begin{eqnarray}&&N,{{N}_{\text{f}}}\to \infty \end{eqnarray} \tag{ 4.17a }$$

holding the fermion density

$$\begin{eqnarray}&&{{N}_{\text{f}}}/N=1\end{eqnarray} \tag{ 4.17b }$$

fixed, we find the non-vanishing winding numbers

$$\begin{eqnarray}&&W_{\text{FS}}^{\left(+\right)}=-W_{\text{FS}}^{\left(-\right)}:=\frac{\text{i}}{2\pi}{\oint}^{}\text{d}k\chi _{-}^{\left(+\right)?}\left(k\right)\left(\frac{\partial \chi _{-}^{\left(+\right)}}{\partial k}\right)~\left(k\right)\end{eqnarray} \tag{ 4.18 }$$

for any non-vanishing dimerization δt. Owing to the reducibility of ${{\mathcal{H}}_{k}}$, the winding number W_FS for the single-particle eigenstates making up the Fermi sea of ${{\mathcal{H}}_{k}}$ is

$$\begin{eqnarray}&&{{W}_{\text{FS}}}=W_{\text{FS}}^{\left(+\right)}+W_{\text{FS}}^{\left(-\right)}=0\end{eqnarray} \tag{ 4.19 }$$

for any non-vanishing dimerization δt. The non-vanishing values of the winding numbers endow each of the single-particle Hamiltonians H⁽⁺⁾ and H⁽⁻⁾ with a topological attribute. The single-particle Hamiltonian $\mathcal{H}$ is topologically trivial as its winding number vanishes.

The very definition of the winding numbers (4.18) and (4.19) requires twisted boundary conditions and a spectral gap between the Fermi sea and all many-body excitations.

With open boundary conditions, two zero-dimensional boundaries at r − 1 and r follow from setting all hopping amplitudes between site r − 1 and r in figure 2 to zero, i.e., erasing the connecting full and dashed lines from figure 2 that intersect the vertical line RVE. In such an open geometry, the winding numbers (4.18) and (4.19) are ill defined. However, the bulk-edge correspondence implies the existence of mid-gap states (zero modes) that are localized at the boundaries (if the thermodynamic limit is taken with open boundary conditions), whenever the winding numbers (4.18) are non-vanishing (if the thermodynamic limit is taken with twisted boundary conditions). There are four such zero modes, a pair of zero modes for each boundary. On any given boundary, one of the zero modes localized on this boundary originates from $H_{r,{{r}^{\prime}}}^{\left(+\right)}$, while the other originates from $H_{r,{{r}^{\prime}}}^{\left(-\right)}$. These four zero modes are eigenstates of either ${{\mathcal{X}}_{03}}+{{\mathcal{X}}_{33}}$ or ${{\mathcal{X}}_{03}}-{{\mathcal{X}}_{33}}$. As a set, they are protected against any perturbation that anti-commutes with ${{\mathcal{X}}_{03}}$ and ${{\mathcal{X}}_{33}}$, i.e., they are protected against any linear combination of ${{\mathcal{X}}_{01}}$, ${{\mathcal{X}}_{02}}$, ${{\mathcal{X}}_{31}}$, and ${{\mathcal{X}}_{32}}$. (We have verified by exact diagonalization that the zero modes in $\sigma \left(\mathcal{H}\right)$ are indeed robust to the perturbations ${{\mathcal{X}}_{01}}$, ${{\mathcal{X}}_{02}}$, ${{\mathcal{X}}_{31}}$, or ${{\mathcal{X}}_{32}}$.)

It is believed that the existence of protected gapless boundary states, when taking the thermodynamic limit with open boundary conditions for a topological band insulator with Hamiltonian $\mathcal{H}$, implies the existence of protected gapless boundary states in the entanglement spectrum of Q_A for a suitable partition of the form (2.9), when taking the thermodynamic limit with closed boundary conditions for the equal-time one-point correlation matrix $\mathcal{Q}$ [3, 8, 12, 13].

We have verified by exact diagonalization that this is also the case when we take the thermodynamic limit (4.17) of $\mathcal{H}$ defined by equation (4.14) with an even number of sites N = 2 M and with the partition defined by equation (4.15). The spectra for $\mathcal{H}$ and Q_A are shown in figure 3(a) for δt = t and N = 12. In both spectra, there are four zero modes within an exponential accuracy resulting from finite-size corrections. In the case of $\mathcal{H}$, a pair of zero modes is exponentially localized at one physical boundary a distance 2M apart from the second pair of zero modes localized on the opposite boundary. In the case of Q_A, a pair of zero modes is exponentially localized at the boundary with B to the left of A a distance M apart from the second pair of zero modes exponentially localized at the boundary with A to the left of B. The exponential decay of the zero modes away from their boundary is inversely proportional to the band gap of $\mathcal{H}$ when periodic boundary conditions are imposed.

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We observe that the existence of zero modes in the spectrum of Q_A does not imply the existence of zero modes in the spectrum of $\mathcal{H}$. For example, if we shift the spectrum of $\mathcal{H}$ defined in equation (4.14) in a uniform way by adding a chemical potential smaller than the band gap, we immediately lose the zero modes. However, all single-particle eigenstates are unperturbed by the chemical potential, for it enters as a perturbation that commutes with $\mathcal{H}$. Consequently, neither $\mathcal{Q}$, nor Q_A, nor their spectra depend on the chemical potential. In particular, the spectrum of Q_A with a non-vanishing chemical potential contains the very same four zero modes that are present when the chemical potential vanishes.

The question we are after is the following. Are the four zero modes of ${{\mathcal{H}}_{r,{{r}^{\prime}}}} ~\left(t,\delta t\right)$, that are localized at both ends of an open chain, stable against perturbations that (i) commute with a reflection from table 4.2 and (ii) have a characteristic energy that is small relative to the unperturbed band gap? Similarly, how stable are the four zero modes of Q_A?

The topological classification of non-interacting fermionic insulators based on the presence or absence of the discrete symmetries under the operations of time reversal, charge conjugation, and chirality satisfies a bulk-edge (holography) correspondence principle. According to this principle a non-vanishing value for a certain topological index defined for the bulk is equivalent to the existence of extended boundary states on physical boundaries. This correspondence is lost when crystalline symmetries such as inversion about a point or reflection about a mirror plane are also imposed. What remains is a symmetry-protected topological classification of non-interacting bulk fermionic insulators that obey a crystalline symmetry [15, 16]. This classification has two distinctive features. First, a symmetry-protected topologically phase is not required to support extended boundary states localized at the physical edges. Second, a bulk-edge correspondence principle is nevertheless believed to hold for the entanglement spectrum.

We are going to verify how this correspondence principle for symmetry-protected topologically phases of non-interacting fermionic insulating phases holds for the one-dimensional model with the elementary building block defined by equation (4.1) and explain why. When (μ, ν) = (0, 0);(1, 1);(2, 1);(3, 0), we are going to show that ${{\tilde{\mathcal{H}}}_{\mu \nu}}$ defined in equation (4.23) fails to support robust edge states for an open geometry, whereas ${{\tilde{\mathcal{Q}}}_{\mu \nu A}}$ supports robust edge states at the entangling boundaries.

To this end, our strategy is going to be to first explain the rows of table 4.2. We shall then choose a combination of symmetry under time reversal and spectral symmetry under charge conjugation that puts ${{\tilde{\mathcal{H}}}_{\mu \nu}}$ in the Cartan symmetry class CI. This symmetry class is a topologically trivial one, a generic perturbation of ${{\tilde{\mathcal{H}}}_{\mu \nu}}$ destroys any boundary state that ${{\mathcal{H}}_{\mu \nu}}$ supports at a physical boundary. However, we shall show that imposing a suitable symmetry under parity guarantees the existence of protected boundary states in the entanglement spectrum $\sigma \left({{\tilde{\mathcal{Q}}}_{\mu \nu\ A}}\right)$.

If periodic boundary conditions are imposed, a perturbation that commutes with the parity transformation generated by ${{\mathcal{X}}_{10}}$ is of the general form

$$\begin{eqnarray}&&{{\mathcal{V}}_{k}}:=\underset{\nu =0}{\overset{3}{\sum}} \left[\underset{\mu =0,1}{\sum} {{f}_{\mu \nu ;k}}+\underset{\mu =2,3}{\sum} {{g}_{\mu \nu ;k}}\right]{{\mathcal{X}}_{\mu \nu}},\end{eqnarray} \tag{ 4.20 }$$

where the functions f_μν;k and g_μν;k are even and odd under the inversion k → −k, respectively (recall equation (4.10b)). For simplicity, the perturbation ${{\mathcal{V}}_{k}}$ is taken independent of k, i.e.,

$$\begin{eqnarray}&&{{\mathcal{V}}_{k}}:=\underset{\nu =0}{\overset{3}{\sum}} \underset{\mu =0,1}{\sum} {{v}_{\mu \nu}}{{\mathcal{X}}_{\mu \nu}}={{\mathcal{X}}_{10}} {{V}_{-k}}\ {{\mathcal{X}}_{10}}\end{eqnarray} \tag{ 4.21 }$$

with the eight parameters v_μν real valued. In the orbital basis, we have

$$\begin{eqnarray}\begin{array}{c} {{\mathcal{V}}_{r,{{r}^{\prime}}}} & = & {{\delta}_{r,{{r}^{\prime}}}}~\left[\underline{\underline{{{v}_{01}} {{\mathcal{X}}_{01}}}}+\underline{\underline{{{v}_{02}} {{\mathcal{X}}_{02}}}}+{{v}_{00}} {{\mathcal{X}}_{00}}+{{v}_{03}} {{\mathcal{X}}_{03}}\right] \\ {} & {} & +{{\delta}_{r,{{r}^{\prime}}}}~\left[\underline{{{v}_{10}} {{\mathcal{X}}_{10}}}+\underline{{{v}_{11}} {{\mathcal{X}}_{11}}}+\underline{{{v}_{12}} {{\mathcal{X}}_{12}}}+\underline{{{v}_{13}} {{\mathcal{X}}_{13}}}\right]. \\ \end{array}\end{eqnarray} \tag{ 4.22 }$$

The terms that have been underlined twice anti-commute with the two commuting matrices ${{\mathcal{X}}_{03}}$ and ${{\mathcal{X}}_{33}}$ that share the zero modes as eigenstates. Hence, the zero modes are protected as a set against the parity-preserving perturbations that are linear combinations of ${{\mathcal{X}}_{01}}$ and ${{\mathcal{X}}_{02}}$. All other terms in equation (4.22) fail to anti-commute with both ${{\mathcal{X}}_{03}}$ and ${{\mathcal{X}}_{33}}$. The zero modes are not necessarily protected as a set under these parity-preserving perturbations. The terms in equation (4.22) that are underlined once either anti-commute with ${{\mathcal{X}}_{03}}$ or ${{\mathcal{X}}_{33}}$ but not with both. The terms in equation (4.22) that are not underlined fail to anti-commute with both ${{\mathcal{X}}_{03}}$ and ${{\mathcal{X}}_{33}}$.

The same exercise can be repeated for the parity transformations generated by ${{\mathcal{X}}_{01}}$, ${{\mathcal{X}}_{20}}$, and ${{\mathcal{X}}_{31}}$. This deliver the first fourteen rows in table 1.

For any μ, ν = 0, 1, 2, 3, we define the single-particle Hamiltonian ${{\tilde{\mathcal{H}}}_{\mu \nu}}\equiv \mathcal{H}+{{\mathcal{V}}_{\mu \nu}}$ by its matrix elements

$$\begin{eqnarray}&&{{\tilde{\mathcal{H}}}_{\mu \nu\ r,{{r}^{\prime}}}}:={{\mathcal{H}}_{r,{{r}^{\prime}}}}+{{\mathcal{V}}_{\mu \nu\ r,{{r}^{\prime}}}},\end{eqnarray} \tag{ 4.23a }$$

$$\begin{eqnarray}&&{{\mathcal{V}}_{\mu \nu\ r,{{r}^{\prime}}}}:={{\delta}_{r,{{r}^{\prime}}}}{{v}_{\mu \nu}}\ {{\mathcal{X}}_{\mu \nu}}.\end{eqnarray} \tag{ 4.23b }$$

The single-particle Hamiltonian ${{\mathcal{H}}_{r,{{r}^{\prime}}}}$ was defined in equation (4.1e) for r, r ' = 1, ··· , N − 1. We choose between imposing open boundary conditions by setting the hopping amplitudes to zero between sites N and N + 1 or periodic boundary conditions. The perturbation strength v_μν is real-valued. It will be set to t/10. The corresponding equal-time one-point correlation matrix defined by equation (2.8) is denoted ${{\tilde{\mathcal{Q}}}_{\mu \nu}}$.

For illustrative purposes, we plot in figures 4(a)–(d) the energy eigenvalue spectrum $\sigma \left({{\tilde{H}}_{\mu \nu}}\right)$ of ${{\tilde{H}}_{\mu \nu}}$ obeying open boundary conditions by an exact diagonalization with N = 12 that can be extrapolated to the thermodynamic limit. Energy eigenvalues are measured in units of 2t. In panels (a)–(c), δt = t implies that all energy eigenstates have wave-functions that are localized on a pair of consecutive sites for which the hopping amplitude is t + δt = 2t. In the remaining panel (d), a δt ≠ t delocalizes bulk energy eigenstates that acquire a dispersion, i.e., a band width. The zero modes in panel (a) are four edge states. They are protected by the chiral symmetry in that they are eigenstates of either ${{\mathcal{X}}_{03}}$ or ${{\mathcal{X}}_{33}}$. For example, they are robust to changing the value of δt away from t. However, these zero modes are shifted to non-vanishing energies by the perturbations ${{\mathcal{V}}_{12}}$ for panel (b) and ${{\mathcal{V}}_{11}}$ for panels (c) and (d). The spectrum in panel (b) is unchanged by the substitutions ${{\mathcal{V}}_{12}}\to {{\mathcal{V}}_{03}}\to {{\mathcal{V}}_{33}}$. The presence or absence of protected (against the perturbation from the second column) zero modes localized on the physical boundaries has been verified in this way for all rows and is reported in the penultimate column of table 4.2.

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We plot in figures 4(e)–(h) the eigenvalue spectrum $\sigma \left({{\tilde{Q}}_{\mu \nu A}}\right)$ of the upper-left block ${{\tilde{Q}}_{\mu \nu A}}$ in the equal-time one-point correlation matrix ${{\tilde{\mathcal{Q}}}_{\mu \nu}}$ corresponding to ${{\tilde{\mathcal{H}}}_{\mu \nu}}$ obeying periodic boundary conditions by an exact diagonalization with N = 12 that can be extrapolated to the thermodynamic limit. Panels (e)–(g) have δt = t. Panel (h) has δt = 2 t/3. There is no perturbation in panel (e), in which case four zero modes are present in the spectrum $\sigma \left({{\tilde{Q}}_{\mu \nu A}}\right)$. The perturbation ${{\mathcal{V}}_{12}}$ splits the four zero modes into two pairs of degenerate eigenstates with eigenvalues only differing by their sign in panel (f), as was the case for the Hamiltonian in panel (b). The spectrum in panel (f) is unchanged by the substitutions ${{\mathcal{V}}_{12}}\to {{\mathcal{V}}_{03}}\to {{\mathcal{V}}_{33}}$. Unlike in panels (c) and (d) for the Hamiltonian, panels (g) and (h) show that the four zero modes of ${{\tilde{Q}}_{\mu \nu A}}$ are robust to the perturbation ${{\mathcal{V}}_{11}}$.

The lesson from figure 3 and table 4.2 is that of all the seven

$$\begin{eqnarray}&&{{\mathcal{V}}_{00}},{{\mathcal{V}}_{03}},{{\mathcal{V}}_{11}},{{\mathcal{V}}_{12}},{{\mathcal{V}}_{21}},{{\mathcal{V}}_{22}},{{\mathcal{V}}_{30}}\end{eqnarray} \tag{ 4.24 }$$

out of fourteen parity-preserving perturbations that gap the zero modes of $\mathcal{H}$, only four, namely

$$\begin{eqnarray}&&{{\mathcal{V}}_{00}},{{\mathcal{V}}_{11}},{{\mathcal{V}}_{21}},{{\mathcal{V}}_{30}},\end{eqnarray} \tag{ 4.25 }$$

fail to also gap the zero modes of Q_A.

To explain this observation, we rely on the reasoning that delivers table 2. We assume that the underlying microscopic model has the triplet of symmetries $\left(\mathscr{P};\mathscr{T};\mathscr{C}\right)\sim \left({{\mathcal{P}}_{\mu \nu}},{{\mathcal{T}}_{\mu \nu}},{{\mathcal{C}}_{\mu \nu}}\right)$ where the doublet $\left(\mathcal{T},\mathcal{C}\right)\sim \left({{\mathcal{T}}_{\mu \nu}},{{\mathcal{C}}_{\mu \nu}}\right)$ defines the symmetry class CI, i.e., $\mathcal{T}_{\mu \nu}^{2}=+\mathbb{I}$ and $\mathcal{C}_{\mu \nu}^{2}=-\mathbb{I}$. We denote the most general perturbation that is compliant with the triplet of symmetries $\left(\mathscr{P};\mathscr{T};\mathscr{C}\right)$ defining a given row of table 2 by ${{\mathcal{V}}_{\mathscr{P};\mathscr{T};\mathscr{C}}}$. The explicit form of this perturbation is to be found in the second column of table 2 as one varies $\left(\mathscr{P};\mathscr{T};\mathscr{C}\right)$. For a given row in table 2, ${{\mathcal{V}}_{\mathscr{P};\mathscr{T};\mathscr{C}}}$ is contained in the most general perturbation ${{\mathcal{V}}_{\mathscr{T};\mathscr{C}}}$ that is compliant with the doublet of symmetries $\left(\mathcal{T},\mathcal{C}\right)$. We define the single-particle Hamiltonian ${{\mathcal{H}}_{\mathscr{P};\mathscr{T};\mathscr{C}}}\equiv \mathcal{H}+{{\mathcal{V}}_{\mathscr{P};\mathscr{T};\mathscr{C}}}$ by its matrix elements

$$\begin{eqnarray}&&{{\mathcal{H}}_{\mathscr{P};\mathscr{T};\mathscr{C} r,r{{\prime}^{{}}}}} :={{\mathcal{H}}_{r,r{{\prime}^{{}}}}}+{{\mathcal{V}}_{\mathscr{P};\mathscr{T};\mathscr{C} r,r{{\prime}^{{}}}}},\end{eqnarray} \tag{ 4.26a }$$

$$\begin{eqnarray}&&{{\mathcal{V}}_{\mathscr{P};\mathscr{T};\mathscr{C} r,{{r}^{\prime}}}}:={{\delta}_{r,{{r}^{\prime}}}}\underset{\mu ,\nu \in \text{row}}{\sum} {{v}_{\mu \nu}}~{{\mathcal{X}}_{\mu \nu}}.\end{eqnarray} \tag{ 4.26a }$$

The corresponding equal-time one-point correlation matrix is ${{\mathcal{Q}}_{\mathscr{P};\mathscr{T};\mathscr{C}}}$ and its upper-left block is ${{Q}_{\mathscr{P};\mathscr{T};\mathscr{C} A}}$. The third column in table 2 provides two signs for each row. The first sign ${{\eta}_{\mathscr{T}}}$ is positive if $\mathscr{P}$ commutes with $\mathscr{T}$ and negative if $\mathscr{P}$ anti-commutes with $\mathscr{T}$. The second sign ${{\eta}_{\mathscr{C}}}$ is positive if $\mathcal{P}$ commutes with $\mathscr{C}$ and negative if $\mathcal{P}$ anti-commutes with $\mathscr{C}$. The information contained with the doublet $\left({{\eta}_{\mathscr{T}}},{{\eta}_{\mathscr{C}}}\right)$ is needed to read from table VI of [16] the bulk topological index of the single-particle Hamiltonian (4.26). This topological index does not guarantee that ${{\mathcal{H}}_{\mathscr{P};\mathscr{T};\mathscr{C}}}$ supports boundary states in an open geometry. In fact, ${{\mathcal{H}}_{\mathscr{P};\mathscr{T};\mathscr{C}}}$ does not support boundary states in an open geometry, for the physical boundaries are interchanged under the operation of parity $\mathscr{P}$. On the other hand, ${{Q}_{\mathscr{P};\mathscr{T};\mathscr{C} A}}$ supports boundary states on the entangling boundaries under the following conditions.

Hamiltonian ${{\mathcal{H}}_{\mathscr{P};\mathscr{T};\mathscr{C}}}$ is local by assumption and gapped if periodic boundary conditions are imposed. As explained in section 3.2, ${{\mathcal{Q}}_{\mathscr{P};\mathscr{T};\mathscr{C}}}$ and all its four blocks inherit this locality. We have shown with equation (2.38) that the upper-left block ${{Q}_{\mathscr{P};\mathscr{T};\mathscr{C} A}}$ inherits the symmetry $\mathscr{T}$ and the spectral symmetry $\mathscr{C}$ of ${{\mathcal{H}}_{\mathscr{P};\mathscr{T};\mathscr{C}}}$. We have also shown with equation (2.45) that the symmetry $\mathscr{P}$ of ${{\mathcal{H}}_{\mathscr{P};\mathscr{T};\mathscr{C}}}$ is turned into a spectral symmetry of ${{Q}_{\mathscr{P};\mathscr{T};\mathscr{C} A}}$ under ${{\Gamma}_{\mathscr{P} A}}:={{C}_{\mathscr{P};\mathscr{T};\mathscr{C} AB}} P_{AB}^{\dagger}$. The unperturbed upper-left block Q_A has two zero modes per entangling boundary. Because of locality, the perturbation

$$\begin{eqnarray}&&\delta {{Q}_{\mathscr{P};\mathscr{T};\mathscr{C} A}}:={{Q}_{\mathscr{P};\mathscr{T};\mathscr{C} A}}-{{Q}_{A}}\end{eqnarray} \tag{ 4.27 }$$

only mixes the two members of a doublet of boundary states of Q_A on a given entangling boundary. Hence, we may represent the effect of the perturbation $\delta {{Q}_{\mathscr{P};\mathscr{T};\mathscr{C} A}}$ by imposing on the Hermitian 2 × 2 matrix

$$\begin{eqnarray}&&\delta {{Q}_{\text{boundary}}}:=\underset{\mu =0}{\overset{3}{\sum}} {{a}_{\mu}}~{{\rho}_{\mu}},\quad {{a}_{\mu}}\in \mathbb{R} \end{eqnarray} \tag{ 4.28 }$$

(ρ₀ is the unit 2 × 2 matrix and ρ₁, ρ₂, ρ₃ are the Pauli matrices), the condition imposed by the symmetry $\mathscr{T}\sim {{\rho}_{0}}~K$ and the spectral symmetries $\mathscr{C}\sim {{\rho}_{2}}~K$ and ${{\Gamma}_{\mathscr{P} A}}$. The first two symmetries imply that

$$\begin{eqnarray}&&\delta {{Q}_{\text{boundary}}}={{a}_{1}}~{{\rho}_{1}}+{{a}_{3}}~{{\rho}_{3}}.\end{eqnarray} \tag{ 4.29 }$$

A doublet of zero modes is thus protected if and only if

$$\begin{eqnarray}&&{{\Gamma}_{\mathscr{P}\ A}}:={{C}_{\mathscr{P};\mathscr{T};\mathscr{C}\ AB}}\ P_{AB}^{\dagger}\sim {{\rho}_{0}},\end{eqnarray} \tag{ 4.30 }$$

for {ρ₀, δQ_boundary} = 0 can then only be satisfied if a₁ = a₃ = 0. We now show that condition (4.30) is only met for the first three rows of table 2, the only rows from table 2 with $\left({{\eta}_{\mathscr{T}}},{{\eta}_{\mathscr{C}}},\right)=\left(+,-\right)$, i.e., the only choice for the triplet $\left({{\mathcal{P}}_{\mu \nu}},{{\mathcal{T}}_{\mu \nu}},{{\mathcal{C}}_{\mu \nu}}\right)$ for which

$$\begin{eqnarray}&&\left[\mathscr{P},\mathscr{T}\right]=0,\quad \left\{\mathscr{P},\mathscr{C}\right\}=0.\end{eqnarray} \tag{ 4.31 }$$

To see this, we are going to combine equation (4.31) with

$$\begin{eqnarray}&&\left[{{C}_{\mathscr{P};\mathscr{T};\mathscr{C}\ AB}},{{T}_{\mathscr{T}}}\right]=0,\quad \left\{{{C}_{\mathscr{P};\mathscr{T};\mathscr{C}\ AB}},{{C}_{\mathscr{C}}}\right\}=0,\end{eqnarray} \tag{ 4.32 }$$

where ${{T}_{\mathscr{T}}}$ and ${{C}_{\mathscr{C}}}$ represent the actions of time reversal and charge conjugation on the partition grading of the equal-time one-point correlation matrix. If we use the algebraic identity

$$\begin{eqnarray}&&\left[{{\Gamma}_{\mathscr{P}\ A}},{{T}_{\mathscr{T}}}\right]={{C}_{\mathscr{P};\mathscr{T};\mathscr{C}\ AB}}\left[P_{AB}^{\dagger},{{T}_{\mathscr{T}}}\right]+\left[{{C}_{\mathscr{P};\mathscr{T};\mathscr{C}\ AB}},{{T}_{\mathscr{T}}}\right]P_{AB}^{\dagger},\end{eqnarray} \tag{ 4.33a }$$

$$\begin{eqnarray}&&\left[{{\Gamma}_{\mathscr{P}\ A}},{{C}_{\mathcal{C}}}\right]={{C}_{\mathscr{P};\mathscr{T};\mathscr{C}\ AB}}\left\{P_{AB}^{\dagger},{{C}_{\mathscr{C}}}\right\}-\left\{{{C}_{\mathscr{P};\mathscr{T};\mathscr{C}\ AB}},{{C}_{\mathscr{C}}}\right\}P_{AB}^{\dagger},\end{eqnarray} \tag{ 4.33b }$$

equation (4.33), when combined with equations (4.32) and (4.31), simplifies to

$$\begin{eqnarray}&&\left[{{\Gamma}_{\mathscr{P}\ A}},{{T}_{\mathscr{T}}}\right]=\left[{{\Gamma}_{\mathscr{P}\ A}},{{C}_{\mathscr{C}}}\right]=0.\end{eqnarray} \tag{ 4.34 }$$

Equation (4.34) allows us to deduce that ${{\Gamma}_{\mathscr{P} A}}$ must be represented by ρ₀ on the two-dimensional Hilbert space spanned by the boundary states on an entangling boundary,

$$\begin{eqnarray}&&{{\Gamma}_{\mathscr{P} A}}\sim {{\rho}_{0}}.\end{eqnarray} \tag{ 4.35 }$$

Hence, the only δQ_boundary in equation (4.29) that anti-commutes with ρ₀ is δQ_boundary = 0, thereby proving the stability of the boundary states on an entangling boundary for the first three rows of table 2.

For completeness, we verify numerically the prediction from section 3.2 that ${{\Gamma}_{\mathscr{P} A}}$ is local in that it does not mix zero modes localized on boundaries separated by a bulk-like distance. To this end, we consider Hamiltonian (4.1c) with or without perturbations for N_tot = 48 orbitals (twelve repeat unit cells with four orbitals per repeat unit cell, N = 12 and N_orb = 4). Plotted in figure 4 as columnar vectors in the orbital basis are the zero modes φ that are localized at the entangling boundaries.

When periodic boundary conditions are imposed, there are two entangling boundaries separated by the bulk-like distance of order N/2 = 6. Correspondingly, there are two zero modes φ_L1 and φ_L2 localized on the left entangling boundary and there are two zero modes φ_R1 and φ_R2 localized on the right entangling boundary. They are represented in figures 4(a), (b), and (d) as columnar vectors in the orbital basis. One verifies that the overlap of any pair of zero modes with one zero mode localized on the left entangling boundary and the other zero mode localized on the right entangling boundary are exponentially suppressed in magnitude by a factor of order exp(−brΔ) with b a number or order unity and the separation r of order N/2 = 6.

When open boundary conditions are imposed, there are two physical boundaries separated by the bulk-like distance of order N/2 = 6. and one entangling boundary a distance of order N/4 = 3 away from either physical boundaries. The perturbation $0.05\ t\ {{\mathcal{X}}_{11}}$ has been added to the Hamiltonian (4.1c). According to table 4.2, this perturbation gaps the zero modes localized on the physical boundaries. Correspondingly, there are two zero modes φ₁ and φ₂ localized on the entangling boundary represented in figures 4(c) and (e) by columnar vectors in the orbital basis.

Our second example is defined by choosing d = 2 and N_orb = 4 in equation (2.1). We represent the action on the orbital degrees of freedom by the 4 × 4 matrices defined in equation (4.9a). The BZ is two-dimensional and the single-particle Hamiltonian admits the direct-sum decomposition [14]

$$\begin{eqnarray}&&\mathcal{H}=\underset{k\in \text{BZ}}{{\oplus}} {{\mathcal{H}}_{k}},\end{eqnarray} \tag{ 5.1a }$$

$$\begin{eqnarray}&&{{\mathcal{H}}_{k}}:=\left[2\ t\ \left( \text{cos}\ {{k}_{1}}+ \text{cos}\ {{k}_{2}}\right)-\mu \right] {{\mathcal{X}}_{03}}-2\ \Delta\ \text{sin}\ {{k}_{1}}\ {{\mathcal{X}}_{31}}-2\ \Delta\ \text{sin}\ {{k}_{2}}\ {{\mathcal{X}}_{02}},\end{eqnarray} \tag{ 5.1b }$$

with the real-valued characteristic energy scales t, Δ, and μ.

Hamiltonian $\mathcal{H}$ can be interpreted as the direct sum

$$\begin{eqnarray}&&{{\mathcal{H}}_{k}}=\mathcal{H}_{k}^{\left(+\right)}\oplus \mathcal{H}_{k}^{\left(-\right)}\end{eqnarray} \tag{ 5.2a }$$

of the Bogoliubov–de-Gennes Hamiltonian

$$\begin{eqnarray}&&\mathcal{H}_{k}^{\left(+\right)}\sim \left[2\ t\ \left(\text{cos}\ {{k}_{1}}+ \text{cos}\ {{k}_{2}}\right)-\mu \right] {{\tau}_{3}}-2\ \Delta \left[\text{sin}\ {{k}_{1}}\ {{\tau}_{1}}+ \text{sin}\ {{k}_{2}}\ {{\tau}_{2}}\right]\end{eqnarray} \tag{ 5.2b }$$

describing p₁ + ip₂ superconducting order and the Bogoliubov–de-Gennes Hamiltonian

$$\begin{eqnarray}&&\mathcal{H}_{k}^{\left(-\right)}\sim \left[2\ t\ \left(\text{cos}\ {{k}_{1}}+ \text{cos}\ {{k}_{2}}\right)-\mu \right] {{\tau}_{3}}+2\ \Delta \left[\text{sin}\ {{k}_{1}}\ {{\tau}_{1}}- \text{sin}\ {{k}_{2}}\ {{\tau}_{2}}\right]\end{eqnarray} \tag{ 5.2c }$$

describing p₁ − ip₂ superconducting order. A gap is present when |μ| < 4t, an assumption that is made throughout in section 5.

The bundle of single-particle eigenstates obtained by collecting the lower band from the bundle $\left\{\mathcal{H}_{k}^{\left(+\right)}, k\in \text{BZ}\right\}$ has the opposite non-vanishing Chern number to that of the bundle of single-particle eigenstates obtained by collecting the lower band from the bundle $\left\{\mathcal{H}_{k}^{\left(-\right)}, k\in \text{BZ}\right\}$. The bundle of single-particle eigenstates obtained by collecting the lower band from the bundle $\left\{{{\mathcal{H}}_{k}}, k\in \text{BZ}\right\}$ is topologically trivial.

The symmetries of the single-particle Hamiltonian defined by equation (5.1) are the following.

• 1  
  There are two symmetries of the inversion type. If $\mathscr{I}$ denotes the inversion of space

  $$\begin{eqnarray}&&\mathscr{I}:r\mapsto -r,\end{eqnarray} \tag{ 5.3a }$$

  then

  $$\begin{eqnarray}&&\mathcal{O}_{\mathscr{I}}^{\dagger} {{\mathcal{H}}_{-k}}{{\mathcal{O}}_{\mathscr{I}}}={{\mathcal{H}}_{+k}}\end{eqnarray} \tag{ 5.3a }$$

  with

  $$\begin{eqnarray}&&{{\mathcal{O}}_{\mathscr{I}}}\in \left\{{{\mathcal{X}}_{03}},{{\mathcal{X}}_{33}}\right\}.\end{eqnarray} \tag{ 5.3c }$$
• 2  
  There are two symmetries of the reflection about the horizontal axis k = (k₁, 0) type. If ${{\mathscr{R}}_{1}}$ denotes the reflection

  $$\begin{eqnarray}&&{{\mathscr{R}}_{1}}:{{r}_{1}}\mapsto +{{r}_{1}},\quad {{\mathscr{R}}_{1}}:{{r}_{2}}\mapsto -{{r}_{2}},\end{eqnarray} \tag{ 5.4a }$$

  then

  $$\begin{eqnarray}&&\mathcal{O}_{{{\mathcal{R}}_{1}}}^{\dagger} {{\mathcal{H}}_{+{{k}_{1}},-{{k}_{2}}}}~{{\mathcal{O}}_{{{\mathcal{R}}_{1}}}}={{\mathcal{H}}_{+k}}\end{eqnarray} \tag{ 5.4b }$$

  with

  $$\begin{eqnarray}&&{{\mathcal{O}}_{{{\mathcal{R}}_{1}}}}\in \left\{{{\mathcal{X}}_{13}},{{\mathcal{X}}_{23}}\right\}.\end{eqnarray} \tag{ 5.4c }$$
• 3  
  There are two symmetries of the reflection about the vertical axis k = (0, k₂) type. If ${{\mathcal{R}}_{2}}$ denotes the reflection

  $$\begin{eqnarray}&&{{\mathscr{R}}_{2}}:{{r}_{1}} \mapsto -{{r}_{1}},\quad {{\mathscr{R}}_{2}}:{{r}_{2}} \mapsto +{{r}_{2}},\end{eqnarray} \tag{ 5.5a }$$

  then

  $$\begin{eqnarray}&&\mathcal{O}_{{{\mathcal{R}}_{2}}}^{\dagger} {{\mathcal{H}}_{-{{k}_{1}},+{{k}_{2}}}}~{{\mathcal{O}}_{{{\mathcal{R}}_{2}}}}={{\mathcal{H}}_{+k}}\end{eqnarray} \tag{ 5.5b }$$

  with

  $$\begin{eqnarray}&&{{\mathcal{O}}_{{{\mathcal{R}}_{2}}}}\in \left\{{{\mathcal{X}}_{10}},{{\mathcal{X}}_{20}}\right\}.\end{eqnarray} \tag{ 5.5c }$$
• 4  
  There are two symmetries of the time-reversal type. If $\mathscr{T}$ denotes the reversal of time

  $$\begin{eqnarray}&&\mathscr{T}:t\mapsto -t,\end{eqnarray} \tag{ 5.6a }$$

  then

  $$\begin{eqnarray}&&\mathcal{O}_{\mathscr{T}}^{\dagger} \mathcal{H}_{-k}^{*} {{\mathcal{O}}_{\mathscr{T}}}={{\mathcal{H}}_{+k}}\end{eqnarray} \tag{ 5.6b }$$

  with

  $$\begin{eqnarray}&&{{\mathcal{O}}_{\mathscr{T}}}\in \left\{{{\mathcal{X}}_{10}},{{\mathcal{X}}_{20}}\right\}.\end{eqnarray} \tag{ 5.6c }$$
• 5  
  There are two spectral symmetries of the charge-conjugation type,

  $$\begin{eqnarray}&&\mathcal{O}_{\mathscr{C}}^{\dagger} \mathcal{H}_{-k}^{\text{T}} {{\mathcal{O}}_{\mathscr{C}}}=-{{\mathcal{H}}_{+k}}\end{eqnarray} \tag{ 5.7a }$$

  with

  $$\begin{eqnarray}&&{{\mathcal{O}}_{\mathscr{C}}}\in \left\{{{\mathcal{X}}_{01}},{{\mathcal{X}}_{31}}\right\}.\end{eqnarray} \tag{ 5.7b }$$
• 6  
  There are two spectral symmetries of the chiral type,

  $$\begin{eqnarray}&&\mathcal{O}_{\mathscr{S}}^{\dagger} {{\mathcal{H}}_{k}} {{\mathcal{O}}_{\mathscr{S}}}=-{{\mathcal{H}}_{k}}\end{eqnarray} \tag{ 5.8a }$$

  with

  $$\begin{eqnarray}&&{{\mathcal{O}}_{\mathscr{C}}}\in \left\{{{\mathcal{X}}_{11}},{{\mathcal{X}}_{21}}\right\}.\end{eqnarray} \tag{ 5.8b }$$

From now on, we adopt a cylindrical geometry instead of the torus geometry from section 5.1, i.e., we compactify two-dimensional space $\left\{x:=\left({{x}_{1}},{{x}_{2}}\right)\in {{\mathbb{R}}^{2}}\right\}$ only along one Cartesian coordinate. Consequently, only one component of the two-dimensional momentum $\left\{k:=\left({{k}_{1}},{{k}_{2}}\right)\in {{\mathbb{R}}^{2}}\right\}$ is chosen to be a good quantum number in equation (5.1).

For any choice of the direction i = 1, 2 along which periodic boundary conditions are imposed while open boundary conditions are imposed along the orthogonal direction i + 1 modulo 2, for any positive integer M_i, and for any good momentum quantum number

$$\begin{eqnarray}&&{{k}_{i}}=\frac{2\pi}{2{{M}_{i}}~\mathfrak{a}}{{n}_{i}},\quad {{n}_{i}}=1,\cdots ,2{{M}_{i}},\quad i=1,2,\end{eqnarray} \tag{ 5.9a }$$

we do the following. First, we denote by ${{\mathcal{H}}_{{{k}_{i}}}}$ the 8M_i+1 × 8M_i+1 Hermitian matrix such that the one-dimensional Fourier transform of

$$\begin{eqnarray}&&{{\mathcal{H}}_{{{k}_{i}} {{r}_{i+1}},r_{i+1}^{\prime}}}:=\langle {{r}_{i+1}}|{{\mathcal{H}}_{{{k}_{i}}}}|r_{i+1}^{\prime}\rangle ,\end{eqnarray} \tag{ 5.9b }$$

delivers equation (5.1b). Here,

$$\begin{eqnarray}&&{{r}_{i+1}}={{n}_{i+1}}~\mathfrak{a},\quad {{n}_{i+1}}=1,\cdots ,2{{M}_{i+1}},\end{eqnarray} \tag{ 5.9c }$$

and

$$\begin{eqnarray}&&r_{i+1}^{\prime}=n_{i+1}^{\prime}~\mathfrak{a},\quad n_{i+1}^{\prime}=1,\cdots ,2{{M}_{i+1}},\end{eqnarray} \tag{ 5.9d }$$

are the lattice sites from an open chain along the direction i + 1 (defined modulo 2) of a rectangular lattice with the lattice spacing $\mathfrak{a}$ and the number of lattice sites M_i + 1 along the direction i + 1 modulo 2. Second, we define the k_i-dependent partition

$$\begin{eqnarray}&&\mathfrak{H}:=\underset{{{k}_{i}}}{{\oplus}} ~{{\mathfrak{H}}_{{{k}_{i}}}},~{{\mathfrak{H}}_{{{k}_{i}}}}:=~{{\mathfrak{H}}_{{{A}_{i+1}}}}\oplus ~{{\mathfrak{H}}_{{{B}_{i+1}}}},\end{eqnarray} \tag{ 5.10a }$$

where

$$\begin{eqnarray}&&{\mathfrak{H}_{{A_{i + 1}}}}: = \mathop {\mathop \oplus \limits^{{M_{i + 1}}} }\limits_{{n_{i + 1}} = 1} \mathop {\mathop \oplus \limits^4 }\limits_{\alpha = 1} |{k_i},{n_{i + 1}},\alpha \rangle \langle {k_i},{n_{i + 1}},\alpha |,\end{eqnarray} \tag{ 5.10b }$$

$$\begin{eqnarray}&&{{\mathfrak{H}}_{{{B}_{i+1}}}}:=\underset{n_{i+1}={M_{i+1}}+1}{\overset{2{M_{i+1}}}{\oplus}} \underset{\alpha =1}{\overset{4}{\oplus}} |{{k}_{i}},{{n}_{i+1}},\alpha \rangle \langle {{k}_{i}},{{n}_{i+1}},\alpha |,\end{eqnarray} \tag{ 5.10c }$$

and the ket |k_i, n_i + 1, α〉 denotes the single-particle state with the Bloch index k_i, for it is extended along the i = 1, 2 direction, the unit repeat cell index n_i + 1, for it is localized at the site n_i + 1 $\mathfrak{a}$ along the i + 1 modulo 2 direction, and the orbital index α = 1, 2, 3, 4. If we denote by ${{\mathcal{R}}_{i}}$ the reflection that leaves k_i unchanged but reverses the sign of k_i + 1, i.e.,

$$\begin{eqnarray}&&{{\mathcal{R}}_{i}}\ {{k}_{i}}=+{{k}_{i}},\quad {{\mathcal{R}}_{i}}\ {{k}_{i+1}}=-{{k}_{i+1}},\end{eqnarray} \tag{ 5.11 }$$

we then have that

$$\begin{eqnarray}&&{{\mathcal{R}}_{i}}\ {{A}_{i+1}}={{B}_{i+1}},\ \ {{\mathcal{R}}_{i}}\ {{B}_{i+1}}={{A}_{i+1}},\end{eqnarray} \tag{ 5.12a }$$

$$\begin{eqnarray}&&\mathscr{C}{{A}_{i+1}}={{A}_{i+1}},\quad \mathscr{C}{{B}_{i+1}}={{B}_{i+1}},\end{eqnarray} \tag{ 5.12b }$$

$$\begin{eqnarray}&&\mathscr{T}{{A}_{i+1}}={{A}_{i+1}},\quad \mathscr{T}{{B}_{i+1}}={{B}_{i+1}},\end{eqnarray} \tag{ 5.12c }$$

$$\begin{eqnarray}&&\mathscr{S}{{A}_{i+1}}={{A}_{i+1}},\quad \mathscr{S}{{B}_{i+1}}={{B}_{i+1}}.\end{eqnarray} \tag{ 5.12d }$$

The thermodynamic limit is defined by

$$\begin{eqnarray}&&N,{{N}_{\text{f}}}\to \infty ,\end{eqnarray} \tag{ 5.13a }$$

with N = (2M₁) × (2M₂), holding the fermion density

$$\begin{eqnarray}&&{{N}_{\text{f}}}/N=1\end{eqnarray} \tag{ 5.13b }$$

fixed. In the thermodynamic limit, the single-particle energy eigenvalue spectrum $\sigma \left({{\mathcal{H}}_{i}}\right)$ of

$$\begin{eqnarray}&&{{\mathcal{H}}_{i}}:=\underset{{{k}_{i}}}{{\oplus}} {{\mathcal{H}}_{{{k}_{i}}}}\end{eqnarray} \tag{ 5.14a }$$

supports two pairs of zero modes, each of which are localized at the opposite (physical) boundaries on the cylinder whose symmetry axis coincides with the direction i + 1 (defined modulo 2) along which open boundary conditions have been imposed. Similarly, the entanglement spectrum $\sigma \left({{Q}_{{{A}_{i+1}}}}\right)$ of the equal-time one-point correlation matrix

$$\begin{eqnarray}&&{{Q}_{{{A}_{i+1}}}}:=\mathbb{I}-2{{C}_{{{A}_{i+1}}}}\end{eqnarray} \tag{ 5.14b }$$

supports a pair of zero modes localized at the entangling boundary.

In order not to confuse gapless boundary modes originating from the physical boundaries with gapless boundary modes originating from the entangling boundary, whenever the physical boundaries support gapless boundary states in the energy spectrum, we opt for a torus geometry when computing the entanglement spectrum.

Desired is a study of the stability of these zero modes under all generic perturbations of ${{\mathcal{H}}_{i}}$ that respect any one $\left(\mathscr{S}\right)$ of the two chiral symmetry from equation (5.8) and any one $\left(\mathcal{R}\right)$ of the four reflection symmetries from equations (5.4) and (5.5).

5.4.1. Definitions of ${{\tilde{\mathcal{H}}}_{i\ \mu \nu}}$ and ${{\tilde{\mathcal{Q}}}_{{{k}_{i}}\ \mu \nu {{A}_{i+1}}}}$.

To begin with the stability analysis, we choose i = 1, 2 and perturb Hamiltonian (5.9b) by adding locally any one of the sixteen matrices ${{\mathcal{X}}_{\mu \nu}}\equiv {{\sigma}_{\mu}}\otimes {{\tau}_{\nu}}$ parametrized by μ, ν = 0, 1, 2, 3. Thus, we define

$$\begin{eqnarray}&&{{\tilde{\mathcal{H}}}_{i\ \mu \nu}}:=\underset{{{k}_{i}}}{{\oplus}} {{\tilde{\mathcal{H}}}_{{{k}_{i}}\ \mu \nu}}\equiv \underset{{{k}_{i}}}{{\oplus}} \left({{\mathcal{H}}_{{{k}_{i}}}}+{{\mathcal{V}}_{\mu \nu}}\right)\end{eqnarray} \tag{ 5.15a }$$

by the matrix elements

$$\begin{eqnarray}&&{{\tilde{\mathcal{H}}}_{{{k}_{i}}\ \mu \nu {{r}_{i+1}},r_{i+1}^{\prime}}}:={{\mathcal{H}}_{{{k}_{i}}\ {{r}_{i+1}},r_{i+1}^{\prime}}}+{{\mathcal{V}}_{\mu \nu\ {{r}_{i+1}},r_{i+1}^{\prime}}},\end{eqnarray} \tag{ 5.15b }$$

$$\begin{eqnarray}&&{{\mathcal{V}}_{\mu \nu\ {{r}_{i+1}},r_{i+1}^{\prime}}}:={{\delta}_{{{r}_{i+1}},r_{i+1}^{\prime}}}{{v}_{\mu \nu}} {{\mathcal{X}}_{\mu \nu}},\end{eqnarray} \tag{ 5.15c }$$

where the matrix elements of ${{\mathcal{H}}_{{{k}_{i}}}}$ were defined in equation (5.9) and the perturbation strength v_μν is real valued. Equipped with ${{\tilde{\mathcal{H}}}_{{{k}_{i}}\ \mu \nu}}$, we define ${{\tilde{\mathcal{Q}}}_{{{k}_{i}}\ \mu \nu {{A}_{i+1}}}}$ from equation (2.11) with the caveat that periodic boundary conditions are imposed along the direction i + 1 (modulo 2).

5.4.2. Spectra $\sigma \left({{\tilde{\mathcal{H}}}_{i\ \mu \nu}}\right)$.

For any i = 1, 2, the spectra $\sigma \left({{\tilde{\mathcal{H}}}_{i \mu \nu}}\right)$ for μ, ν = 0, 1, 2, 3 were obtained from exact diagonalization with the unperturbed energy scales t = Δ = − μ = 1. These spectra are characterized by two continua of single-particle excitations separated by an energy gap as is illustrated in figures 5(a)–(d). A discrete number (four) of branches are seen to peel off from these continua, some of which eventually cross the band gap. These mid-gap branches disperse along the physical boundaries (edges) while they decay exponentially fast away from the edges. They are thus called edge states. Their crossings, if any, define degenerate edge states with group velocities of opposite signs along the edges. Their crossings at vanishing energy and momentum, if any, define degenerate non-propagating zero modes with group velocities of opposite signs along the edges. Their crossings at vanishing energy and non-vanishing momenta, if any, define degenerate propagating zero modes with group velocities of opposite signs along the edges. Their crossings at non-vanishing energies and vanishing momentum, if any, define degenerate non-propagating edge states with group velocities of opposite signs along the edges. The entries ○ in the last two columns of table 5.14 accounts for the existence of at least one crossing at vanishing energy and momentum. The entries ⊗ in the last two columns of table 5.14 account for the existence of crossings not necessarily at vanishing energy and momentum. The entries × in the last two column of table 5.14 accounts for the absence of crossings.

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The origin of the mid-gap branches in figures 5(a)–(f) is the following. The single-particle Hamiltonian defined in equation (5.1b) for a torus geometry supports four boundary states for a cylindrical geometry that are extended along the boundary but exponentially localized away from the boundary due to the non-vanishing Chern numbers for each of $\mathcal{H}_{k}^{\left(+\right)}$ and $\mathcal{H}_{k}^{\left(-\right)}$. These dispersing boundary states show up in the single-particle energy spectrum as four branches of mid-gap states that crosses at vanishing energy and momentum, thereby defining four degenerate non-propagating zero modes. In the presence of the perturbation $0.3t{{\mathcal{X}}_{13}}$ when the boundary dictates that k₁ is a good quantum number, the energy spectrum shows a four-fold degenerate crossing at vanishing energy and momentum in figure 5(a). However, the very same perturbation, when the boundary dictates that k₂ is a good quantum number, gaps the unperturbed crossing at vanishing energy and momentum in figure 5(b). In the presence of the perturbation $0.3t{{\mathcal{X}}_{02}}$ when the boundary dictates that k₂ is a good quantum number, the energy spectrum shows a four-fold-degenerate crossing at vanishing energy but away from k₂ = 0 in figure 5(c)]. In the presence of the perturbation $0.3t{{\mathcal{X}}_{32}}$ when the boundary dictates that k₂ is a good quantum number, the energy spectrum shows four non-degenerate crossings away from vanishing energy and vanishing momentum in figure 5(d). Even though all the crossings in the unperturbed entanglement spectra are robust to the perturbations in figures 5(e)–(h), they can be shifted away from vanishing energy as in figure 5(h) or momentum as in figure 5(g).

To understand the effect of any one of the sixteen perturbations from table 5.14 on the zero modes of Hamiltonian ${{\mathcal{H}}_{i}}$ defined in equation (5.14a), where i = 1, 2 is the choice made along the direction for which periodic boundary conditions is imposed (recall equation (5.9), it is useful to consider figure 5. The physical boundaries are the two circles centered at L and R on the symmetry axis i + 1 modulo 2 of the cylinder in figure 5. In the limit for which the band gap is taken to infinity, the effective theory for the single-particle eigenstates of ${{\mathcal{H}}_{i}}$ defined in equation (5.14a) that are propagating along the edges L and R with the same group velocity but localized away from them takes the form

$$\begin{eqnarray}&&{{\mathcal{H}}_{\text{edges} {{k}_{i}}}}={{k}_{i}}\ {{Y}_{33}}\quad i=1,2.\end{eqnarray} \tag{ 5.16a }$$

Here, we have set the group velocity to unity and

$$\begin{eqnarray}&&{{Y}_{\mu \nu}}:={{\rho}_{\mu}}\otimes {{\varrho}_{\nu}},\end{eqnarray} \tag{ 5.16b }$$

where we have introduced the two sets of unit 2 × 2 matrix and Pauli matrices ρ_μ with μ = 0, 1, 2, 3 and ϱ_ν with ν = 0, 1, 2, 3. The eigenvalue −1 of ϱ₃ is interpreted as the left edge L in figure 5. The eigenvalue +1 of ϱ₃ is interpreted as the right edge R in figure 5. Hence, the matrices ϱ₁ and ϱ₂ mix edge states localized on opposite edges. The eigenstates of ρ₃ describe right and left movers on a given edge. The matrices ρ₁ and ρ₂ mix left and right movers on a given edge. Hamiltonian (5.16) describes four single-particle states propagating along the two edges with the momentum ± k_i. These single-particle states are solely supported on the edges, reason for which we call them edge states. Their direction of propagation originates from the relative sign in p₁ ± ip₂ if ${{\mathcal{H}}_{i}}$ is interpreted as a Bogoliubov–de-Gennes superconductor.

A generic perturbation to the effective Hamiltonian (5.16) that allows mixing between all four edge states is

$$\begin{eqnarray}&&{{\mathcal{V}}_{\text{edges}}}=\underset{\mu ,\nu =0}{\overset{3}{\sum}} {{v}_{\text{edges}\mu\ \nu}} {{Y}_{\mu \nu}},\quad {{v}_{\text{edges} \mu \nu}}\in \mathbb{R} ,\end{eqnarray} \tag{ 5.17 }$$

to lowest order in a gradient expansion. For any given μ, ν = 0, 1, 2, 3, the perturbation

$$\begin{eqnarray}&&{{\mathcal{V}}_{\text{edges}\ \mu \nu}}:={{v}_{\text{edges}\ \mu \nu}}\ {{Y}_{\mu \nu}}\end{eqnarray} \tag{ 5.18 }$$

opens a gap in the spectrum of

$$\begin{eqnarray}&&{{\tilde{\mathcal{H}}}_{\text{edges}\ {{k}_{i}}\ \mu \nu}}:={{\mathcal{H}}_{\text{edges}\ {{k}_{i}}}}+{{\mathcal{V}}_{\text{edges}\ \mu \nu}}\end{eqnarray} \tag{ 5.19 }$$

if it anti-commutes with Y₃₃, thereby forbidding the crossing of the bulk gap by any one of the four mid-gap branches from the effective Hamiltonian (5.16). This is what happens in figure 5(b). A perturbation ${{\mathcal{V}}_{\text{edges}\ \mu \nu}}$ that commutes with Y₃₃ can shift the location of the two crossings of the four mid-gap branches from the effective Hamiltonian (5.16) away from either vanishing energy or vanishing momentum. For example, the perturbation ${{\mathcal{V}}_{\text{edges}\ 33}}$ with v_{edges 33} < 0 moves the crossing in the effective Hamiltonian (5.16) to the right, as happens in figure 5(c). Figure 5(d) is realized when only one of v_{edges 03} and v_{edges 30} are non-vanishing in the perturbation (5.18). Evidently, figure 5(a) suggests that not all perturbations parametrized by (5.17) move the degenerate crossing of the effective Hamiltonian equation (5.16).

To understand figure 5(a), we assume that the perturbation ${{\mathcal{V}}_{\text{edges}}}$ is local and that the ratio ξ/L, where L is the length of the cylinder while ξ ∝ 1/Δ is the bulk correlation length associated to the bulk gap Δ, is taken to zero. Hence, the effective edge theory for

$$\begin{eqnarray}&&{{\tilde{\mathcal{H}}}_{i\ \mu \nu}}:={{\mathcal{H}}_{i}}+{{\mathcal{V}}_{\mu \nu}}\end{eqnarray} \tag{ 5.20 }$$

remains block diagonal, with one of the block given by

$$\begin{eqnarray}&&{{\tilde{\mathcal{H}}}_{\text{edge}\ {{k}_{i}}}}={{v}_{0}}{{\rho}_{0}}+{{v}_{1}}{{\rho}_{1}}+{{v}_{2}}{{\rho}_{2}}+\left({{k}_{i}}+{{v}_{3}}\right) {{\rho}_{3}}\end{eqnarray} \tag{ 5.21a }$$

for some effective real-valued couplings v₀, v₁, v₂, and v₃. As was the case for the bulk Hamiltonian (5.1),

$$\begin{eqnarray}&&{{\mathcal{H}}_{\text{edge}\ {{k}_{i}}}}:={{k}_{i}}\ {{\rho}_{3}}=:{{\tilde{\mathcal{H}}}_{\text{edge}\ {{k}_{i}}\ \mu \nu}}-{{\tilde{\mathcal{V}}}_{\text{edge}\ \mu \nu}}\end{eqnarray} \tag{ 5.21b }$$

obeys two spectral chiral symmetries, for it anti-commutes with ρ₁ and ρ₂. Moreover, ${{\mathcal{H}}_{\text{edge}\ {{k}_{i}}}}$ commutes with the operation by which k_i → − k_i is composed with the conjugation of ${{\mathcal{H}}_{\text{edge}\ {{k}_{i}}}}$ with either ρ₁ and ρ₂. Imposing simultaneously this pair of chiral symmetries and this pair of reflection symmetries restricts the effective edge theory to equation (5.21b). However, these symmetry constraints are redundant to enforce the form (5.21b) as we now show.

The crossing at vanishing energy and momentum of Hamiltonian (5.21b) is not generically robust to the perturbations in the perturbed Hamiltonian (5.21a). However, if we impose that the perturbed Hamiltonian (5.21a) anti-commutes with either ρ₁ or ρ₂, i.e., the effective perturbed Hamiltonian belongs to the symmetry class AIII, then the edge theory simplifies to the direct sum over blocks of the form

$$\begin{eqnarray}&&\mathcal{H}_{\text{edge}\ {{k}_{i}}\ \mu \nu}^{\text{AIII}}={{v}_{2}}\ {{\rho}_{2}}+\left({{k}_{i}}+{{v}_{3}}\right)\ {{\rho}_{3}}.\end{eqnarray} \tag{ 5.22 }$$

Here, we have chosen, without loss of generality, to implement the chiral symmetry by demanding that $\mathcal{H}_{\text{edge}\ {{k}_{i}}\ \mu \nu}^{\text{AIII}}$ anti-commutes with ρ₁. As is, the perturbations in Hamiltonian (5.22) still gaps the unperturbed dispersion (5.21b). However, if we impose the symmetry constraint

$$\begin{eqnarray} &&{{\rho}_{1}}{{\mathcal{H}}_{\text{edge}\ \left(-{{k}_{i}}\right)\ \mu \nu \mathscr{S}\mathscr{R}}}\ {{\rho}_{1}}={{\mathcal{H}}_{\text{edge} \left(+{{k}_{i}}\right) \mu \nu\ \mathscr{S}\mathscr{R}}} \end{eqnarray} \tag{ 5.23 }$$

to define ${{\mathcal{H}}_{\text{edge}\ {{k}_{i}}\ \mu \nu\ \mathscr{S}\mathscr{R}}}$, we find that

$$\begin{eqnarray}&&{{\mathcal{H}}_{\text{edge}\ {{k}_{i}}\ \mu \nu\ \mathscr{S}\mathscr{R}}}={{k}_{i}}\ {{\rho}_{3}}.\end{eqnarray} \tag{ 5.24 }$$

In other words, if we impose, in addition to the chiral symmetry, the reflection symmetry generated by ρ₁ and k_i → −k_i, we find that no perturbation can gap the unperturbed dispersion (5.21b). Had we chosen to impose the reflection symmetry generated by ρ₂ instead of ρ₁, i.e., a realization of parity that anti-commutes with the choice ρ₁ we made to implement the chiral transformation, then the perturbation v₂ ρ₂ in equation (5.22) would not be prevented from removing the crossing of the edge states through a spectral gap.

Now, the reflection symmetry obeyed by the effective edge Hamiltonian (5.21b) can only originate from a reflection about the plane frame in blue that contains the cylinder axis in figure 5. A reflection about the plane framed in red that is orthogonal to the cylinder axis in figure 5 exchanges the edges L and R. As such, it can only be represented within the representation defined by equations (5.16) and (5.17) of the effective edge Hamiltonian that allows mixing of the two opposite edges.

5.4.3. Spectra $\sigma \left({{\tilde{\mathcal{Q}}}_{\mu \nu\ {{A}_{i+1}}}}\right)$.

For any i = 1, 2, the spectra $\sigma \left({{\tilde{\mathcal{Q}}}_{\mu \nu\ {{A}_{i+1}}}}\right)$ for μ, ν = 0, 1, 2, 3 were also obtained from exact diagonalization with the unperturbed energy scales t = Δ = −μ = 1. These spectra are characterized by two nearly flat bands at ±1 from which pairs of mid-gap branches occasionally peel off as is illustrated in figures 5(e)–(f ). We observe in both figures 5(e) and (f ) one crossing of the mid-gap branches at vanishing entanglement eigenvalue and momentum. The crossing in 5.2(g) takes place at vanishing entanglement eigenvalue but non-vanishing momentum. The crossing in 5.2(h) takes place at non-vanishing entanglement but vanishing momentum. Crossings of the mid-gap branches taking place at vanishing entanglement eigenvalue and momentum are indicated by ○ in the last two columns of table 5.14. The absence of any crossing of the mid-gap branches is indicated by × in the last two columns of table 5.14. Crossings of the mid-gap branches taking place away from vanishing entanglement eigenvalue and momentum are indicated by ⊗ in the last two columns of table 5.14.

Comparison of figures 5(a)–(d) and (e)–(f ) suggests that the existence of crossings in the entanglement spectrum defined by equation (5.14b) is more robust to perturbations than that for the Hamiltonian spectrum. Choose i = 1, 2 (i + 1 modulo 2), μ = 0, 1, 2, 3, and ν = 0, 1, 2, 3. According to table 5.14 any crossing in the perturbed Hamiltonian spectrum $\sigma \left({{\tilde{\mathcal{H}}}_{i \mu \nu}}\right)$ implies a crossing in the perturbed entanglement spectrum $\sigma \left({{\tilde{\mathcal{Q}}}_{\mu \nu\ {{A}_{i+1}}}}\right)$. The converse is not true. There are crossings in the perturbed entanglement spectrum $\sigma \left({{\tilde{\mathcal{Q}}}_{\mu \nu\ {{A}_{i+1}}}}\right)$ but no crossings in the perturbed Hamiltonian spectrum $\sigma \left({{\tilde{\mathcal{H}}}_{i \mu\ \nu}}\right)$, as is shown explicitly when comparing figure 5(f ) to (b). In fact, table 5.14 shows the following difference between the perturbed Hamiltonian and entanglement spectra. On the one hand, the existence of crossings in $\sigma \left({{\tilde{\mathcal{H}}}_{i\ \mu \nu}}\right)$ does not necessarily implies the existence of crossings in $\sigma \left({{\tilde{\mathcal{H}}}_{\left(i+1\right)\ \mu \nu}}\right)$. On the one hand, the existence of crossings in $\sigma \left({{\tilde{\mathcal{Q}}}_{\mu \nu\ {{A}_{i}}}}\right)$ implies the existence of crossings in $\sigma \left({{\tilde{\mathcal{Q}}}_{\mu \nu {{A}_{i+1}}}}\right)$ and vice versa.

To understand the effect of any one of the sixteen perturbations from table 5.14 on the zero modes of ${{\mathcal{Q}}_{{{A}_{i+1}}}}$ defined in equation (5.14b), where i + 1 modulo 2 is the choice made for the direction along which the partition is made (recall equation (5.9)), it is useful to consider the compactification along the cylinder axis of figure 5 consisting in identifying the circles at L and R. We thereby obtain a torus, whose intersection with the plane framed in red from figure 5 defines two entangling boundaries separated by the distance L/2. The spectrum $\sigma \left({{\mathcal{Q}}_{{{A}_{i+1}}}}\right)$ supports four mid-gap branches, two per entangling boundaries. In the limit for which the band gap is taken to infinity, the effective theory for the eigenstates of ${{\mathcal{Q}}_{{{A}_{i+1}}}}$ that propagate along the entangling boundaries but are localized away from them is given by

$$\begin{eqnarray}&&{{\mathcal{Q}}_{\text{edges}\ {{k}_{i}}\ {{A}_{i+1}}}}={{k}_{i}}\ {{Y}_{33}}.\end{eqnarray} \tag{ 5.25 }$$

The interpretation of the 4 × 4 matrices Y_μν with μ, ν = 0, 1, 2, 3 is the same as the one given below equation (5.16b).

A generic perturbation to the effective Hamiltonian (5.25) that allows mixing between all four entangling states is

$$\begin{eqnarray}&&\delta {{\cal Q}_{{\rm{edges}}\:{A_{i + 1}}}} = \mathop {\mathop \sum \limits^3 }\limits_{\mu ,\nu = 0} {v_{{\rm{edges}}\:\mu \nu \:{A_{i + 1}}}}{Y_{\mu \nu }},{v_{{\rm{edges}}\:\mu \nu {A_{i + 1}}}} \in \mathbb{R},\end{eqnarray} \tag{ 5.26 }$$

to lowest order in a gradient expansion. For any given μ, ν = 0, 1, 2, 3, the perturbation

$$\begin{eqnarray}&&\delta {{\mathcal{Q}}_{\text{edges}\ \mu \nu\ {{A}_{i+1}}}}:={{v}_{\text{edges}\ \mu \nu\ {{A}_{i+1}}}}{{Y}_{\mu \nu}}\end{eqnarray} \tag{ 5.27 }$$

opens a gap in the spectrum of

$$\begin{eqnarray}&&{{\tilde{\mathcal{Q}}}_{\text{edges}\ {{k}_{i}}\ \mu \nu\ {{A}_{i+1}}}}:={{\mathcal{Q}}_{\text{edges}\ {{k}_{i}}\ {{A}_{i+1}}}}+\delta {{\mathcal{Q}}_{\text{edges}\ \mu \nu\ {{A}_{i+1}}}}\end{eqnarray} \tag{ 5.28 }$$

if it anti-commutes with Y₃₃, thereby forbidding the crossing of the bulk gap by any one of the four mid-gap branches from the effective Hamiltonian (5.25). A perturbation ${{\mathcal{V}}_{\text{edges}\ \mu \nu\ {{A}_{i+1}}}}$ that commutes with Y₃₃ can shift the location of the two crossings of the four mid-gap branches from the effective Hamiltonian (5.25) away from either vanishing energy or vanishing momentum.

To understand the robustness of the effective (5.25) to perturbations, we assume that the perturbation ${{\mathcal{V}}_{\text{edges}\ {{A}_{i+1}}}}$ is local and that the ratio ξ/L, where L is the length of the cylinder while ξ ∝ 1/Δ is the bulk correlation length associated to the bulk gap Δ, is taken to zero. Hence, the effective edge theory for

$$\begin{eqnarray}&&{{\tilde{\mathcal{Q}}}_{\mu \nu {{A}_{i+1}}}}:={{\mathcal{Q}}_{{{A}_{i+1}}}}+\delta {{\mathcal{Q}}_{\mu \nu {{A}_{i+1}}}}\end{eqnarray} \tag{ 5.29 }$$

remains block diagonal, with one of the blocks given by

$$\begin{eqnarray}&&{{\tilde{\mathcal{Q}}}_{\text{edge}\ {{k}_{i}}\ {{A}_{i+1}}}}={{v}_{0}}\ {{\rho}_{0}}+{{v}_{1}}\ {{\rho}_{1}}+{{v}_{2}}\ {{\rho}_{2}}+\left({{k}_{i}}+{{v}_{3}}\right)\ {{\rho}_{3}}\end{eqnarray} \tag{ 5.30 }$$

for some effective real-valued couplings v₀, v₁, v₂, and v₃.

To proceed, we recall that if $\mathcal{S}$ is a unitary spectral symmetry of $\mathcal{H}$ in that ${{\mathcal{S}}^{-1}} \mathcal{H}\mathcal{S}=-\mathcal{H}$ and if $\mathcal{S}$ is block diagonal with respect to the partition into A and B, we then deduce from

$$\begin{eqnarray}\mathcal{Q}={{\mathcal{S}}^{-1}} \left(-1\right)\mathcal{Q}\mathcal{S} = \left(\begin{array}{c} S_{A}^{-1} & 0 \\ 0 & S_{B}^{-1} \\ \end{array}\right)\left(\begin{array}{c} -{{Q}_{A}} & +2{{C}_{AB}} \\ +2{{C}_{BA}} & -{{Q}_{B}} \\ \end{array}\right)\left(\begin{array}{c} {{S}_{A}} & 0 \\ 0 & {{S}_{B}} \\ \end{array}\right)\end{eqnarray} \tag{ 5.31 }$$

that

$$\begin{eqnarray}&&S_{A}^{-1}\ {{Q}_{A}}\ {{S}_{A}}=-{{Q}_{A}},\quad S_{B}^{-1}\ {{Q}_{B}}\ {{S}_{B}}=-{{Q}_{B}},\end{eqnarray} \tag{ 5.32a }$$

and

$$\begin{eqnarray}&&S_{A}^{-1}\ {{C}_{AB}}\ {{S}_{B}}=-{{C}_{AB}},\quad S_{B}^{-1}\ {{C}_{BA}}\ {{S}_{A}}=-{{C}_{BA}}.\end{eqnarray} \tag{ 5.32b }$$

As was the case for the bulk Hamiltonian (5.1),

$$\begin{eqnarray}&&{{\mathcal{Q}}_{\text{edge}\ {{k}_{i}}\ {{A}_{i+1}}}}:={{k}_{i}}\ {{\rho}_{3}}\end{eqnarray} \tag{ 5.33 }$$

obeys two spectral chiral symmetries, for it anti-commutes with ρ₁ and ρ₂, and two spectral symmetries under charge conjugation, for it anti-commutes with the composition of k_i → −k_i with conjugation by ρ₀ or ρ₃.

The crossing at vanishing energy in equation (5.33) is not generically robust to the perturbations in equation (5.30). However, if we impose that equation (5.30) anti-commutes with either ρ₁ or ρ₂, then the entangling theory simplifies to the direct sum over blocks of the form

$$\begin{eqnarray}&&\mathcal{Q}_{\text{edge}\ {{k}_{i}}\ \mu \nu\ {{A}_{i+1}}}^{\text{AIII}}={{v}_{2}}\ {{\rho}_{2}}+\left({{k}_{i}}+{{v}_{3}}\right)\ {{\rho}_{3}}.\end{eqnarray} \tag{ 5.34 }$$

Here and without loss of generality, we have chosen to implement the chiral symmetry by demanding that $\mathcal{Q}_{\text{edge}\ {{k}_{i}}\ \mu \nu\ {{A}_{i+1}}}^{\text{AIII}}$ anti-commutes with ρ₁. As is, one perturbation in Hamiltonian (5.34) still gaps the unperturbed dispersion (5.33). However, if we impose the symmetry constraint

$$\begin{eqnarray}&&{{\rho}_{2}}{{\tilde{\mathcal{Q}}}_{\text{edge}\ {{k}_{i}}\ \mu \nu\ {{A}_{i+1}}\mathscr{S}{{\mathcal{R}}_{i}}}}\ {{\rho}_{2}}={{\tilde{\mathcal{Q}}}_{\text{edge}\ {{k}_{i}}\ \mu \nu\ {{A}_{i+1}}\mathscr{S}{{\mathcal{R}}_{i}}}}\end{eqnarray} \tag{ 5.35 }$$

to define ${{\tilde{\mathcal{Q}}}_{\text{edge}\ {{k}_{i}}\ \mu \nu\ {{A}_{i+1}}\mathscr{S}{{\mathcal{R}}_{i}}}}$, we find that

$$\begin{eqnarray}&&{{\tilde{\mathcal{Q}}}_{\text{edge}\ {{k}_{i}}\ \mu \nu\ {{A}_{i+1}}\mathscr{S}{{\mathcal{R}}_{i}}}}=\left({{k}_{i}}+{{v}_{3}}\right)\ {{\rho}_{3}}\end{eqnarray} \tag{ 5.36 }$$

displays a crossing at vanishing energy and momentum k_i = − v₃. In other words, if we impose, in addition to the chiral symmetry, the effective chiral symmetry generated by ρ₂, we find that no perturbation can gap the unperturbed dispersion (5.33), although it can move the momentum of the zero mode away from vanishing energy. As is implied by the notation, the effective chiral symmetry generated by ρ₂ originates from protecting the symmetry class AIII by demanding that reflection symmetry about the plane defining the entangling boundaries, i.e., the plane framed in red in figure 5, holds. According to section 3.2 the reflection about the plane framed in red in figure 5 induces a local effective chiral symmetry of the form given in equation (2.45), i.e., there exists a

$$\begin{eqnarray}&&{{\Gamma}_{{{\mathcal{R}}_{i}}}}:={{C}_{{{A}_{i+1}}{{B}_{i+1}}{{k}_{i}}}}{{\mathcal{R}}_{i}}\end{eqnarray} \tag{ 5.37 }$$

such that

$$\begin{eqnarray}&&\left\{{{\Gamma}_{{{\mathcal{R}}_{i}}}},{{\tilde{\mathcal{Q}}}_{\text{edge}\ {{k}_{i}}\ \mu \nu\ {{A}_{i+1}}\mathscr{S}{{\mathcal{R}}_{i}}}}\right\}=0.\end{eqnarray} \tag{ 5.38 }$$

The choice to represent ${{\Gamma}_{{{\mathcal{R}}_{i}}}}$ by ρ₂, given the choice to represent the chiral symmetry $\mathscr{S}$ by ρ₁, is a consequence of the following assumption and the following fact. First, we demand that $\mathscr{S}$ and ${{\mathcal{R}}_{i}}$ commute, i.e., we demand that

$$\begin{eqnarray}&&\left[\mathcal{S},{{\mathcal{R}}_{i}}\right]=0.\end{eqnarray} \tag{ 5.39 }$$

Second, the identity

$$\begin{eqnarray}\begin{array}{l} \left\{\mathcal{S},{{\mathcal{R}}_{i}}\right\} & = & \left\{\mathcal{S},{{C}_{{{A}_{i+1}}{{B}_{i+1}} {{k}_{i}}}}{{\mathcal{R}}_{i}}\right\} \\ {} & = & \left\{\mathcal{S},{{C}_{{{A}_{i+1}}{{B}_{i+1}} {{k}_{i}}}}\right\}{{\mathcal{R}}_{i}}-{{C}_{{{A}_{i+1}}{{B}_{i+1}} {{k}_{i}}}}\left[\mathcal{S},{{\mathcal{R}}_{i}}\right] \\ \end{array}\end{eqnarray} \tag{ 5.40 }$$

holds.

5.4.4. Spectra $\sigma \left({{\mathcal{H}}_{{{k}_{i}} \mathcal{R};\mathscr{S}}}\right)$ and $\sigma \left({{Q}_{{{k}_{i}} \mathcal{R};\mathscr{S}{{A}_{i+1}}}}\right)$.

It is time to present the reasoning that delivers table 5.41. Let i = 1, 2 and define i + 1 modulo 2. We assume that the underlying microscopic model has the doublet of symmetries $\left(\mathcal{R};\mathscr{S}\right)\sim \left({{\mathcal{R}}_{\mu \nu}},{{\mathcal{S}}_{\mu \nu}}\right)$ where $\mathscr{S}\sim {{\mathcal{S}}_{\mu \nu}}$ defines the symmetry class AIII. We denote the most general perturbation that is compliant with the doublet of symmetries $\left(\mathcal{R};\mathscr{S}\right)$ defining a given row of table 2 by ${{\mathcal{V}}_{\mathcal{R};\mathscr{S}}}$. The explicit form of this perturbation is to be found in the second column of table 5.41 as one varies $\left(\mathcal{R};\mathscr{S}\right)$. For a given row in table 5.41, ${{\mathcal{V}}_{\mathcal{R};\mathscr{S}}}$ is contained in the most general perturbation ${{\mathcal{V}}_{\mathscr{S}}}$ that is compliant with the AIII symmetry. We define the single-particle Hamiltonian ${{\mathcal{H}}_{i\mathcal{R};\mathscr{S}}}\equiv {{\mathcal{H}}_{i}}+{{\mathcal{V}}_{\mathcal{R};\mathscr{S}}}$ by its matrix elements

$$\begin{eqnarray}&&{{\mathcal{H}}_{{{k}_{i}} \mathcal{R};\mathscr{S}{{r}_{i+1}},r_{i+1}^{\prime}}}:={{\mathcal{H}}_{{{k}_{i}} {{r}_{i+1}},r_{i+1}^{\prime}}}+{{\mathcal{V}}_{\mathcal{R};\mathscr{S}{{r}_{i+1}},r_{i+1}^{\prime}}},\end{eqnarray} \tag{ 5.41a }$$

$$\begin{eqnarray}&&{{\mathcal{V}}_{\mathcal{R};\mathscr{S}{{r}_{i+1}},r_{i+1}^{\prime}}}:={{\delta}_{{{r}_{i+1}},r_{i+1}^{\prime}}}\underset{\mu ,\nu \in \text{row}}{\sum} {{v}_{\mu \nu}}~{{\mathcal{X}}_{\mu \nu}}.\end{eqnarray} \tag{ 5.41b }$$

The corresponding equal-time one-point correlation matrix is ${{\mathcal{Q}}_{\mathcal{R};\mathscr{S}\ \left(i+1\right)}}$ and its upper-left block is ${{Q}_{\mathcal{R};\mathscr{S} {{A}_{i+1}}}}$. The third column in table 5.41 provides one sign for each row. The sign ${{\eta}_{\mathscr{S}}}$ is positive if $\mathcal{R}$ commutes with $\mathscr{S}$ and negative if $\mathcal{R}$ anti-commutes with $\mathscr{S}$. The information contained in ${{\eta}_{\mathscr{S}}}$ is needed to read from table VI of [16] the bulk topological index of the single-particle Hamiltonian (5.41). This topological index does not guarantee that ${{\mathcal{H}}_{\mathcal{R};\mathscr{S}}}$ supports boundary states in an open geometry. On the other hand, ${{Q}_{\mathcal{R};\mathscr{S} {{A}_{i+1}}}}$ supports boundary states on the entangling boundaries. The entries ○ for the columns $\sigma \left({{H}_{{{k}_{i}} \mathcal{R};\mathscr{S}}}\right)$ and $\sigma \left({{Q}_{{{k}_{i}} \mathcal{R};\mathscr{S}{{A}_{i+1}}}}\right)$ are a consequence of our stability analysis. On the one hand, table 5.41 demonstrates explicitly that the presence of a reflection symmetry in addition to the chiral symmetry does not guarantee that a non-vanishing bulk topological index implies protected edge states in the spectrum $\sigma \left({{H}_{{{k}_{i}} \mathcal{R};\mathscr{S}}}\right)$ for both edges i = 1, 2. On the other hand, table 5.41 demonstrates explicitly that a non-vanishing bulk topological index always implies protected entangling states in the spectrum $\sigma \left({{Q}_{{{k}_{i}} \mathcal{R};\mathscr{S}\ {{A}_{i+1}}}}\right)$ irrespective of the choice i = 1, 2 made for the entangling boundary.

We are going to study the dependences of the spectra $\sigma \left({{\mathcal{H}}_{{{k}_{i}} \mathcal{R};\mathscr{S}}}\right)$ and $\sigma \left({{Q}_{{{k}_{i}} \mathcal{R};\mathscr{S}\ {{A}_{i+1}}}}\right)$ (with i + 1 defined modulo 2) on the system sizes for the single-particle Hamiltonian ${{\mathcal{H}}_{{{k}_{i}} \mathcal{R};\mathscr{S}}}$ defined by the matrix elements (5.41) and the corresponding upper-left block ${{Q}_{{{k}_{i}} \mathcal{R};\mathscr{S}\ {{A}_{i+1}}}}$ from the equal-time one-point correlation matrix. Without loss of generality, we will present numerical results obtained by choosing the fifth row in table 5.41 to define ${{\mathcal{H}}_{{{k}_{i}} {{\mathcal{R}}_{2}};\mathscr{S}}}$ and ${{Q}_{{{k}_{i}} {{\mathcal{R}}_{2}};\mathscr{S}\ {{A}_{i+1}}}}$ with (v₀₂, v₀₃, v₁₂, v₁₃) = (0.2, −0.1, 0.05, 0.3).

The question we want to address is the following. On the one hand, figures 5(a)–(d) suggest that the mid-gap branches merge into the continuum when π/2 < |k_i| < π. On the other hand, the mid-gap branches in figure 5(e) seem to be separated by a very small gap from all other bands in the vicinity of |k₁| = π, while no gap is resolved in energy between the mid-gap branches and all other branches in the vicinity of |k₁| = π for figures 5(f)–(h).

The first question we are going to address is how to interpret the 'peeling of' the mid-gap branches from the continua in figures 5(a)–(d). To this end, we present in figure 5(a) the spectrum $\sigma \left({{\mathcal{H}}_{{{k}_{1}}{{\mathcal{R}}_{2}};\mathscr{S}}}\right)$ obtained by imposing periodic boundary conditions along the direction i = 1 and open boundary conditions along the direction i + 1 = 2 for the linear system sizes M₁ = 128 and M₂ = 64. We then compute the spectrum of ${{\mathcal{H}}_{{{k}_{1}}{{\mathcal{R}}_{2}};\mathscr{S}}}$ for each value M₂ = 16, 32, 48, 64, 80, 96 of the cylinder height. At last, we compute the minimal value Δ_min(π/2, M₂) > 0 taken by the difference in energy between the positive energy eigenvalues at k₁ = π/2 from any one of the bulk bands and the mid-gap branch with positive energy eigenvalue at k₁ = π/2. The dependence of this direct gap Δ_min(π/2, M₂) > 0 on M₂ = 16, 32, 48, 64, 80, 96 holding M₁ = 128 fixed is plotted in figure 5(b). The fast decrease of Δ_min(π/2, M₂) > 0 with increasing values of M₂ = 16, 32, 48, 64, 80, 96 holding M₁ = 128 fixed is interpreted as the merging of the mid-gap branch into the continuum of conduction bulk states in the quasi-one-dimensional limit M₂ → ∞ holding M₁ = 128 and π/2 ⩽̸ k₁ ⩽̸ π fixed. Similarly, one may verify that the mid-gap branches merge into the valence and conduction bulk continua in figures 5(a)–(d) for a non-vanishing interval of momenta k_i in the quasi-one-dimensional limit M_i + 1 → ∞ holding M_i fixed. If we change the conserved momentum k_i adiabatically, say by imposing twisted boundary conditions instead of periodic ones along the i direction, a charge can be transferred from the valence bulk continuum to the conduction bulk continuum through the mid-gap branches. This is an example of spectral flow induced by mid-gap states crossing a bulk gap.

The same quasi-one-dimensional scaling analysis, when performed on the spectra $\sigma \left({{\mathcal{H}}_{{{k}_{2}}{{\mathcal{R}}_{2}};\mathscr{S}}}\right)$ and $\sigma \left({{Q}_{{{k}_{1}}{{\mathcal{R}}_{2}};\mathscr{S}\ {{A}_{2}}}}\right)$, delivers a very different result. First, the mid-gap branches of ${{\mathcal{H}}_{{{k}_{2}}{{\mathcal{R}}_{2}};\mathscr{S}}}$ fail to cross. Second, the minimal value Δ_min(π, M₂) > 0 taken by the difference between the positive eigenvalues at k₁ = π from any one of the bulk bands and the mid-gap branch with positive energy eigenvalues at k₁ = π converge to a non-vanishing value upon increasing M₂ = 16, 32, 48, 64, 80, 96 holding M₁ = 128 fixed in figure 5(c). Instead of plotting the evidence for the saturation value lim_M2 → ∞Δ_min(π, M₂) > 0, we plot in figure 5(d) the entanglement spectrum

$$\begin{eqnarray}&&{{\varpi}_{{{k}_{1}}}}:= \ln \left(\frac{1}{{{\zeta}_{{{k}_{1}}}}}-1\right),\end{eqnarray} \tag{ 5.42 }$$

which we already encountered in equation (2.13).

We now reproduce figures 5(c) and (d) for M₁ = 64 and M₂ = 128 with the single change that we choose the geometry in which it is k₂ instead of k₁ that is chosen to be the good quantum number. There follow figures 5(e) and (f ) for $\sigma \left({{Q}_{{{k}_{2}}{{\mathcal{R}}_{2}};\mathscr{S} {{A}_{1}}}}\right)$ and $\sigma \left(\ln \left(Q_{{{k}_{2}}{{\mathcal{R}}_{2}};\mathscr{S}\ {{A}_{1}}}^{-1}-1\right)\right)$, respectively. We have verified that, if we increase simultaneously M₁ and M₂, there is a level crossing between the mid-gap branches and the bulk branches in the neighborhood of ± π. The existence of this level crossing is implied by the coloring of the bands in figure 5(f ). This coloring has the following origin.

For any row from table 5.41, we may construct the operator

$$\begin{eqnarray}&&{{\Lambda}_{{{k}_{i}} \mathcal{R}\ {{A}_{i+1}}}}:={{\Gamma}_{{{k}_{i}} \mathcal{R}}}{{S}_{{{A}_{i+1}}}}\end{eqnarray} \tag{ 5.43 }$$

whenever the reflection $\mathcal{R}$ is block off-diagonal with respect to the partition. Here, ${{\Gamma}_{{{k}_{i}} \mathcal{R}}}$ was defined in equation (2.45), while the representation $\mathcal{S}$ of the chiral operation $\mathscr{S}$ is diagonal in the partition with the upper-left block ${{S}_{{{A}_{i+1}}}}$.

For the fifth row from table 5.41, ${{\Gamma}_{{{k}_{1}}{{\mathcal{R}}_{2}}}}$ does not exist in the geometry that defines ${{Q}_{{{k}_{1}}{{\mathcal{R}}_{2}};\mathscr{S}\ {{A}_{2}}}}$, since ${{\mathcal{R}}_{2}}$ is block diagonal for this partition. For the fifth row from table 5.41, ${{\Gamma}_{{{k}_{2}}{{\mathcal{R}}_{2}}}}$ exists in the geometry that defines ${{Q}_{{{k}_{2}}{{\mathcal{R}}_{2}};\mathscr{S}\ {{A}_{1}}}}$ and anti-commutes with ${{Q}_{{{k}_{2}}{{\mathcal{R}}_{2}};\mathscr{S}\ {{A}_{1}}}}$ according to equation (2.52), since ${{\mathcal{R}}_{2}}$ is block off-diagonal for this partition. Hence, ${{\Lambda}_{{{k}_{2}}{{\mathcal{R}}_{2}}\ {{A}_{1}}}}$ commutes with ${{Q}_{{{k}_{2}}{{\mathcal{R}}_{2}};\mathscr{S}\ {{A}_{1}}}}$ and we may associate to any eigenstate of ${{Q}_{{{k}_{2}}{{\mathcal{R}}_{2}};\mathscr{S}\ {{A}_{1}}}}$ an eigenvalue of ${{\Lambda}_{{{k}_{2}}{{\mathcal{R}}_{2}}\ {{A}_{1}}}}$. This eigenvalue of ${{\Lambda}_{{{k}_{2}}{{\mathcal{R}}_{2}} {{A}_{1}}}}$ is complex valued as ${{\Lambda}_{{{k}_{2}}{{\mathcal{R}}_{2}}\ {{A}_{1}}}}$ is not necessarily Hermitian. It turns out that ${{\Lambda}_{{{k}_{2}}{{\mathcal{R}}_{2}}\ {{A}_{1}}}}$ is purely imaginary according to our exact diagonalizations. The coloring in figure 5(f) corresponds to the sign of the imaginary eigenvalue of ${{\Lambda}_{{{k}_{2}}{{\mathcal{R}}_{2}}\ {{A}_{1}}}}$. This quantum number rules out an avoided level crossing of the mid-gap branches in the neighborhood of k₂ = 0 in figure 5(f). This quantum number also rules out an avoided level crossing at the zone boundary k₂ = π. Inspection of the colored dispersions in figure 5(f) suggests that twisting boundary conditions induces a spectral flow from the valence to the conduction bands through the mid-gap branches. This is the basis of the distinction between crossings of mid-gap branches that are compatible with a spectral flow and are denoted by ⊙ in table 5.41 and crossing of mid-gap branches that are not compatible with a spectral flow and are denoted by ⊗ in table 5.41.