Detection of Chern numbers and entanglement in topological two-species systems through subsystem winding numbers

To introduce the basic concepts of our argument and to motivate for overcoming the problems which arise when the system grows in complexity, we start by briefly reviewing the detection of Chern numbers from time-of-flight images. Let us consider a translationally invariant non-interacting system of fermions in two spatial dimensions described by the Hamiltonian $H={{\int }_{{\boldsymbol{p}} \in BZ}}{{{\rm d}}^{2}}p\;{\boldsymbol{ \psi }} _{{\boldsymbol{p}} }^{\dagger }h({\boldsymbol{p}} ){{{\boldsymbol{ \psi }} }_{{\boldsymbol{p}} }}$, where BZ denotes the Brillouin zone. We assume that the system has two components, i.e. that the Hamiltonian is given in the basis ${{{\boldsymbol{ \psi }} }_{{\boldsymbol{p}} }}={{({{a}_{{\boldsymbol{p}} }},{{b}_{{\boldsymbol{p}} }})}^{T}}$ for some fermionic operators ${{a}_{{\boldsymbol{p}} }}$ and ${{b}_{{\boldsymbol{p}} }}$. The ground state of the system is given by $|\Psi \rangle ={{\prod }_{{\boldsymbol{p}} }}|{{\psi }_{{\boldsymbol{p}} }}\rangle$ which is defined in terms of the fermionic operators $a_{{\boldsymbol{p}} }^{\dagger }$ and $b_{{\boldsymbol{p}} }^{\dagger }$ acting on the vacuum, $|00\rangle$. Furthermore, we require that the system has an energy gap. The kernel Hamiltonian $h({\boldsymbol{p}} )$ has two energy bands ${{E}_{1}}({\boldsymbol{p}} )\leqslant 0$ and ${{E}_{2}}({\boldsymbol{p}} )\geqslant 0$ such that if ${{{\rm min} }_{{\boldsymbol{p}} }}|{{E}_{2}}({\boldsymbol{p}} )|-{{{\rm max} }_{{\boldsymbol{p}} }}|{{E}_{1}}({\boldsymbol{p}} )|\ne 0$ then the system has a gap. The Chern number characterizing the ground state of such a system is formally defined as

$$\begin{eqnarray}&&\nu =-\frac{{\rm i}}{2\pi }{{\int }_{BZ}}{{{\rm d}}^{2}}p\;\;{\rm tr}\left( {{P}_{{\boldsymbol{p}} }}\left[ {{\partial }_{{{p}_{x}}}}{{P}_{{\boldsymbol{p}} }},{{\partial }_{{{p}_{y}}}}{{P}_{{\boldsymbol{p}} }} \right] \right),\end{eqnarray} \tag{ 1 }$$

where ${{P}_{{\boldsymbol{p}} }}=|\psi ({\boldsymbol{p}} )\rangle \langle \psi ({\boldsymbol{p}} )|$ is the projector onto the ground state of $h({\boldsymbol{p}} )$, $|\psi ({\boldsymbol{p}} )\rangle$. Depending on how the projector is represented (see appendix A for detailed derivations), the Chern number can equivalently be expressed as the Berry phase accrued around the boundary $\partial BZ$ of the two dimensional Brillouin zone

$$\begin{eqnarray}&&\nu =-\frac{{\rm i}}{2\pi }{{\oint }_{\partial BZ}}\langle \psi ({\boldsymbol{p}} )|\partial |\psi ({\boldsymbol{p}} )\rangle \cdot {\rm d}{\boldsymbol{p}} ,\end{eqnarray} \tag{ 2 }$$

or as the winding number

$$\begin{eqnarray}&&\nu =\frac{1}{4\pi }{{\int }_{BZ}}{{{\rm d}}^{2}}p\;{\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )\cdot \left( {{\partial }_{{{p}_{x}}}}{\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )\times {{\partial }_{{{p}_{y}}}}{\boldsymbol{\hat{s}}} ({\boldsymbol{p}} ) \right),\end{eqnarray} \tag{ 3 }$$

that counts how many times the three-vector ${\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )$ winds around the unit sphere as the Brillouin zone is spanned. This normalized vector parametrizes the Hamiltonian through

$$\begin{eqnarray}&&h({\boldsymbol{p}} )=|{\boldsymbol{s}} |{\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )\cdot {\boldsymbol{ \sigma }} ,\end{eqnarray} \tag{ 4 }$$

where ${\boldsymbol{ \sigma }} =({{\sigma }^{x}},{{\sigma }^{y}},{{\sigma }^{z}})$ is the vector of Pauli matrices. It was shown in [22] for Chern insulators, where ${{a}_{{\boldsymbol{p}} }}$ and ${{b}_{{\boldsymbol{p}} }}$ denote the two sublattices of the Haldane model on a honeycomb lattice, and in [21] for topological superconductors, where ${{b}_{{\boldsymbol{p}} }}=a_{-{\boldsymbol{p}} }^{\dagger }$, that the vector ${\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )$ can be constructed from physical observables associated with time-of-flight measurements. As the representation (3) of the Chern number is fully given by ${\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )$, this means that the Chern number can be constructed from such measurements.

To be more precise, by studying how the atom cloud expands as the trap is switched off, one obtains a set of time-of-flight images that amount to measuring density operators of the form $a_{{\boldsymbol{p}} }^{\dagger }{{a}_{{\boldsymbol{p}} }}$ and $b_{{\boldsymbol{p}} }^{\dagger }{{b}_{{\boldsymbol{p}} }}$. As these correspond to different species, one can assume that they can be measured independently by releasing only one species from the trap at a time. By direct evaluation one then finds that ${\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )$ can be written either as the expectation value of the ground state of the Bloch Hamiltonian, ${{h}_{{\boldsymbol{p}} }}$, with the Pauli matrices or as the expectation value of the ground state of the full Hamiltonian, H, with the Pauli operators in the second quantized representation

$$\begin{eqnarray}&&{\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )=\langle \psi ({\boldsymbol{p}} )|{\boldsymbol{ \sigma }} |\psi ({\boldsymbol{p}} )\rangle =\langle {{\psi }_{{\boldsymbol{p}} }}|{\boldsymbol{ \psi }} _{{\boldsymbol{p}} }^{\dagger }{\boldsymbol{ \sigma }} {{{\boldsymbol{ \psi }} }_{{\boldsymbol{p}} }}|{{\psi }_{{\boldsymbol{p}} }}\rangle .\end{eqnarray} \tag{ 5 }$$

These two representations are equivalent. The s_z component is thus given directly by the time-of-flight images, while the other components can be obtained by suitable Hamiltonian manipulations [21, 22].

This simple relation between physical observables and the Chern number breaks down when the number of degrees of freedom is increased, e.g. through the introduction of spin, staggering or more internal atomic states. The Bloch Hamiltonian, $h({\boldsymbol{p}} )$, is no longer a 2 × 2 matrix and can thus no longer be expanded in the Pauli basis $\{{{\sigma }^{x}},{{\sigma }^{y}},{{\sigma }^{z}}\}$. The Hamiltonian can still be expanded in a basis of some higher dimensional matrices and parametrized by a vector ${\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )$, but this vector will now live in a space of dimension larger than three [5]. It follows that the formal definition of the Chern number (1) no longer reduces to a simple winding number (3). While the components of the higher dimensional ${\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )$ could still be in principle obtained via time-of-flight images following Hamiltonian manipulations, there is no longer a recipe for how the Chern number could be constructed from them.

Our main result is to show that the Chern number can be decomposed into component dependent contributions, which can be evaluated as physically observable winding numbers 3, as in the simple example described above. Our construction generalizes the concept of spin Chern number to general pseudospin degrees of freedom. We demonstrate our method by applying it to spin Hall insulators [12] and a staggered topological superconductor [21]. In each case the phase diagrams are accurately reproduced given that the subsystem components are not highly entangled. This can be diagnosed using the same operators as the ones used to construct the subsystem winding numbers, which means that our method enables us to probe entanglement between different degrees of freedom.

Let us consider a system with four distinct types of fermion, whose annihilation operators we denote by a₁, a₂, b₁ and b₂, where the a priori bipartition between a and b is physically motivated (such as different spin orientations, atomic internal levels or different sectors of some discrete symmetry). Assuming translational invariance with respect to these operators the Hamiltonian can always be given in momentum space as

$$\begin{eqnarray}&&H={{\int }_{{\boldsymbol{p}} \in BZ}}{{{\rm d}}^{2}}p\;\Psi _{{\boldsymbol{p}} }^{\dagger }h({\boldsymbol{p}} ){{\Psi }_{{\boldsymbol{p}} }}\qquad {\rm with}\qquad {{\Psi }_{{\boldsymbol{p}} }}=\left( \begin{array}{ccccccccccccccc} {{a}_{1,{\boldsymbol{p}} }} \\ {{a}_{2,{\boldsymbol{p}} }} \\ {{b}_{1,{\boldsymbol{p}} }} \\ {{b}_{2,{\boldsymbol{p}} }} \\ \end{array} \right).\end{eqnarray} \tag{ 6 }$$

A general state in the Hilbert space of the system can be written as

$$\begin{eqnarray}&&|\Psi \rangle =\mathop{\prod }\limits_{{\boldsymbol{p}} }\left( \mathop{\sum }\limits_{n_{1,{\boldsymbol{p}} }^{a},n_{2,{\boldsymbol{p}} }^{a},n_{1,{\boldsymbol{p}} }^{b},n_{2,{\boldsymbol{p}} }^{b}=0,1}{{\alpha }_{n_{1,{\boldsymbol{p}} }^{a},n_{2,{\boldsymbol{p}} }^{a},n_{1,{\boldsymbol{p}} }^{b},n_{2,{\boldsymbol{p}} }^{b}}}|n_{1,{\boldsymbol{p}} }^{a},n_{2,{\boldsymbol{p}} }^{a},n_{1,{\boldsymbol{p}} }^{b},n_{2,{\boldsymbol{p}} }^{b}\rangle \right)\equiv \mathop{\prod }\limits_{{\boldsymbol{p}} }|{{\psi }_{{\boldsymbol{p}} }}\rangle ,\end{eqnarray} \tag{ 7 }$$

where we have expressed it in the occupation basis

$$\begin{eqnarray}&&\left| n_{1,{\boldsymbol{p}} }^{a},n_{2,{\boldsymbol{p}} }^{a},n_{1,{\boldsymbol{p}} }^{b},n_{2,{\boldsymbol{p}} }^{b} \right\rangle ={{(a_{1,{\boldsymbol{p}} }^{\dagger })}^{n_{1,{\boldsymbol{p}} }^{a}}}{{(a_{2,{\boldsymbol{p}} }^{\dagger })}^{n_{2,{\boldsymbol{p}} }^{a}}}{{(b_{1,{\boldsymbol{p}} }^{\dagger })}^{n_{1,{\boldsymbol{p}} }^{b}}}{{(b_{2,{\boldsymbol{p}} }^{\dagger })}^{n_{2,{\boldsymbol{p}} }^{b}}}\left| 0000 \right\rangle .\end{eqnarray} \tag{ 8 }$$

Here $n_{{\rm i},{\boldsymbol{p}} }^{a,b}=0,1$ are the fermionic occupation numbers and $\left| 0000 \right\rangle$ corresponds to the vacuum state of all fermionic modes. Eigenstates of hamiltonian (6) will be of this form for some set of coefficients ${{\alpha }_{n_{1,{\boldsymbol{p}} }^{a},n_{2,{\boldsymbol{p}} }^{a},n_{1,{\boldsymbol{p}} }^{b}n_{2,{\boldsymbol{p}} }^{b}}}$ that satisfy ${{\sum }_{n_{1,{\boldsymbol{p}} }^{a},n_{2,{\boldsymbol{p}} }^{a},n_{1,{\boldsymbol{p}} }^{b}n_{2,{\boldsymbol{p}} }^{b}=0,1}}|{{\alpha }_{n_{1,{\boldsymbol{p}} }^{a},n_{2,{\boldsymbol{p}} }^{a},n_{1,{\boldsymbol{p}} }^{b}n_{2,{\boldsymbol{p}} }^{b}}}{{|}^{2}}=1$.

Let us now assume that the system conserves particle number, i.e. $[H,N]=0$, where $N={{\sum }_{{\boldsymbol{p}} ,\alpha =1,2}}(a_{\alpha ,{\boldsymbol{p}} }^{\dagger }{{a}_{\alpha ,{\boldsymbol{p}} }}+b_{\alpha ,{\boldsymbol{p}} }^{\dagger }{{b}_{\alpha ,{\boldsymbol{p}} }})$. At half filling the ground state $|{{\psi }_{{\boldsymbol{p}} }}\rangle$ satisfies the condition ${{\sum }_{\alpha =1,2}}(n_{\alpha ,{\boldsymbol{p}} }^{a}+n_{\alpha ,{\boldsymbol{p}} }^{b})=2$. This means that a complete local basis for each momentum component of the ground state is given by

$$\begin{eqnarray}&&\left\{ \left| 1100 \right\rangle ,\left| 1010 \right\rangle ,\left| 1001 \right\rangle ,\left| 0110 \right\rangle ,\left| 0101 \right\rangle ,\left| 0011 \right\rangle \right\}.\end{eqnarray} \tag{ 9 }$$

We can divide the state into two orthogonal subspaces

$$\begin{eqnarray}\begin{array}{rcl} |{{\psi }_{{\boldsymbol{p}} }}\rangle & = & A|\psi (n_{1,{\boldsymbol{p}} }^{a}+n_{2,{\boldsymbol{p}} }^{a}={\rm even};n_{1,{\boldsymbol{p}} }^{b}+n_{2,{\boldsymbol{p}} }^{b}={\rm even})\rangle \\ {} & {} & +B|\psi (n_{1,{\boldsymbol{p}} }^{a}+n_{2,{\boldsymbol{p}} }^{a}={\rm odd};n_{1,{\boldsymbol{p}} }^{b}+n_{2,{\boldsymbol{p}} }^{b}={\rm odd})\rangle \\ {} & \equiv & A\left| \psi (e;e) \right\rangle +B\left| \psi (o;o) \right\rangle , \\ \end{array}\end{eqnarray} \tag{ 10 }$$

where the populations $n_{1,{\boldsymbol{p}} }^{a}+n_{2,{\boldsymbol{p}} }^{a}$ and $n_{1,{\boldsymbol{p}} }^{b}+n_{2,{\boldsymbol{p}} }^{b}$ are either both even or both odd for some A and B, with $|A{{|}^{2}}+|B{{|}^{2}}=1$. This partitioning of the state in this particular way facilitates our derivation.

We now perform a Schmidt decomposition on each part of the state, the even and the odd. We can write

$$\begin{eqnarray}\begin{array}{rcl} |\psi (e;e)\rangle & = & {\rm cos} {{\theta }_{e}}|{{a}_{e}}\rangle \otimes |{{b}_{e}}\rangle +{\rm sin} {{\theta }_{e}}|{{\tilde{a}}_{e}}\rangle \otimes |{{\tilde{b}}_{e}}\rangle , \\ |\psi (o;o)\rangle & = & {\rm cos} {{\theta }_{o}}|{{a}_{o}}\rangle \otimes |{{b}_{o}}\rangle +{\rm sin} {{\theta }_{o}}|{{\tilde{a}}_{o}}\rangle \otimes |{{\tilde{b}}_{o}}\rangle , \\ \end{array}\end{eqnarray} \tag{ 11 }$$

where ${{\theta }_{e}},{{\theta }_{o}}\in [0,\pi /2)$ such that all the Schmidt coefficients are non-negative. It is understood that the $\left| {{a}_{e/o}} \right\rangle \left| {{b}_{e/o}} \right\rangle$ states written in the population basis have the fermionic operators ordered as in (8). The vectors $|{{\tilde{a}}_{e/o}}\rangle |{{\tilde{b}}_{e/o}}\rangle$ obey the orthogonality conditions $\langle {{a}_{e/o}}|{{\tilde{a}}_{e/o}}\rangle =0$ and $\langle {{b}_{e/o}}|{{\tilde{b}}_{e/o}}\rangle =0$. More explicitly we have

$$\begin{eqnarray}\begin{array}{rcl} |{{a}_{o}}\rangle & = & ({{\alpha }_{01}}a_{2,{\boldsymbol{p}} }^{\dagger }+{{\alpha }_{10}}a_{1,{\boldsymbol{p}} }^{\dagger })|00\rangle ,\quad |{{\tilde{a}}_{o}}\rangle =(\alpha _{10}^{*}a_{2,{\boldsymbol{p}} }^{\dagger }-\alpha _{01}^{*}a_{1,{\boldsymbol{p}} }^{\dagger })|00\rangle , \\ |{{b}_{o}}\rangle & = & ({{\beta }_{01}}b_{2,{\boldsymbol{p}} }^{\dagger }+{{\beta }_{10}}b_{1,{\boldsymbol{p}} }^{\dagger })|00\rangle ,\quad |{{\tilde{b}}_{o}}\rangle =(\beta _{10}^{*}b_{2,{\boldsymbol{p}} }^{\dagger }-\beta _{01}^{*}b_{1,{\boldsymbol{p}} }^{\dagger })|00\rangle \\ \end{array}\end{eqnarray} \tag{ 12 }$$

and

$$\begin{eqnarray}\begin{array}{rcl} |{{a}_{e}}\rangle & = & {{{\rm e}}^{{\rm i}{{\phi }_{a}}}}|00\rangle ,\quad |{{\tilde{a}}_{e}}\rangle ={{{\rm e}}^{{\rm i}{{\tilde{\phi }}_{a}}}}a_{1,{\boldsymbol{p}} }^{\dagger }a_{2,{\boldsymbol{p}} }^{\dagger }|00\rangle , \\ |{{b}_{e}}\rangle & = & {{{\rm e}}^{{\rm i}{{\phi }_{b}}}}b_{1,{\boldsymbol{p}} }^{\dagger }b_{2,{\boldsymbol{p}} }^{\dagger }|00\rangle ,\quad |{{\tilde{b}}_{e}}\rangle ={{{\rm e}}^{{\rm i}{{\tilde{\phi }}_{b}}}}|00\rangle , \\ \end{array}\end{eqnarray} \tag{ 13 }$$

where $|{{\alpha }_{01}}{{|}^{2}}+|{{\alpha }_{10}}{{|}^{2}}=|{{\beta }_{01}}{{|}^{2}}+|{{\beta }_{10}}{{|}^{2}}=1$. The phases ${{\phi }_{a/b}}$ and ${{\tilde{\phi }}_{a/b}}$ are in general non-zero. However, after multiplying $|{{\psi }_{{\boldsymbol{p}} }}\rangle$ by a global phase of ${{{\rm e}}^{-{\rm i}({{\phi }_{a}}+{{\phi }_{b}})}}$, we can transfer them to the odd-odd subspace through a $U(1)$ gauge transformation $a_{1,{\boldsymbol{p}} }^{\dagger }\to {{{\rm e}}^{-{\rm i}({{\tilde{\phi }}_{a}}+{{\tilde{\phi }}_{b}}-{{\phi }_{a}}-{{\phi }_{b}})}}a_{1,{\boldsymbol{p}} }^{\dagger }$. Then the only momentum dependence in the Schmidt decomposed even-even subspace is in the real and positive Schmidt coefficients, ${\rm cos} {{\theta }_{e}}$ and ${\rm sin} {{\theta }_{e}}$.

We now show that we can write the Chern number of the system as a sum of the Berry phases accrued by each subsystem. To this end, we evaluate the Chern number of the ground state of the system, (10), as the Berry phase (2). Without loss of generality we can take A and B to be real and non-negative. This is achieved by absorbing possible complex phases into the states $|{{a}_{e/o}}\rangle |{{b}_{e/o}}\rangle$ and $|{{\tilde{a}}_{e/o}}\rangle |{{\tilde{b}}_{e/o}}\rangle$. Employing relation (10) we can write the Chern number as

$$\begin{eqnarray}&&\nu =-\frac{{\rm i}}{2\pi }{{\oint }_{\partial BZ}}{{A}^{2}}\langle \psi (e;e)|\partial |\psi (e;e)\rangle \cdot {\rm d}{\boldsymbol{p}} -\frac{{\rm i}}{2\pi }{{\oint }_{\partial BZ}}{{B}^{2}}\langle \psi (o;o)|\partial |\psi (o;o)\rangle \cdot {\rm d}{\boldsymbol{p}} ,\end{eqnarray} \tag{ 14 }$$

where terms of the form $A\partial A$ do not contribute as, due to the reality condition on A and B, $A\partial A+B\partial B=\partial ({{A}^{2}}+{{B}^{2}})/2=0$. Direct evaluation of the integrand in the even-even case finds it to be zero due to ${\rm cos} {{\theta }_{e}}\partial {\rm cos} {{\theta }_{e}}+{\rm sin} {{\theta }_{e}}\partial {\rm sin} {{\theta }_{e}}=\partial ({{{\rm cos} }^{2}}{{\theta }_{e}}+{{{\rm sin} }^{2}}{{\theta }_{e}})/2=0$. Thus the first term on the right hand side of 14 does not contribute to the Berry phase. Noting that $\langle {{i}_{o}}|\partial |{{i}_{o}}\rangle =-\langle {{\tilde{i}}_{o}}|\partial |{{\tilde{i}}_{o}}\rangle$ in (12) and using the positivity and normalization of the Schmidt coefficients, a direct evaluation gives

$$\begin{eqnarray}&&\nu =-\frac{{\rm i}}{2\pi }{{\sum }_{i=a,b}}{{\oint }_{\partial BZ}}S\langle {{{\rm i}}_{o}}|\partial |{{{\rm i}}_{o}}\rangle \cdot {\rm d}{\boldsymbol{p}} ,\qquad S=|B{{|}^{2}}T,\end{eqnarray} \tag{ 15 }$$

where $T={{{\rm cos} }^{2}}{{\theta }_{o}}-{{{\rm sin} }^{2}}{{\theta }_{o}}$ is a measure of entanglement between the subsystems.

Thus we have succeeded in decomposing the Chern number into a sum of exclusive contributions from the a or b subsystems. Due to appearance of the function S, these contributions are not Berry phases which one could equivalently evaluate as winding numbers of vectors ${{{\boldsymbol{\hat{s}}} }_{a}}({\boldsymbol{p}} )$ and ${{{\boldsymbol{\hat{s}}} }_{b}}({\boldsymbol{p}} )$. However, the decomposition holds for any $S\ne 0$. For S = 1 the two subsystems are unentangled and the Chern number can be directly written as a sum of Berry phase contributions. But the decomposition also holds for $S\ne 1$ and fails only when $S\to 0$ where the system is maximally entangled. This can be seen in the examples of section 4 where our method diverges from the theoretical predictions only when the system approaches maximal entanglement.

In direct analogy with the single component case presented in the introduction, we define the observables for the a and b subsystems as

$$\begin{eqnarray}\begin{array}{rcl} \Sigma _{a}^{x} & = & a_{1,{\boldsymbol{p}} }^{\dagger }{{a}_{2,{\boldsymbol{p}} }}+a_{2,{\boldsymbol{p}} }^{\dagger }{{a}_{1,{\boldsymbol{p}} }},\Sigma _{a}^{y}=-{\rm i}a_{1,{\boldsymbol{p}} }^{\dagger }{{a}_{2,{\boldsymbol{p}} }}+{\rm i}a_{2,{\boldsymbol{p}} }^{\dagger }{{a}_{1,{\boldsymbol{p}} }},\Sigma _{a}^{z}=a_{1,{\boldsymbol{p}} }^{\dagger }{{a}_{1,{\boldsymbol{p}} }}-a_{2,{\boldsymbol{p}} }^{\dagger }{{a}_{2,{\boldsymbol{p}} }} \\ \Sigma _{b}^{x} & = & b_{1,{\boldsymbol{p}} }^{\dagger }{{b}_{2,{\boldsymbol{p}} }}+b_{2,{\boldsymbol{p}} }^{\dagger }{{b}_{1,{\boldsymbol{p}} }},\Sigma _{b}^{y}=-{\rm i}b_{1,{\boldsymbol{p}} }^{\dagger }{{b}_{2,{\boldsymbol{p}} }}+{\rm i}b_{2,{\boldsymbol{p}} }^{\dagger }{{b}_{1,{\boldsymbol{p}} }},\Sigma _{b}^{z}=b_{1,{\boldsymbol{p}} }^{\dagger }{{b}_{1,{\boldsymbol{p}} }}-b_{2,{\boldsymbol{p}} }^{\dagger }{{b}_{2,{\boldsymbol{p}} }}. \\ \end{array}\end{eqnarray} \tag{ 16 }$$

Since these operators conserve the number of particles there are no cross-terms between the even and odd subspaces of 10, and thus the expectation values of these operators ${{{\boldsymbol{ \Sigma }} }_{i,{\boldsymbol{p}} }}=\left( \Sigma _{i}^{x},\Sigma _{i}^{y},\Sigma _{i}^{z} \right)$ read

$$\begin{eqnarray}&&\langle {{\psi }_{{\boldsymbol{p}} }}\mid {{{\boldsymbol{ \Sigma }} }_{i,p}}\mid {{\psi }_{{\boldsymbol{p}} }}\rangle =\mid A{{\mid }^{2}}\langle \psi (e;e)\mid {{{\boldsymbol{ \Sigma }} }_{i,p}}\mid \psi (e;e)\rangle +\mid B{{\mid }^{2}}\langle \psi (o;o)\mid {{{\boldsymbol{ \Sigma }} }_{i,p}}\mid \psi (o;o)\rangle .\end{eqnarray} \tag{ 17 }$$

By direct evaluation we find that the contribution from the even subspace vanishes. On the other hand, the odd subspace component gives

$$\begin{eqnarray}&&\langle \psi (o;o)|{{{\boldsymbol{ \Sigma }} }_{i,{\boldsymbol{p}} }}|\psi (o;o)\rangle ={{{\rm cos} }^{2}}{{\theta }_{o}}\langle {{i}_{o}}|{{{\boldsymbol{ \Sigma }} }_{i,{\boldsymbol{p}} }}|{{i}_{o}}\rangle +{{{\rm sin} }^{2}}{{\theta }_{o}}\langle {{\tilde{i}}_{o}}|{{{\boldsymbol{ \Sigma }} }_{i,{\boldsymbol{p}} }}|{{\tilde{i}}_{o}}\rangle =T\langle {{i}_{o}}|{{{\boldsymbol{ \Sigma }} }_{i,{\boldsymbol{p}} }}|{{i}_{o}}\rangle ,\end{eqnarray} \tag{ 18 }$$

where we have used the tracelessness of the ${{{\boldsymbol{ \Sigma }} }_{i,{\boldsymbol{p}} }}$ operators that implies $\langle {{\tilde{i}}_{o}}|{{{\boldsymbol{ \Sigma }} }_{i,{\boldsymbol{p}} }}|{{\tilde{i}}_{o}}\rangle =-\langle {{i}_{o}}|{{{\boldsymbol{ \Sigma }} }_{i,{\boldsymbol{p}} }}|{{i}_{o}}\rangle$. Altogether we then obtain

$$\begin{eqnarray}\begin{array}{rcl} \langle {{\psi }_{{\boldsymbol{p}} }}\mid {{{\boldsymbol{ \Sigma }} }_{a,{\boldsymbol{p}} }}\mid {{\psi }_{{\boldsymbol{p}} }}\rangle & = & S\langle {{\psi }_{a}}({\boldsymbol{p}} )\mid {\boldsymbol{ \sigma }} \mid {{\psi }_{a}}({\boldsymbol{p}} )\rangle , \\ \langle {{\psi }_{{\boldsymbol{p}} }}\mid {{{\boldsymbol{ \Sigma }} }_{b,{\boldsymbol{p}} }}\mid {{\psi }_{{\boldsymbol{p}} }}\rangle & = & S\langle {{\psi }_{b}}({\boldsymbol{p}} )\mid {\boldsymbol{ \sigma }} \mid {{\psi }_{b}}({\boldsymbol{p}} )\rangle \\ \end{array}\end{eqnarray} \tag{ 19 }$$

where $\left| {{\psi }_{a}}({\boldsymbol{p}} ) \right\rangle ={{\left( {{\alpha }_{01}},{{\alpha }_{10}} \right)}^{T}}$ and $\left| {{\psi }_{b}}({\boldsymbol{p}} ) \right\rangle ={{\left( {{\beta }_{01}},{{\beta }_{10}} \right)}^{T}}$. In other words, we obtain two three-vectors ${{{\boldsymbol{\hat{s}}} }_{a}}({\boldsymbol{p}} )$ and ${{{\boldsymbol{\hat{s}}} }_{b}}({\boldsymbol{p}} )$ whose normalized components are given by

$$\begin{eqnarray}&&{{{\boldsymbol{\hat{s}}} }_{i}}({\boldsymbol{p}} )=\frac{{{{\boldsymbol{s}} }_{i}}({\boldsymbol{p}} )}{|{{{\boldsymbol{s}} }_{i}}({\boldsymbol{p}} )|}=\langle {{\psi }_{i}}({\boldsymbol{p}} )|{\boldsymbol{ \sigma }} |{{\psi }_{i}}({\boldsymbol{p}} )\rangle ,\qquad |{{{\boldsymbol{s}} }_{i}}({\boldsymbol{p}} )|=|B{{|}^{2}}|T|.\end{eqnarray} \tag{ 20 }$$

The fact that the norm of these vectors is equal to $|S|$ implies that the degree of entanglement between the subsystems can be probed by using the operators (16).

Exactly like the Berry phase (2) and the winding number (3) were equivalent representations of the Chern number for two component systems, we can define a subsystem Chern number ${{\nu }_{i}}$ for the state $|{{i}_{o}}\rangle$ and represent it equivalently either as the subsystem Berry phase

$$\begin{eqnarray}&&{{\nu }_{i}}=\frac{i}{2\pi }{{\oint }_{\partial BZ}}\langle {{i}_{o}}|\partial |{{i}_{o}}\rangle \cdot {\rm d}{\boldsymbol{p}} ,\end{eqnarray} \tag{ 21 }$$

or as the subsystem winding numbers

$$\begin{eqnarray}&&{{\nu }_{i}}=\frac{1}{4\pi }{{\int }_{BZ}}{{{\rm d}}^{2}}p\;{{{\boldsymbol{\hat{s}}} }_{i}}({\boldsymbol{p}} )\cdot \left( {{\partial }_{{{p}_{x}}}}{{{{\boldsymbol{\hat{s}}} }}_{i}}({\boldsymbol{p}} )\times {{\partial }_{{{p}_{y}}}}{{{{\boldsymbol{\hat{s}}} }}_{i}}({\boldsymbol{p}} ) \right).\end{eqnarray} \tag{ 22 }$$

Formally this comes about by viewing $\left| {{\psi }_{i}}({\boldsymbol{p}} ) \right\rangle$ as the ground state of a fictitious Hamiltonian ${{H}_{i}}={{{\boldsymbol{\hat{s}}} }_{i}}({\boldsymbol{p}} )\cdot {\boldsymbol{ \sigma }}$ with eigenvalues $E=\pm 1$. As we have shown above, the components of the vectors ${{{\boldsymbol{\hat{s}}} }_{i}}({\boldsymbol{p}} )$ can be obtained from the observables (16) and hence the subsystem Chern number is an observable.

We have shown that each subsystem Berry phase (21) is proportional to a winding number(22) that is physically observable. When S = 1, by (15), the Chern number becomes additive in the subsystem winding numbers. However, the method also works for $S\ne 1$ and examples in section 4 show that the Chern number is returned as long as the system is not maximally entangled, or $S\nrightarrow 0$. We will now show that with small modifications the same method applies directly also to topological superconductors that conserve only the particle parity.

The generalization to topological superconductors is straightforward. We take the Hamiltonian to be of the same form as (6) with the basis given now by $\Psi _{{\boldsymbol{p}} }^{\dagger }=\left( a_{{\boldsymbol{p}} }^{\dagger },{{a}_{-{\boldsymbol{p}} }},b_{{\boldsymbol{p}} }^{\dagger },{{b}_{-{\boldsymbol{p}} }} \right)$. The general state can be written as (7) with the Fock space ordered as

$$\begin{eqnarray}&&\left| n_{{\boldsymbol{p}} }^{a},n_{-{\boldsymbol{p}} }^{a},n_{{\boldsymbol{p}} }^{b},n_{-{\boldsymbol{p}} }^{b} \right\rangle ={{(a_{{\boldsymbol{p}} }^{\dagger })}^{n_{{\boldsymbol{p}} }^{a}}}{{(a_{-{\boldsymbol{p}} }^{\dagger })}^{n_{-{\boldsymbol{p}} }^{a}}}{{(b_{{\boldsymbol{p}} }^{\dagger })}^{n_{{\boldsymbol{p}} }^{b}}}{{(b_{-{\boldsymbol{p}} }^{\dagger })}^{n_{-{\boldsymbol{p}} }^{b}}}\left| 0000 \right\rangle .\end{eqnarray} \tag{ 23 }$$

A superconducting system conserves only the total parity, i.e. $\left[ H,P \right]=0$ with $P={\rm exp} \left( {\rm i}\pi {{\sum }_{{\boldsymbol{p}} }}\left( a_{{\boldsymbol{p}} }^{\dagger }{{a}_{{\boldsymbol{p}} }}+b_{{\boldsymbol{p}} }^{\dagger }{{b}_{{\boldsymbol{p}} }} \right) \right)={{P}_{a}}{{P}_{b}}$, while component parities P_a and P_b are not independently conserved. Without loss of generality we assume that the ground state resides in the even total parity sector. This means parities in the subsystems a and b are correlated such that ${{P}_{a}}={{P}_{b}}$, which in turn means that the ground state complies with the condition of zero overall momentum. In this parity sector the ground state is thus given in the basis spanned by the states

$$\begin{eqnarray}&&\left\{ \left| 0000 \right\rangle ,\left| 0011 \right\rangle ,\left| 1100 \right\rangle ,\left| 1111 \right\rangle ,\left| 0110 \right\rangle ,\left| 1001 \right\rangle \right\}.\end{eqnarray} \tag{ 24 }$$

As with the topological insulators, we split the Hilbert space into even and odd occupation subspaces and write

$$\begin{eqnarray}&&|{{\psi }_{{\boldsymbol{p}} }}\rangle =A\left| \psi (e;e) \right\rangle +B\left| \psi (o;o) \right\rangle .\end{eqnarray} \tag{ 25 }$$

Performing a Schmidt decomposition between the subsystems a and b in this parity sector, we obtain a general expression which has the same form as (11), but with the Schmidt bases now being given by

$$\begin{eqnarray}\begin{array}{rcl} |{{a}_{e}}\rangle & = & ({{\alpha }_{00}}+{{\alpha }_{11}}a_{{\boldsymbol{p}} }^{\dagger }a_{-{\boldsymbol{p}} }^{\dagger })|00\rangle ,\quad |{{\tilde{a}}_{e}}\rangle =(\alpha _{11}^{*}-\alpha _{00}^{*}a_{{\boldsymbol{p}} }^{\dagger }a_{-{\boldsymbol{p}} }^{\dagger })|00\rangle , \\ |{{b}_{e}}\rangle & = & ({{\beta }_{00}}+{{\beta }_{11}}b_{{\boldsymbol{p}} }^{\dagger }b_{-{\boldsymbol{p}} }^{\dagger })|00\rangle ,\quad |{{\tilde{b}}_{e}}\rangle =(\beta _{11}^{*}-\beta _{00}^{*}b_{{\boldsymbol{p}} }^{\dagger }b_{-{\boldsymbol{p}} }^{\dagger })|00\rangle , \\ \end{array}\end{eqnarray} \tag{ 26 }$$

and

$$\begin{eqnarray}\begin{array}{rcl} |{{a}_{o}}\rangle & = & {{{\rm e}}^{{\rm i}{{\phi }_{a}}}}a_{-{\boldsymbol{p}} }^{\dagger }|00\rangle ,\quad |{{\tilde{a}}_{o}}\rangle ={{{\rm e}}^{{\rm i}{{\tilde{\phi }}_{a}}}}a_{{\boldsymbol{p}} }^{\dagger }\left| 00 \right\rangle , \\ |{{b}_{o}}\rangle & = & {{{\rm e}}^{{\rm i}{{\phi }_{b}}}}b_{{\boldsymbol{p}} }^{\dagger }|00\rangle ,\quad |{{\tilde{b}}_{o}}\rangle ={{{\rm e}}^{{\rm i}{{\tilde{\phi }}_{b}}}}b_{-{\boldsymbol{p}} }^{\dagger }|00\rangle . \\ \end{array}\end{eqnarray} \tag{ 27 }$$

As in the case of topological insulators where all coefficients except those in the odd subspace could be made real through $U(1)$ gauge transformations, we can now take only the coefficients in the even subspace to be complex and all other coefficients to be real. The decomposition of the Berry phase proceeds in similar steps to the insulating case. The only difference is that it is now the odd subspace contribution that vanishes in (14), with the Chern number being now given by

$$\begin{eqnarray}&&\nu =-\frac{{\rm i}}{2\pi }{{\sum }_{i=a,b}}{{\oint }_{\partial BZ}}S\langle {{i}_{e}}|\partial |{{i}_{e}}\rangle \cdot {\rm d}{\boldsymbol{p}} ,\qquad S=|A{{|}^{2}}T.\end{eqnarray} \tag{ 28 }$$

The relevant observables ${{{\boldsymbol{ \Sigma }} }_{i,{\boldsymbol{p}} }}=\left( \Sigma _{i}^{x},\Sigma _{i}^{y},\Sigma _{i}^{z} \right)$ to evaluate the subsystem winding numbers are now defined by

$$\begin{eqnarray}\begin{array}{rcl} \Sigma _{a}^{x} & = & a_{{\boldsymbol{p}} }^{\dagger }a_{-{\boldsymbol{p}} }^{\dagger }+{{a}_{-{\boldsymbol{p}} }}{{a}_{{\boldsymbol{p}} }},\Sigma _{a}^{y}=-{\rm i}a_{{\boldsymbol{p}} }^{\dagger }a_{-{\boldsymbol{p}} }^{\dagger }+{\rm i}{{a}_{-{\boldsymbol{p}} }}{{a}_{{\boldsymbol{p}} }},\Sigma _{a}^{z}=a_{{\boldsymbol{p}} }^{\dagger }{{a}_{{\boldsymbol{p}} }}-a_{-{\boldsymbol{p}} }^{\dagger }{{a}_{-{\boldsymbol{p}} }}, \\ \Sigma _{b}^{x} & = & b_{{\boldsymbol{p}} }^{\dagger }b_{-{\boldsymbol{p}} }^{\dagger }+{{b}_{-{\boldsymbol{p}} }}{{b}_{{\boldsymbol{p}} }},\Sigma _{b}^{y}=-{\rm i}b_{{\boldsymbol{p}} }^{\dagger }b_{-{\boldsymbol{p}} }^{\dagger }+{\rm i}{{b}_{-{\boldsymbol{p}} }}{{b}_{{\boldsymbol{p}} }},\Sigma _{b}^{z}=b_{{\boldsymbol{p}} }^{\dagger }{{b}_{{\boldsymbol{p}} }}-b_{-{\boldsymbol{p}} }^{\dagger }{{b}_{-{\boldsymbol{p}} }}. \\ \end{array}\end{eqnarray} \tag{ 29 }$$

Computing their expectation values we find that now only the even subspace contributes. The precise expressions are given by

$$\begin{eqnarray}&&\langle \psi (e;e)|{{{\boldsymbol{ \Sigma }} }_{a,{\boldsymbol{p}} }}|\psi (e;e)\rangle =S\langle {{\psi }_{a}}({\boldsymbol{p}} )|{\boldsymbol{ \sigma }} |{{\psi }_{a}}({\boldsymbol{p}} )\rangle ,\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}&&\langle \psi (e;e)|{{{\boldsymbol{ \Sigma }} }_{b,{\boldsymbol{p}} }}|\psi (e;e)\rangle =S\langle {{\psi }_{a}}({\boldsymbol{p}} )|{\boldsymbol{ \sigma }} |{{\psi }_{a}}({\boldsymbol{p}} )\rangle ,\end{eqnarray} \tag{ 31 }$$

where now $|{{\psi }_{a}}({\boldsymbol{p}} )\rangle ={{\left( {{\alpha }_{00}},{{\alpha }_{11}} \right)}^{T}}$ and $|{{\psi }_{b}}({\boldsymbol{p}} )\rangle ={{\left( {{\beta }_{00}},{{\beta }_{11}}) \right)}^{T}}$. Thus the expectation values can again be used to define two three-vectors ${{{\boldsymbol{\hat{s}}} }_{a}}({\boldsymbol{p}} )$ and ${{{\boldsymbol{\hat{s}}} }_{b}}({\boldsymbol{p}} )$, that can be used to evaluate the subsystem winding numbers (22). Under the same assumption of non-maximal entanglement between the subsystems, the Chern number will be shown to be additive in these winding numbers.

As the first example, we consider the quantum spin-Hall insulator defined on a honeycomb lattice [12]. The Hamiltonian of this model is given by

$$\begin{eqnarray}&&\begin{array}{l} H=t\mathop{\sum }\limits_{\langle {\boldsymbol{ij}} \rangle }a_{{\boldsymbol{i}} }^{\dagger }{{b}_{{\boldsymbol{j}} }}+{{\lambda }_{v}}\mathop{\sum }\limits_{{\boldsymbol{i}} }\left( a_{{\boldsymbol{i}} }^{\dagger }{{a}_{{\boldsymbol{i}} }}-b_{{\boldsymbol{i}} }^{\dagger }{{b}_{{\boldsymbol{i}} }} \right) \\ \quad +{\rm i}{{\lambda }_{SO}}\mathop{\sum }\limits_{\langle \langle {\boldsymbol{ij}} \rangle \rangle }{{\xi }_{SO}}\left( a_{{\boldsymbol{i}} }^{\dagger }{{\sigma }_{z}}{{a}_{{\boldsymbol{j}} }}+b_{{\boldsymbol{i}} }^{\dagger }{{\sigma }_{z}}{{b}_{{\boldsymbol{j}} }} \right)+{\rm i}{{\lambda }_{R}}\mathop{\sum }\limits_{\langle {\boldsymbol{ij}} \rangle }a_{{\boldsymbol{i}} }^{\dagger }{{\left( {\boldsymbol{ \sigma }} \times {{{\hat{d}}}_{{\boldsymbol{i}} }} \right)}_{z}}{{b}_{{\boldsymbol{j}} }}, \\ \end{array}\end{eqnarray} \tag{ 37 }$$

where the spinors $a_{{\boldsymbol{i}} }^{\dagger }=(a_{{\boldsymbol{i}} ,\uparrow }^{\dagger },a_{{\boldsymbol{i}} ,\downarrow }^{\dagger })$ and $b_{{\boldsymbol{i}} }^{\dagger }=(b_{{\boldsymbol{i}} ,\uparrow }^{\dagger },b_{{\boldsymbol{i}} ,\downarrow }^{\dagger })$ denote the two sublattice degrees of freedom of the honeycomb lattice. The first two terms of magnitudes t and λ_v describe spin-independent nearest-neighbour tunnelling and a sublattice energy imbalance, respectively. The other two terms proportional to ${{\lambda }_{R}}$ and ${{\lambda }_{SO}}$ are nearest and next-nearest neighbour spin-orbit couplings, respectively. In the notation used ${{\xi }_{SO}}={\rm sign}({{\hat{d}}_{1}}\times {{\hat{d}}_{2}})$ with ${{\hat{d}}_{1}}$ and ${{\hat{d}}_{2}}$ being vectors that connect the next-to-nearest neighbour sites.

By Fourier transforming the Hamiltonian (37) it takes the Bloch form (6) in the basis ${{\Psi }_{{\boldsymbol{p}} }}={{({{a}_{\uparrow ,{\boldsymbol{p}} }},{{a}_{\downarrow ,{\boldsymbol{p}} }},{{b}_{\uparrow ,{\boldsymbol{p}} }},{{b}_{\downarrow ,{\boldsymbol{p}} }})}^{T}}$. By diagonalizing this Hamiltonian Kane and Mele showed that it supports a trivial insulator and a quantum spin Hall phase, which are distinguished by a ${{\mathbb{Z}}_{2}}$ valued topological invariant [12]. While in a time-reversal symmetric system the Chern number is zero in all phases, the ${{\mathbb{Z}}_{2}}$ invariant was shown to be related to the so called spin Chern numbers that are quantized for each spin component [33]. More precisely, the ${{\mathbb{Z}}_{2}}$ invariant was defined as the difference of the spin Chern numbers, ${{\nu }_{S}}=({{\tilde{\nu }}_{\uparrow }}-{{\tilde{\nu }}_{\downarrow }})/2$, that only takes non-zero value in the quantum spin Hall phase. The phase diagram as a function of the microscopic parameters is shown in figure 1.

Zoom In Zoom Out Reset image size

The spin Chern number has a natural counterpart in our constrution if we identify the spin up and spin down components as the two subsystems with respect to which the ground state is Schmidt decomposed. The corresponding operators for evaluating the subsystem winding numbers are

$$\begin{eqnarray}\begin{array}{rclrclrcl} \Sigma _{\uparrow }^{x} & = & a_{\uparrow ,{\boldsymbol{p}} }^{\dagger }{{b}_{\uparrow ,{\boldsymbol{p}} }}+b_{\uparrow ,{\boldsymbol{p}} }^{\dagger }{{a}_{\uparrow ,{\boldsymbol{p}} }}, & \Sigma _{\uparrow }^{y} & = & -{\rm i}a_{\uparrow ,{\boldsymbol{p}} }^{\dagger }{{b}_{\uparrow ,{\boldsymbol{p}} }}+{\rm i}b_{\uparrow ,{\boldsymbol{p}} }^{\dagger }{{a}_{\uparrow ,{\boldsymbol{p}} }}, & \Sigma _{\uparrow }^{z} & = & a_{\uparrow ,{\boldsymbol{p}} }^{\dagger }{{a}_{\uparrow ,{\boldsymbol{p}} }}-b_{\uparrow ,{\boldsymbol{p}} }^{\dagger }{{b}_{\uparrow ,{\boldsymbol{p}} }} \\ \end{array}\end{eqnarray} \tag{ 38 }$$

and similarly for the $\downarrow$-spin component. We construct the vectors ${{{\boldsymbol{\hat{s}}} }_{\uparrow }}({\boldsymbol{p}} )$ and ${{{\boldsymbol{\hat{s}}} }_{\downarrow }}({\boldsymbol{p}} )$ from these observables and by inserting them into (22), calculate the corresponding subsystem winding numbers ${{\nu }_{\uparrow }}$ and ${{\nu }_{\downarrow }}$. We find that these observables give then precisely the spin Chern numbers ${{\tilde{\nu }}_{\uparrow }}$ and ${{\tilde{\nu }}_{\downarrow }}$, with figure 1 showing that the phase diagram is precisely reproduced. The figure also shows that the subsystem entanglement measure $|S|$ remains large within the QSH phase, which confirms that the spin components are minimally entangled in this phase. We take this as confirming the reliability of our method for non-maximally entangled states.

The second example we consider is a recently introduced topological superconductor with a staggered chemical potential [21] that enables the system to support topological phases characterized by Chern numbers $\nu =0,\pm 1$ and $\pm 2$. The model is defined on a square lattice by the Hamiltonian

$$\begin{eqnarray}&&\begin{array}{l} H=\mathop{\sum }\limits_{{\boldsymbol{j}} }\left[ (\mu -\delta )a_{{\boldsymbol{j}} }^{\dagger }{{a}_{{\boldsymbol{j}} }}+(\mu +\delta )b_{{\boldsymbol{j}} }^{\dagger }{{b}_{{\boldsymbol{j}} }}+t\left( ia_{{\boldsymbol{j}} }^{\dagger }{{b}_{{\boldsymbol{j}} }}-ib_{{\boldsymbol{j}} }^{\dagger }{{a}_{{\boldsymbol{j}} +{\boldsymbol{\hat{x}}} }}+a_{{\boldsymbol{j}} }^{\dagger }{{a}_{{\boldsymbol{j}} +{\boldsymbol{\hat{y}}} }}+b_{{\boldsymbol{j}} }^{\dagger }{{b}_{{\boldsymbol{j}} +{\boldsymbol{\hat{y}}} }} \right) \right. \\ \quad +\left. \Delta \left( a_{{\boldsymbol{j}} }^{\dagger }b_{{\boldsymbol{j}} }^{\dagger }+b_{{\boldsymbol{j}} }^{\dagger }a_{{\boldsymbol{j}} +{\boldsymbol{\hat{x}}} }^{\dagger }+a_{{\boldsymbol{j}} }^{\dagger }a_{{\boldsymbol{j}} +{\boldsymbol{\hat{y}}} }^{\dagger }+b_{{\boldsymbol{j}} }^{\dagger }b_{{\boldsymbol{j}} +{\boldsymbol{\hat{y}}} }^{\dagger } \right)+{\rm h}.{\rm c}. \right], \\ \end{array}\end{eqnarray} \tag{ 39 }$$

where $a_{{\boldsymbol{j}} }^{\dagger }$ and $b_{{\boldsymbol{j}} }^{\dagger }$ denote the two sublattice degrees of freedom that are distinguished by the staggered offset δ in the chemical potential μ and δ. The other two coefficients t and Δ correspond to the nearest neighbour hopping and pairing, respectively.

We partition the system based on the sublattices, which implies that the relevant operators to evaluate the corresponding subsystem winding numbers are given by (29). Calculating the observable vectors ${{{\boldsymbol{\hat{s}}} }_{a}}({\boldsymbol{p}} )$ and ${{{\boldsymbol{\hat{s}}} }_{b}}({\boldsymbol{p}} )$ through evaluation of (30) and (31), figure 2 shows that the phase diagram is in general faithfully reproduced. Discrepancies between the Chern number and the sum of the sublattice winding numbers occur only in regions where the sublattices become (close to) maximally entangled. Thus as we analytically argued above, caution should be taken when trusting the results in these regimes. Still, we note that the discrepancies are only in the signs, which means that all distinct types of topological phases are distinguished.

Zoom In Zoom Out Reset image size

Consider the Hamiltonian of a 2-dimensional many body system of non-interacting fermions given by

$$\begin{eqnarray}&&H={{\int }_{{\boldsymbol{p}} }}{{{\rm d}}^{2}}p\;\;{\boldsymbol{ \psi }} _{{\boldsymbol{p}} }^{\dagger }h({\boldsymbol{p}} ){{{\boldsymbol{ \psi }} }_{{\boldsymbol{p}} }},\end{eqnarray} \tag{ A.1 }$$

where ${\boldsymbol{p}} =({{p}_{x}},{{p}_{y}})$ is an element of the Brillouin zone, $h({\boldsymbol{p}} )$ is the Bloch Hamiltonian and ${{{\boldsymbol{ \psi }} }_{{\boldsymbol{p}} }}$ is the two dimensional spinor containing the relevant field mode operators. We denote the ground state of the Bloch Hamiltonian by $\left| \psi ({\boldsymbol{p}} ) \right\rangle$. Let us define a set of projectors ${{P}_{{\boldsymbol{p}} }}=\left| \psi ({\boldsymbol{p}} ) \right\rangle \left. \psi ({\boldsymbol{p}} ) \right|$. The Chern number ν, of the first Chern class, is one of many topological invariants that define an order that cannot be destroyed by local unitary transformations. The Chern number has three representations of interest, illustrated in figure 3.

Zoom In Zoom Out Reset image size

The most fundamental representation is in terms of projectors ${{P}_{{\boldsymbol{p}} }}$. It is written as

$$\begin{eqnarray}&&\nu =-\frac{{\rm i}}{2\pi }{{\int }_{BZ}}{{{\rm d}}^{2}}p\;\;{\rm tr}\left( {{P}_{{\boldsymbol{p}} }}\left[ {{\partial }_{{{p}_{x}}}}{{P}_{{\boldsymbol{p}} }},{{\partial }_{{{p}_{y}}}}{{P}_{{\boldsymbol{p}} }} \right] \right).\end{eqnarray} \tag{ A.2 }$$

From this expression we can retrieve both the Berry phase and winding number representations.

We can write the Chern number as the Berry phase that the ground state accrues around the boundary of the Brillouin zone. To see this we take the projector representation (A.2) and substitute in the definition ${{P}_{{\boldsymbol{p}} }}=\left| \psi ({\boldsymbol{p}} ) \right\rangle \left. \psi ({\boldsymbol{p}} ) \right|$

$$\begin{eqnarray}\begin{array}{rcl} \nu & = & -\frac{{\rm i}}{2\pi }\int _{BZ}^{{}}{{{\rm d}}^{2}}p\langle \psi \left( {\boldsymbol{p}} \right)|[{\mkern 1mu} |{{\partial }_{{{p}_{x}}}}\psi \left( {\boldsymbol{p}} \right)\rangle \langle \psi \left( {\boldsymbol{p}} \right)| \\ {} & {} & +\;|\psi \left( {\boldsymbol{p}} \right)\rangle \langle {{\partial }_{{{p}_{x}}}}\psi \left( {\boldsymbol{p}} \right)|,|{{\partial }_{{{p}_{y}}}}\psi \left( {\boldsymbol{p}} \right)\rangle \langle \psi \left( {\boldsymbol{p}} \right)| \\ {} & {} & +\;|\psi \left( {\boldsymbol{p}} \right)\rangle \langle {{\partial }_{{{p}_{y}}}}\psi \left( {\boldsymbol{p}} \right)|{\mkern 1mu} ]|\psi \left( {\boldsymbol{p}} \right)\rangle . \\ \end{array}\end{eqnarray} \tag{ A.3 }$$

The ground state is normalized so we have ${{\partial }_{\mu }}\left\langle \psi ({\boldsymbol{p}} )|\psi ({\boldsymbol{p}} ) \right\rangle =0$, where $\mu ={{p}_{x}},{{p}_{y}}$, such that

$$\begin{eqnarray}&&\langle {{\partial }_{\mu }}\psi ({\boldsymbol{p}} )|\psi ({\boldsymbol{p}} )\rangle =-\langle \psi ({\boldsymbol{p}} )|{{\partial }_{\mu }}\psi ({\boldsymbol{p}} )\rangle .\end{eqnarray} \tag{ A.4 }$$

Expanding (A.3) and using relation (A.4) we obtain

$$\begin{eqnarray}&&\nu =-\frac{{\rm i}}{2\pi }{{\int }_{BZ}}{{{\rm d}}^{2}}p(\langle {{\partial }_{{{p}_{x}}}}\psi ({\boldsymbol{p}} )|{{\partial }_{{{p}_{y}}}}\psi ({\boldsymbol{p}} )\rangle -\langle {{\partial }_{{{p}_{y}}}}\psi ({\boldsymbol{p}} )|{{\partial }_{{{p}_{x}}}}\psi ({\boldsymbol{p}} )\rangle ).\end{eqnarray} \tag{ A.5 }$$

We now recognize that the integrand is the Berry curvature, ${{F}_{{{p}_{x}}{{p}_{y}}}}$,

$$\begin{eqnarray}&&{{F}_{{{p}_{x}}{{p}_{y}}}}={{\partial }_{{{p}_{x}}}}{{A}_{{{p}_{y}}}}-{{\partial }_{{{p}_{y}}}}{{A}_{{{p}_{x}}}}=\langle {{\partial }_{{{p}_{x}}}}\psi ({\boldsymbol{p}} )|{{\partial }_{{{p}_{y}}}}\psi ({\boldsymbol{p}} )\rangle -\langle {{\partial }_{{{p}_{y}}}}\psi ({\boldsymbol{p}} )|{{\partial }_{{{p}_{x}}}}\psi ({\boldsymbol{p}} )\rangle ,\end{eqnarray} \tag{ A.6 }$$

where ${\bf A}=\left\langle \psi ({\boldsymbol{p}} )|\partial |\psi ({\boldsymbol{p}} ) \right\rangle$ is the Berry connection. Using Stokes' theorem, the Chern number can be written

$$\begin{eqnarray}&&\nu =-\frac{{\rm i}}{2\pi }{{\int }_{BZ}}{{{\rm d}}^{2}}p\;\;{{F}_{{{p}_{x}}{{p}_{y}}}}=-\frac{{\rm i}}{2\pi }{{\oint }_{\partial BZ}}{\rm d}{\boldsymbol{p}} \cdot {\bf A},\end{eqnarray} \tag{ A.7 }$$

where $\partial BZ$ is the boundary of the Brillouin zone. Hence, the Chern number can be written in terms of the Berry phase accumulated by the ground state around the boundary of the Brillouin zone.

When the Bloch Hamiltonian $h({\boldsymbol{p}} )$ is 2-dimensional the ground state $\left| \psi ({\boldsymbol{p}} ) \right\rangle$ lies on the Bloch sphere. Therefore we can write down an explicit form of the projector in terms of a normalized 3-vector ${\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )$

$$\begin{eqnarray}&&{{P}_{{\boldsymbol{p}} }}=|\psi ({\boldsymbol{p}} )\rangle \langle \psi ({\boldsymbol{p}} )|=\frac{1}{2}\left( 11-{\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )\cdot {\boldsymbol{ \sigma }} \right),\end{eqnarray} \tag{ A.8 }$$

where $11$ is the identity matrix and ${\boldsymbol{ \sigma }} =\left( {{\sigma }^{x}},{{\sigma }^{y}},{{\sigma }^{z}} \right)$ is the vector of Pauli matrices. Substituting the preceeding definition into the definition of the Chern number in terms of projectors (A.2) and employing the identities

$$\begin{eqnarray}&&({\bf a}\cdot {\boldsymbol{ \sigma }} )({\bf b}\cdot {\boldsymbol{ \sigma }} )=11{\bf a}\cdot {\bf b}+{\rm i}{\boldsymbol{ \sigma }} \cdot {\bf a}\times {\bf b},\end{eqnarray} \tag{ A.9 }$$

for two 3-vectors ${\bf a}$ and ${\bf b}$, and

$$\begin{eqnarray}&&{\rm tr}\left( {{\sigma }^{\alpha }}{{\sigma }^{\beta }} \right)=2{{\delta }^{\alpha \beta }},\qquad \alpha ,\beta =x,y,z,\end{eqnarray} \tag{ A.10 }$$

we obtain

$$\begin{eqnarray}\begin{array}{rcl} \nu & = & -\frac{{\rm i}}{2\pi }{{\int }_{BZ}}{{{\rm d}}^{2}}p\;\;{\rm tr}({{P}_{{\boldsymbol{p}} }}[{{\partial }_{{{p}_{x}}}}{{P}_{{\boldsymbol{p}} }},{{\partial }_{{{p}_{y}}}}{{P}_{{\boldsymbol{p}} }}]) \\ {} & {} & =-\frac{{\rm i}}{2\pi }{{\int }_{BZ}}{{{\rm d}}^{2}}p\;{\rm tr}(|\psi ({\boldsymbol{p}} )\rangle \langle \psi ({\boldsymbol{p}} )|\frac{1}{4}[{{\partial }_{{{p}_{x}}}}{\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )\cdot {\boldsymbol{ \sigma }} ,{{\partial }_{{{p}_{y}}}}{\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )\cdot {\boldsymbol{ \sigma }} ]) \\ {} & {} & =\frac{1}{4\pi }{{\int }_{BZ}}{{{\rm d}}^{2}}p\;{\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )\cdot ({{\partial }_{{{p}_{x}}}}{\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )\times {{\partial }_{{{p}_{y}}}}{\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )). \\ \end{array}\end{eqnarray} \tag{ A.11 }$$

The vector ${\boldsymbol{\hat{s}}} ({\boldsymbol{p}} ):{{T}^{2}}\to {{S}^{2}}$ is a map between the toroidal Brillouin zone and the unit 2-sphere as shown in figure 4. The expression for the Chern number derived above (A.11) describes the winding of this map around the unit sphere when the Brillouin zone is spanned.

Zoom In Zoom Out Reset image size

To complete the picture outlined in figure 3 we now derive the winding number directly from the Berry phase. We start by writing the Chern number as the integral of the Berry curvature

$$\begin{eqnarray}&&\nu =-\frac{{\rm i}}{2\pi }{{\int }_{BZ}}{{{\rm d}}^{2}}p(\langle {{\partial }_{{{p}_{x}}}}\psi ({\boldsymbol{p}} )|{{\partial }_{{{p}_{y}}}}\psi ({\boldsymbol{p}} )\rangle -\langle {{\partial }_{{{p}_{y}}}}\psi ({\boldsymbol{p}} )|{{\partial }_{{{p}_{x}}}}\psi ({\boldsymbol{p}} )\rangle ).\end{eqnarray} \tag{ A.12 }$$

Assuming the Bloch Hamiltonian is 2-dimensional we can write the projector as (A.8). We substitute the projector into (A.12)

$$\begin{eqnarray}&&\nu =-\frac{{\rm i}}{2\pi }{{\int }_{BZ}}{{{\rm d}}^{2}}p({{\partial }_{{{p}_{x}}}}(\langle \psi ({\boldsymbol{p}} )|{{P}_{{\boldsymbol{p}} }}){{\partial }_{{{p}_{y}}}}({{P}_{{\boldsymbol{p}} }}\left| \psi ({\boldsymbol{p}} ) \right\rangle )-{{\partial }_{{{p}_{y}}}}(\langle \psi ({\boldsymbol{p}} )|{{P}_{{\boldsymbol{p}} }}){{\partial }_{{{p}_{x}}}}({{P}_{{\boldsymbol{p}} }}\left| \psi ({\boldsymbol{p}} ) \right\rangle )).\end{eqnarray} \tag{ A.13 }$$

Using the fact that ${{P}^{2}}=P$, $\left\{ {{\partial }_{\mu }}{\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )\cdot {\boldsymbol{ \sigma }} ,{\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )\cdot {\boldsymbol{ \sigma }} \right\}=0$ and relation (A.9) we find that

$$\begin{eqnarray}\begin{array}{rcl} \nu & = & \frac{1}{4\pi }{{\int }_{BZ}}{{{\rm d}}^{2}}p\;\;\langle \psi ({\boldsymbol{p}} )|{\boldsymbol{ \sigma }} \cdot ({{\partial }_{{{p}_{x}}}}{\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )\times {{\partial }_{{{p}_{y}}}}{\boldsymbol{\hat{s}}} ({\boldsymbol{p}} ))|\psi ({\boldsymbol{p}} )\rangle \\ {} & {} & =\frac{1}{4\pi }{{\int }_{BZ}}{{{\rm d}}^{2}}p\;{\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )\cdot ({{\partial }_{{{p}_{x}}}}{\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )\times {{\partial }_{{{p}_{y}}}}{\boldsymbol{\hat{s}}} ({\boldsymbol{p}} )). \\ \end{array}\end{eqnarray} \tag{ A.14 }$$