Classification and surface anomaly of glide symmetry protected topological phases in three dimensions

We start from a generic gapped system with the full symmetry group

$$\begin{eqnarray}&&{G}_{s}={{\mathbb{Z}}}^{{{ \mathcal G }}_{y}}\times {G}_{0}=\{{({{ \mathcal G }}_{y})}^{n}| n\in {\mathbb{Z}}\}\times {G}_{0}\end{eqnarray} \tag{ 1 }$$

generated by global (on-site) symmetry G₀ and non-symmorphic glide symmetry ${{ \mathcal G }}_{y}=\left\{{M}_{[010]}| \left(\tfrac{1}{2},0,0\right)\right\}$

$$\begin{eqnarray}&&(x,y,z)\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}\left(x+\displaystyle \frac{{a}_{x}}{2},-y,z\right).\end{eqnarray} \tag{ 2 }$$

as illustrated in figure 1, where the arrows point to $\pm \hat{y}$ directions and a_x represents the length of Bravais lattice primitive vector along $\hat{x}$ direction. The orientation-reversing glide operation combines mirror reflection w.r.t. [010] plane and translation along $\hat{x}$ direction by half a Bravais lattice vector. Two glide operation leads to a Bravais translation ${T}_{x}={({{ \mathcal G }}_{y})}^{2}$ along $\hat{x}$ direction.

As mentioned earlier, GSPT phases are 'weak' crystalline SPT states, which can be adiabatically connected to trivial product states if glide symmetry is absent. They are different from the 'strong' SPT phases, which cannot be adiabatically connected to a product state irrespective of any crystal symmetries, as long as certain on-site/global symmetries are preserved. Mathematically this means there exists a finite depth quantum circuit U^loc that preserves all global symmetries but not the glide symmetry, such that

$$\begin{eqnarray}&&{U}^{{\rm{loc}}}| \psi \rangle =| 0\rangle ,\end{eqnarray} \tag{ 3 }$$

where $| \psi \rangle$ represents any GSPT state and $| 0\rangle$ is the trivial product state. Hereafter we assume a finite correlation length $\xi \ll {a}_{x}$ in GSPT state $| \psi \rangle$. This assumption can always be satisfied by choosing a large primitive unit cell along $\hat{x}$ direction, and we expect the topological classification of long-wavelength (infrared) physics to be independent of these microscopic (ultraviolet) details. We divide the physical Hilbert space of this GSPT state into four types of regions: $\{{A}_{m},{B}_{m},{C}_{m},{D}_{m}\}$ as shown in figure 1. In particular these regions are related pairwise by the glide operation:

$$\begin{eqnarray}&&{A}_{m}\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}{B}_{m}\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}{A}_{m+1},\\ &&{C}_{m}\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}{D}_{m}\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}{C}_{m+1},\end{eqnarray} \tag{ 4 }$$

where $m\in {\mathbb{Z}}$ is the unit cell index along $\hat{x}$ direction. In particular we choose the width of A_n and B_n regions to be $w\gg \xi$ ² , as illustrated in figure 1. Keeping a finite value of width w in the thermodynamic limit allows us to treat each A_n (or B_n) region as a 2D system extensive along $\hat{y}$ and $\hat{z}$ directions.

As shown in figure 1, for a finite depth quantum circuit U^loc, we can always find regions ${C}_{n}^{\prime }\supset {C}_{n}$ and ${D}_{n}^{\prime }\supset {D}_{n}$, such that

$$\begin{eqnarray}&&{U}_{{C}_{n}^{\prime }}^{{\rm{loc}}}| \psi \rangle ={| 0\rangle }_{{C}_{n}}\otimes | {\psi }_{{\bar{C}}_{n}}\rangle ,\,\,\,{U}_{{C}_{n}^{\prime }}^{{\rm{loc}}}\equiv {{ \mathcal P }}_{{C}_{n}^{\prime }}{U}^{{\rm{loc}}}{{ \mathcal P }}_{{C}_{n}^{\prime }},\end{eqnarray} \tag{ 5 }$$

where ${{ \mathcal P }}_{R}$ is the projection operator into the Hilbert space of region R, and $\bar{R}$ represents the region outside R. In other words, we can always identify local finite depth quantum circuits ${U}_{{C}_{n}^{\prime }}^{{\rm{loc}}}$ that trivializes the a region C_n, which locally transform the GSPT state into a trivial product state ${| 0\rangle }_{{C}_{n}}$ within region C_n. Notice that all these regions do not overlap with each other:

$$\begin{eqnarray}&&{C}_{m}^{\prime }\cap {C}_{n}^{\prime }={D}_{m}^{\prime }\cap {D}_{n}^{\prime }=0,\,\,\,\forall \,m\ne n,{C}_{m}^{\prime }\cap {D}_{n}^{\prime }=0,\,\,\,\forall \,m,n.\end{eqnarray} \tag{ 6 }$$

Therefore we can construct the following symmetric finite depth quantum circuit that preserves glide and all on-site symmetries:

$$\begin{eqnarray}&&{U}_{{CD}}^{{\rm{loc}}}\equiv \displaystyle \prod _{n}{U}_{{C}_{n}^{\prime }}^{{\rm{loc}}}\cdot {U}_{{D}_{n}^{\prime }}^{{\rm{loc}}},{U}_{{D}_{n}^{\prime }}^{{\rm{loc}}}\equiv {{ \mathcal G }}_{y}{U}_{{C}_{n}^{\prime }}^{{\rm{loc}}}{{ \mathcal G }}_{y}^{-1},\,\,\,{U}_{{C}_{n+1}^{\prime }}^{{\rm{loc}}}\equiv {{ \mathcal G }}_{y}{U}_{{D}_{n}^{\prime }}^{{\rm{loc}}}{{ \mathcal G }}_{y}^{-1}\end{eqnarray} \tag{ 7 }$$

such that it trivializes all regions $\{{C}_{n},{D}_{n}\}$ for GSPT state $| \psi \rangle$:

$$\begin{eqnarray}&&| {\psi }_{f.p.}\rangle ={U}_{{CD}}^{{\rm{loc}}}| \psi \rangle =\displaystyle \prod _{n}{| 0\rangle }_{{C}_{n}\cup {D}_{n}}\otimes | {\psi }_{{A}_{n}}\rangle \otimes | {\psi }_{{B}_{n}}\rangle .\end{eqnarray} \tag{ 8 }$$

Clearly the above 'fixed-point' state $| {\psi }_{{\rm{f}}.{\rm{p}}.}\rangle$ for a GSPT phase satisfies all symmetries since

$$\begin{eqnarray}&&| {\psi }_{{B}_{n}}\rangle ={{ \mathcal G }}_{y}| {\psi }_{{A}_{n}}\rangle ,\,\,\,| {\psi }_{{A}_{n+1}}\rangle ={{ \mathcal G }}_{y}| {\psi }_{{B}_{n}}\rangle ={T}_{x}| {\psi }_{{A}_{n}}\rangle .\end{eqnarray} \tag{ 9 }$$

In order for $| {\psi }_{{\rm{f}}.{\rm{p}}.}\rangle$ to describe a SRE GSPT phase, each layer $| {\psi }_{{A}_{n}}\rangle$ in the fixed-point wavefunction must correspond to a SRE³ 2D phase that preserves global symmetry G₀, with no ground state degeneracy on any closed manifold.

Therefore we have shown that any GSPT wavefunction, through a fully-symmetric finite depth quantum circuit (7), can be reduced to a fixed-point wavefunction (8). This fixed-point wavefunction describes an array of decoupled 2D layers (A_n and B_n in figure 1) arranged in a glide-symmetric way. Although here we focus on 3D SRE phases with non-symmorphic glide symmetry, the above arguments and coupled layer construction can be easily generalized and applied to other spatial dimensions, and to non-symmorphic screw symmetries.

Although here we focus on SPT phases protected by glide symmetry, it is straightforward to see that all above discussions and fixed-point wavefunctions (8) equally apply to any 2-fold non-symmorphic symmetries, such as 2-fold screw symmetry where each layer is perpendicular to the screw axis. Another similar symmetry is the combination ${\tilde{T}}_{x}\equiv {T}_{x}\cdot { \mathcal T }$ of Bravais lattice translation T_x and time reversal ${ \mathcal T }$, which is preserved in anti-ferromagnetic (AFM) TIs [34, 35] and superconductors [36].

As argued above, we have justified the coupled layer construction for any GSPT phase of interacting bosons/fermions. As illustrated in figure 1, each layer ${A}_{n}/{B}_{n}$ is a SRE 2D symmetric phase $| {\psi }_{{A}_{n}/{B}_{n}}\rangle$. Due to glide symmetry condition (9), the SRE 2D layers $\{{A}_{n},{B}_{n}\}$ are not independent of each other: once we fixed the SRE 2D phase $| {\psi }_{{A}_{0}}\rangle$ in one layer, all other layers and hence the 3D GSPT phase $| {\psi }_{{\rm{f}}.{\rm{p}}.}\rangle$ are automatically determined by glide symmetry. Therefore the classification of 3D GSPT phase $| \psi \rangle$ with symmetry group ${G}_{s}={{\mathbb{Z}}}^{{{ \mathcal G }}_{y}}\times {G}_{0}$ is reduced to (but not equivalent to, as will be clear soon) the classification of 2D SRE phase $| {\psi }_{{A}_{0}}\rangle$ preserving on-site symmetry G₀. We call these 2D SRE phases $| {\psi }_{{A}_{0}}\rangle$ on layer A₀ as the '2D root states' of 3D GSPT phases. We can therefore label a 3D GSPT phase ${| {{\rm{\Psi }}}_{0}\rangle }_{{{ \mathcal G }}_{y}}$ in (8) by its associated 2D SRE root phase $| {\psi }_{{A}_{0}}\rangle \simeq | {{\rm{\Psi }}}_{0}\rangle$ by defining

$$\begin{eqnarray}&&{| {{\rm{\Psi }}}_{0}\rangle }_{{{ \mathcal G }}_{y}}\simeq \displaystyle \prod _{n}{| 0\rangle }_{{C}_{n}\cup {D}_{n}}\otimes | {\psi }_{{A}_{n}}={{\rm{\Psi }}}_{0}\rangle \otimes | {\psi }_{{B}_{n}}={\bar{{\rm{\Psi }}}}_{0}\rangle .\end{eqnarray} \tag{ 10 }$$

Here $| {\bar{{\rm{\Psi }}}}_{0}\rangle$ is defined as the mirror-reflection image (equivalent to time reversal image) of a generic 2D (bosonic or fermionic) SRE phase $| {{\rm{\Psi }}}_{0}\rangle$, so that 3D phase ${| {{\rm{\Psi }}}_{0}\rangle }_{{{ \mathcal G }}_{y}}$ defined above preserves glide symmetry ${{ \mathcal G }}_{y}$.

Recent progress on SPT phases [5] leads to a classification of SRE phases with on-site symmetry group G₀. In particular, a SRE phase is gapped with no ground state degeneracy on any closed manifold. In 2D it can either a SPT phase [15] with non-chiral edge modes; or an 'invertible' phase [37, 38] with chiral edge modes, generated by the E₈ state of bosons [39] and the ${p}_{x}+\hspace{1pt}{\rm{i}}\hspace{1pt}{p}_{y}$ chiral superconductor of fermions [40]. The classification leads to an Abelian group structure of 2D SRE phases, where the addition of two elements in this Abelian group corresponds to the tensor product of two SRE phases (hence the Abelian addition rules).

Is there a one-to-one correspondence between 3D GSPT phases ${| {{\rm{\Psi }}}_{0}\rangle }_{{{ \mathcal G }}_{y}}$ in (10) and 2D SRE states $| {{\rm{\Psi }}}_{0}\rangle$ with the same on-site symmetry? This naive expectation turns out to be wrong: in general the 3D GSPT classification is a subgroup of the 2D SRE classification with the same on-site symmetry group G₀, for the following reason. While by definition two distinct 3D GSPT phases cannot share the same 2D SRE root state $| {\psi }_{{A}_{0}}\rangle \simeq | {{\rm{\Psi }}}_{0}\rangle$ in their fixed-point wavefunctions, the reverse statement is not true. In other words, two different 2D SRE root phases $| {{\rm{\Psi }}}_{0}\rangle$ and $| {{\rm{\Psi }}}_{0}^{\prime }\rangle$ can lead to the same 3D GSPT phase ${| {{\rm{\Psi }}}_{0}\rangle }_{{{ \mathcal G }}_{y}}\simeq {| {{\rm{\Psi }}}_{0}^{\prime }\rangle }_{{{ \mathcal G }}_{y}}$. For a simplest example, consider a GSPT phase ${| {{\rm{\Psi }}}_{0}\rangle }_{{{ \mathcal G }}_{y}}$ with

$$\begin{eqnarray}&&| {\psi }_{{A}_{n}}\rangle \simeq | {{\rm{\Psi }}}_{0}\rangle ,\,\,\,| {\psi }_{{B}_{n}}\rangle \simeq | {\bar{{\rm{\Psi }}}}_{0}\rangle \end{eqnarray} \tag{ 11 }$$

and another state ${| {\bar{{\rm{\Psi }}}}_{0}\rangle }_{{{ \mathcal G }}_{y}}$ with

$$\begin{eqnarray}&&| {\psi }_{{A}_{n}}^{\prime }\rangle \simeq | {\bar{{\rm{\Psi }}}}_{0}\rangle ,\,\,\,| {\psi }_{{B}_{n}}^{\prime }\rangle \simeq | {{\rm{\Psi }}}_{0}\rangle .\end{eqnarray} \tag{ 12 }$$

Clearly they describe the same GSPT phase, since they merely differ by a (arbitrary) choice of $\{{A}_{n}\}$ versus $\{{B}_{n}\}$ layers. Therefore we have shown that two SRE 2D root phases that are mirror-reflection image of each other will lead to the same GSPT phase. Notice that 2D SRE phases (hence their 3D GSPT counterparts) generally satisfy the following addition rule:

$$\begin{eqnarray}&&| {{\rm{\Psi }}}_{0}\rangle \simeq | {\bar{{\rm{\Psi }}}}_{0}\rangle \oplus (| {{\rm{\Psi }}}_{0}\rangle \oplus | {{\rm{\Psi }}}_{0}\rangle )\Longleftrightarrow | {{\rm{\Psi }}}_{0}\rangle \oplus | {\bar{{\rm{\Psi }}}}_{0}\rangle \simeq | 0\rangle ,\end{eqnarray} \tag{ 13 }$$

where $| 0\rangle$ represents the trivial product state. Now that we have argued ${| {{\rm{\Psi }}}_{0}\rangle }_{{{ \mathcal G }}_{y}}\simeq {| {\bar{{\rm{\Psi }}}}_{0}\rangle }_{{{ \mathcal G }}_{y}}$, an immediate consequence is the following 'Z₂ addition rule' for any 3D GSPT phases:

$$\begin{eqnarray}&&{{\rm{Z}}}_{2}\,{\rm{addition}}\ {\rm{rule}}\ {\rm{of}}\ 3{\rm{d}}\ {\rm{GSPT}}\ {\rm{phases}}:\\ &&\quad {| {{\rm{\Psi }}}_{0}\rangle }_{{{ \mathcal G }}_{y}}\oplus {| {{\rm{\Psi }}}_{0}\rangle }_{{{ \mathcal G }}_{y}}\simeq {| 0\rangle }_{{{ \mathcal G }}_{y}}.\end{eqnarray} \tag{ 14 }$$

In other words, two copies of the same 3D GSPT phases always lead to the trivial product state. This important conclusion for GSPT phases allows us to further reduce 2D SRE classification and to achieve the 3D GSPT classification.

The above Z₂ addition rule (14) for 3D GSPT phases also raises a question: given a fixed-point wavefunction (8) built from 2D SRE root phase $| {\psi }_{{A}_{0}}\rangle =| {{\rm{\Psi }}}_{0}\rangle$, how do we know that it is a nontrivial GSPT phase (i.e. it is not equivalent to the trivial product state $| 0\rangle$) or not? To address this issue we explicitly construct certain topological orders, that symmetrically gap out the glide-invariant [001] side surface ($\hat{x}-\hat{y}$ plane in figure 1) states of the GSPT phase. We further show that these STOs exhibit anomalous symmetry implementations [14], a fingerprint for interacting SPT phases [5] from bulk-boundary correspondence.

The coupled layer construction is particularly powerful for this purpose, when we study interaction effects on the [001] side surface which generally hosts glide-protected gapless surface states for weakly-interacting fermions (e.g. 'hourglass fermions' for ${G}_{0}=U(1)\rtimes {Z}_{4}^{{ \mathcal T }}$). Specifically since each 2D layer intersects with the [001] side surface on its gapless edge, short-range interactions can couple these gapless 1D edge states to form a gapped STO [19, 29, 35, 36, 41, 42]. Such a 'CWC' enables us to construct these 2D STOs and analyze their symmetry properties. The anomalies of STOs can be detected in physical responses of the gapped symmetric surface, such as thermal and electric Hall conductance, or the magnetic flux (in particular π flux) on the surface. On the contrary, the side surface states of a trivial GSPT phase can always be gapped out symmetrically without developing any topological orders. This will be demonstrated in all examples.

As mentioned earlier, we will use the 'CWC' [41, 42] to study anomalous STO on the [001] side surface of a GSPT phase, where each wire is simply the edge states at the intersection of [001] side surface and each 2D SRE layer in the coupled layer construction of 3D GSPT phases. These gapless 1D edge states of 2D SRE bosonic/fermionic phases have been studied extensively in the literature [43, 44]. To construct the anomalous STOs, our strategy is to either couple these 'edge wires' directly, or to deposit extra 1D quantum wires on the side surface and couple them together with the 'edge wires'. This will be demonstrated in the examples.

This CWC for GSPT surface states also provides an alternative argument for the Z₂ addition rule (14) for any bosonic/fermionic GSPT phase, as illustrated in figure 2(b). Consider a 3D GSPT phase ${| {{\rm{\Psi }}}_{0}\rangle }_{{{ \mathcal G }}_{y}}\oplus {| {{\rm{\Psi }}}_{0}\rangle }_{{{ \mathcal G }}_{y}}$, where the protected edge modes of each 2D SRE phase $| {{\rm{\Psi }}}_{0}\rangle$ are labeled by arrowed blue lines in figure 2(b). The edge modes of its mirror-reflection (or time-reversal) image $| {\bar{{\rm{\Psi }}}}_{0}\rangle$ are labeled by red lines with opposite arrows in figure 2(b). Due to addition rule (13) for a pair of 2D SRE phases $\{| {{\rm{\Psi }}}_{0}\rangle ,| {\bar{{\rm{\Psi }}}}_{0}\rangle \}$ that are time-reversal (or mirror-reflection) counterparts of each other, their edge states together (one red plus one blue lines) can be gapped out symmetrically. More concretely, the robust edge states of a 2D SRE phase $| {{\rm{\Psi }}}_{0}\rangle$ can be described by chiral boson fields $\{{\phi }^{I}\}$

$$\begin{eqnarray}&&{{ \mathcal L }}_{{\rm{edge}}}=4\pi \displaystyle \sum _{I,J}{\partial }_{t}{\phi }^{I}(y,t){{\bf{K}}}_{I,J}{\partial }_{y}{\phi }^{J}(y,t)\,+\,\cdots ,\end{eqnarray} \tag{ 15 }$$

where ⋯represents non-universal kinetic energy terms, with commutation relation

$$\begin{eqnarray}&&[{\phi }^{I}({y}^{\prime }),{\partial }_{y}{\phi }^{J}(y)]=2\pi \hspace{1pt}{\rm{i}}\hspace{1pt}{{\bf{K}}}_{I,J}^{-1}\delta (y-{y}^{\prime }).\end{eqnarray} \tag{ 16 }$$

Therefore the gapless surface states on glide-invariant [001] side surface of state ${| {{\rm{\Psi }}}_{0}\rangle }_{{{ \mathcal G }}_{y}}\oplus {| {{\rm{\Psi }}}_{0}\rangle }_{{{ \mathcal G }}_{y}}$ are described by

$$\begin{eqnarray*}&&{{ \mathcal L }}_{0}=\displaystyle \sum _{n\in \tfrac{{\mathbb{Z}}}{2}}\displaystyle \sum _{a=1,2}\displaystyle \frac{{(-1)}^{2n}}{4\pi }\displaystyle \sum _{I,J}{\partial }_{t}{\phi }_{n,a}^{I}(y,t){{\bf{K}}}_{I,J}{\partial }_{y}{\phi }_{n,a}^{J}(y,t)+\cdots .\end{eqnarray*}$$

In particular under glide symmetry operation, the chiral bosons transform as

$$\begin{eqnarray}&&{\phi }_{n,a}^{I}(y)\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}{\phi }_{n+1/2,a}^{I}(-y).\end{eqnarray} \tag{ 17 }$$

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Since $\{{\phi }_{n}^{I}\}$ transform in the same way under global symmetries for $\forall \,n,a$, the following coupling terms that pair up two neighboring edges

$$\begin{eqnarray}&&{{ \mathcal H }}_{{\rm{int}}}=-{C}_{0}\displaystyle \sum _{n\in {\mathbb{Z}}/2}\displaystyle \sum _{I}\cos ({\phi }_{n,2}^{I}-{\phi }_{n+\tfrac{1}{2},1}^{I})\end{eqnarray} \tag{ 18 }$$

trivially gap out the surface states without breaking any symmetry. Therefore the glide-invariant [001] surface states of 3D GSPT phase ${| {{\rm{\Psi }}}_{0}\rangle }_{{{ \mathcal G }}_{y}}\oplus {| {{\rm{\Psi }}}_{0}\rangle }_{{{ \mathcal G }}_{y}}$ can be gapped out without breaking any symmetry, in a pairwise fashion illustrated in figure 2(b). Due to the bulk-boundary correspondence, this 3D GSPT state is equivalent to the trivial product state, reaffirming the Z₂ addition rule (14).

Previously in the coupled layer construction of certain 'weak' SPT phases, their symmetric STOs have been constructed by adding 'plates' of 2D topological orders on the side surface between two neighboring layers, then gapping out the edge states of plates and the layers by certain inter-wire couplings [19, 29, 35]. However in this approach the side surface is by construction not a uniform 2D system, where it is subtle to even define a topological order. This concern motivates us to follow a different approach: instead of depositing 2D plates on the side surfaces, we deposit 1D quantum wires on the side surface between two neighboring layers in a glide-invariant fashion. We further construct symmetric 2D STOs by properly coupling the 1D wires with gapless edge states of each layer.

Below we briefly review the CWC of 2D topological orders, focusing on Abelian Z_k gauge theories that will be frequently encountered in STOs of various GSPT phases. We first show how to construct a 2D Z_k gauge theory by properly coupling a 2D array of boson quantum wires, and then demonstrate the bulk and edge anyons and their fractional statistics in the Z_k gauge theory. The basic ideas and strategy are explained in the appendix. Consider a 2D array of 1D Luttinger liquids $\{{\varphi }_{l},{\theta }_{l}| l\in {\mathbb{Z}}\}$ described by

$$\begin{eqnarray}&&{{ \mathcal L }}_{{Z}_{k}}={{ \mathcal L }}_{0}+{{ \mathcal H }}_{{\rm{int}}},{{ \mathcal L }}_{0}=\displaystyle \frac{1}{2\pi }\displaystyle \sum _{l\in {\mathbb{Z}}}{\partial }_{t}{\varphi }_{l}{\partial }_{y}{\theta }_{l}+\cdots \end{eqnarray} \tag{ 19 }$$

with commutation relation

$$\begin{eqnarray}&&[{\theta }_{{l}_{1}}({y}_{1}),{\varphi }_{{l}_{2}}({y}_{2})]=[{\varphi }_{{l}_{1}}({y}_{1}),{\theta }_{{l}_{2}}({y}_{2})]=\hspace{1pt}{\rm{i}}\hspace{1pt}\pi {\delta }_{{l}_{1},{l}_{2}}\mathrm{Sign}({y}_{1}-{y}_{2}).\end{eqnarray} \tag{ 20 }$$

Physically each quantum wire can be realized by e.g. the Luttinger liquid of a 1D gapless 'XYX' spin chain ${\hat{H}}_{{XYX}}=-{J}_{1}{\sum }_{r}({S}_{r}^{x}{S}_{r+1}^{x}+{S}_{r}^{z}{S}_{r+1}^{z})$, where physical observables can be expressed in terms of boson fields $\{\varphi (r),\theta (r)\}$ as

$$\begin{eqnarray}&&{S}^{+}\equiv {S}^{z}+\hspace{1pt}{\rm{i}}\hspace{1pt}{S}^{x}\sim {{\rm{e}}}^{{\rm{i}}\varphi },\,\,\,{\rho }_{{S}^{y}}\sim \displaystyle \frac{{\partial }_{r}\theta (r)}{2\pi }.\end{eqnarray} \tag{ 21 }$$

Clearly under on-site U(1) charge rotation ($\hat{Q}$ labels the total charge) and time reversal operation, the boson fields $\{{\varphi }_{l},{\theta }_{l}| l\in {\mathbb{Z}}\}$ transform as

$$\begin{eqnarray}&&\left(\begin{array}{c}{\varphi }_{l}\\ {\theta }_{l}\end{array}\right)\mathop{\longrightarrow }\limits^{{{\rm{e}}}^{{\rm{i}}\alpha \hat{Q}}}\left(\begin{array}{c}{\varphi }_{l}\\ {\theta }_{l}\end{array}\right),\,\,\,\left(\begin{array}{c}{\varphi }_{l}\\ {\theta }_{l}\end{array}\right)\mathop{\longrightarrow }\limits^{{ \mathcal T }}\left(\begin{array}{c}{\varphi }_{l}+\pi \\ -{\theta }_{l}\end{array}\right).\end{eqnarray} \tag{ 22 }$$

The following inter-wire terms

$$\begin{eqnarray}&&{{ \mathcal H }}_{{\rm{int}}}=-\displaystyle \sum _{l\in {\mathbb{Z}}}{C}_{l}\cos {\hat{{\rm{\Theta }}}}_{l},\,\,\,{{\rm{\Theta }}}_{l}={\varphi }_{l-1}-k{\theta }_{l}-{\varphi }_{l+1}\end{eqnarray} \tag{ 23 }$$

gap out all wires and lead to an Abelian topological order, described by Abelian Chern–Simon theory with ${\bf{K}}$ matrix

$$\begin{eqnarray}{{\bf{K}}}_{{Z}_{k}}=\left(\begin{array}{cc}0 & k\\ k & 0\end{array}\right).\end{eqnarray} \tag{ 24 }$$

This is the CWC of 2D Z_k gauge theory.

The anyons in this 2D topological order can be identified both on the edge or in the bulk. To see this, we first consider an open boundary between the $(2l-1)$th and $(2l)$th wire by setting ${C}_{2l-1}={C}_{2l}=0$. Zero-modes on the right (or left) edge, i.e. vertex operators commuting with all cosine terms correspond to the anyons on the edge, in this case

$$\begin{eqnarray}&&{{\rm{e}}}^{{\rm{i}}{\phi }_{l}^{{e}}}={{\rm{e}}}^{{\rm{i}}{\varphi }_{2l}/k}\sim {{\rm{e}}}^{{\rm{i}}(\tfrac{1}{k}{\varphi }_{2l-2}-{\theta }_{2l-1})},{{\rm{e}}}^{{\rm{i}}{\phi }_{l}^{m}}={{\rm{e}}}^{{\rm{i}}({\theta }_{2l}+\tfrac{1}{k}{\varphi }_{2l+1})}\sim {{\rm{e}}}^{{\rm{i}}{\varphi }_{2l-1}/k}.\end{eqnarray} \tag{ 25 }$$

We use e and m to label the gauge charge and gauge flux of a Z_k gauge theory, and their fractional (mutual) statistics is indicated by commutation relation

$$\begin{eqnarray}&&[{\phi }_{l}^{e}({y}_{1}),{\phi }_{l}^{m}({y}_{2})]=\hspace{1pt}{\rm{i}}\hspace{1pt}\displaystyle \frac{\pi }{k}\mathrm{Sign}({y}_{1}-{y}_{2}).\end{eqnarray} \tag{ 26 }$$

Meanwhile on a closed manifold with no boundary (${C}_{l}\ne 0,\,\forall \,l$), anyons ${{\rm{e}}}^{{\rm{i}}{\phi }_{l}^{e/m}}$ in (25) are nothing but kinks [42] of cosine terms C_2l and ${C}_{2l-1}$, which create $2\pi$ phase slips in the arguments ${{\rm{\Theta }}}_{2l},{{\rm{\Theta }}}_{2l-1}$ of associated cosine terms One anyon of type a can hop from one wire to another across the bulk, via the following hopping operators

$$\begin{eqnarray}&&{T}_{{l}_{1},{l}_{2}}^{a}={\phi }_{{l}_{2}}^{a}-{\phi }_{{l}_{1}}^{a}+\displaystyle \sum _{l}{t}_{l}{{\rm{\Theta }}}_{l},\,\,\,{t}_{l}^{a}\in {\mathbb{R}}\end{eqnarray} \tag{ 27 }$$

consisting of a string of local operators. For example in our case of Z_k gauge theory (23), the anyon hopping operators are

$$\begin{eqnarray}&&{T}_{l,l+1}^{e}=-{\theta }_{2l+1},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{t}_{2l+1}^{e}=\displaystyle {\textstyle \tfrac{1}{k}};{T}_{l,l+1}^{m}=-{\theta }_{2l},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{t}_{2l+2}^{m}=\displaystyle {\textstyle \tfrac{1}{k}}.\end{eqnarray} \tag{ 28 }$$

As discussed in the appendix, the braiding statistics of anyons can be easily computed based on the anyon hopping operators, and the results are consistent with (26) for the edge anyons.

The construction of STO is very similar to (23) for 2D Z_k gauge theory, where anyons and their statistics can be identified as reviewed in the appendix and demonstrated above. What makes the STO anomalous is one important difference: a part (sometimes all) of the 1D quantum wires $\{{\varphi }_{l},{\theta }_{l}\}$ in (19) will be replaced by the protected edge states of nontrivial 2D SRE phases, whose symmetry implementations cannot be realized in a pure 1D system. This replacement is crucial for realizing an anomalous STO respecting glide symmetry, and will be encountered in many examples.

While the strategy established in this section applies to all global symmetries, in this paper we mostly focus on examples with on-site symmetry group G₀ generated by U(1) charge/spin symmetry and/or time reversal symmetry ${ \mathcal T }$. In particular, we consider 5 on-site symmetry groups for bosonic GSPT phases in section 3

$$\begin{eqnarray}&&\mathrm{No}\ \mathrm{on} \mbox{-} \mathrm{site}\ \mathrm{symmetry}:\,\,{G}_{0}={Z}_{1},\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}&&\mathrm{Time}\ \mathrm{reversal}\ \mathrm{symmetry}:\,\,{G}_{0}={Z}_{2}^{{ \mathcal T }},\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}&&U(1)\,\mathrm{symmetry}:\,\,{G}_{0}=U(1),\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&U(1)\,\mathrm{charge}\ \mathrm{and}\ \mathrm{time}\ \mathrm{reversal}:\,\,{G}_{0}=U(1)\rtimes {Z}_{2}^{{ \mathcal T }},\end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray}&&U(1)\,\mathrm{spin}\ \mathrm{and}\ \mathrm{time}\ \mathrm{reversal}:\,\,{G}_{0}=U(1)\times {Z}_{2}^{{ \mathcal T }}\end{eqnarray} \tag{ 33 }$$

for 3D GSPT phases of interacting bosons. The results are summarized in table 1. For all bosonic GSPT phases studied here, it turns out that their surface states can always be gapped symmetrically by Abelian Z₂ topological orders, with 4-fold ground state degeneracy on torus. The anomalies of these STOs are also summarized in table 1.

The situation is more complicated for interacting fermions, since the microscopic fermions can either be half-integer-spin Kramers doublets with ${{ \mathcal T }}^{2}={{\bf{P}}}_{{\rm{f}}}$ where ${{\bf{P}}}_{{\rm{f}}}={(-1)}^{\hat{F}}$ represents the fermion parity; or integer-spin Kramers singlets with ${{ \mathcal T }}^{2}=1$. Therefore the above five cases for interacting bosons will lead to seven cases for interacting fermions discussed in section 4, where the on-site symmetry can be categorized into Altland-Zirnbauer (AZ) 10-fold way [45]. The corresponding results are summarized in table 2. Unlike in bosonic GSPT case, in certain fermionic GSPT phases, non-Abelian topological orders with a non-integer chiral central charge ${c}_{-}$ of their edge states are necessary to symmetrically gap out the surface states. Examples with these non-Abelian STOs include topological superconductors (TSC) in AZ symmetry class D and DIII, and TI in class A (see table 2).

In the absence of any on-site symmetry, 2D SRE phases of bosons are classified and characterized by an integer-valued index $\nu \in {\mathbb{Z}}$. The 'generator' of this integer group, i.e. corresponding 2D root state is the E₈ state [39, 43]. The definitive physical properties of E₈ state is quantized thermal Hall conductance [46] ${\kappa }_{{\rm{H}}}/T=\tfrac{{\pi }^{2}}{3}\tfrac{{k}_{B}^{2}}{h}{c}_{-}$ with chiral central charge ${c}_{-}=8$. In particular the chiral edge states of E₈ state is described by Lagrangian density

$$\begin{eqnarray}{{ \mathcal L }}_{{\rm{edge}}} & = & \displaystyle \frac{1}{4\pi }\displaystyle \sum _{I,J}{\partial }_{t}{\phi }^{I}(y,t){{\bf{K}}}_{{E}_{8}}^{I,J}{\partial }_{y}{\phi }^{J}(y,t)\,+\,\cdots ,\\ {{\bf{K}}}_{{E}_{8}} & = & \left(\begin{array}{cccccccc}2 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & -1\\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 2\end{array}\right)={{\rm{\Gamma }}}_{8}\cdot {{\rm{\Gamma }}}_{8}^{T},\end{eqnarray} \tag{ 34 }$$

where '⋯' represents non-universal kinetic energy terms ${{\bf{K}}}_{{E}_{8}}$ is the Cartan matrix of Lie group E₈ and

$$\begin{eqnarray}{{\rm{\Gamma }}}_{8}=\left(\begin{array}{cccccccc}1 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0\\ -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2}\\ 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0\end{array}\right)\end{eqnarray} \tag{ 35 }$$

is the simple root matrix of E₈ group. $\{{\phi }^{I}\}$ are chiral boson operators satisfying commutation relation

$$\begin{eqnarray}&&[{\phi }^{I}({y}^{\prime }),{\partial }_{y}{\phi }^{J}(y)]=2\pi \hspace{1pt}{\rm{i}}\hspace{1pt}{{\bf{K}}}_{I,J}^{-1}\delta (y-{y}^{\prime }).\end{eqnarray} \tag{ 36 }$$

According to Z₂ addition rule (14), we know that two copies of E₈ states ($\nu =\pm 2$) in each layer lead to the trivial product state i.e. ${| {E}_{8}\rangle }_{{{ \mathcal G }}_{y}}\oplus {| {E}_{8}\rangle }_{{{ \mathcal G }}_{y}}\simeq {| 0\rangle }_{{{ \mathcal G }}_{y}}$. It turns out a single E₈ state in each layer leads to a nontrivial GSPT phase ${| {E}_{8}\rangle }_{{{ \mathcal G }}_{y}}\ne {| 0\rangle }_{{{ \mathcal G }}_{y}}$, resulting in a ${{\mathbb{Z}}}_{2}$ classification of GSPT phases with no on-site symmetry.

Below we explicitly construct anomalous STOs on the [001] side surface of ${| {E}_{8}\rangle }_{{{ \mathcal G }}_{y}}$ state, in the framework of CWC. We label the E₈ chiral bosons from the 'edge wire' of A_n layer by $\{{\phi }_{n}^{I}| n\in {\mathbb{Z}}\}$, and those of B_n layer by $\{{\phi }_{n}^{I}| n\in {\mathbb{Z}}+\tfrac{1}{2}\}$. The Lagrangian density for decoupled wires on [001] side surface is given by

$$\begin{eqnarray}&&{{ \mathcal L }}_{0}=\displaystyle \sum _{n\in {\mathbb{Z}}/2}\displaystyle \frac{{(-1)}^{2n}}{4\pi }\displaystyle \sum _{I,J=1}^{8}{\partial }_{t}{\phi }_{n}^{I}(y,t){{\bf{K}}}_{{E}_{8}}^{I,J}{\partial }_{y}{\phi }_{n}^{J}(y,t)+\cdots .\end{eqnarray} \tag{ 37 }$$

Clearly under glide symmetry operation, the chiral bosons transform as

$$\begin{eqnarray}&&{\phi }_{n}^{I}(y)\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}{\phi }_{n+1/2}^{I}(-y).\end{eqnarray} \tag{ 38 }$$

The root matrix ${{\rm{\Gamma }}}_{8}$ defines a basis change of chiral bosons [35, 47] with simple commutation relations:

$$\begin{eqnarray}&&{\tilde{\phi }}_{n}^{I}\equiv \displaystyle \sum _{J}{({{\rm{\Gamma }}}_{8})}_{J,I}{\phi }_{n}^{J},[{\tilde{\phi }}_{n}^{I}({y}^{\prime }),{\partial }_{y}{\tilde{\phi }}_{m}^{J}(y)]={(-1)}^{2n}2\pi \hspace{1pt}{\rm{i}}\hspace{1pt}{\delta }_{I,J}{\delta }_{m,n}\delta (y-{y}^{\prime }).\end{eqnarray} \tag{ 39 }$$

In other words, each column of matrix ${{\rm{\Gamma }}}_{8}$ corresponds to a chiral fermion mode on the edge of a ${\sigma }_{{xy}}=1$ integer quantum Hall (IQH) state. These 'chiral fermions' are however non-local operators due to the non-integer ($-\tfrac{1}{2}$) entries in matrix ${{\rm{\Gamma }}}_{8}$. To construct a fully-symmetric STO out of these 'edge wires', our strategy is to couple the chiral fermions $\{{\tilde{\phi }}_{n}^{I}\}$ in pairs between two neighboring wires, as illustrated in figure 2(a). Specifically we consider the following glide-invariant inter-wire couplings (with positive strength ${C}_{I}\gt 0$):

$$\begin{eqnarray}&&{{ \mathcal H }}_{{\rm{int}}}=-\displaystyle \sum _{n\in {\mathbb{Z}}/2}\displaystyle \sum _{I=1}^{4}{C}_{I}\cos \left[\displaystyle \sum _{J=1}^{4}{{\bf{M}}}_{I,J}({\tilde{\phi }}_{n}^{J}+{\tilde{\phi }}_{n+\tfrac{1}{2}}^{J+4})\right]\end{eqnarray} \tag{ 40 }$$

which pair up counter-propagating chiral fermions ${\tilde{\phi }}_{n}^{J}$ and ${\tilde{\phi }}_{n+\tfrac{1}{2}}^{J+4}$. A proper choice of 4 × 4 matrix ${\bf{M}}$

$$\begin{eqnarray}{\bf{M}}=\left(\begin{array}{cccc}1 & 1 & 0 & 0\\ 1 & -1 & 0 & 0\\ 0 & 1 & -1 & 0\\ 0 & 0 & 1 & -1\end{array}\right)\end{eqnarray} \tag{ 41 }$$

can make (40) a local Hamiltonian that fully gap out the surface while preserving glide symmetry (38). Note that all cosine terms commute with each other, and can be minimized simultaneously. The argument of each cosine term is hence pinned at certain classical minimum in the the ground state. Bulk anyons correspond to kinks of these cosine terms [42]. Following the strategy outlined in appendix, it is straightforward to identify the inequivalent bulk anyons:

$$\begin{eqnarray}{e}\sim {{\rm{e}}}^{\tfrac{{\rm{i}}}{2}\displaystyle \sum _{I=1}^{4}{\tilde{\phi }}_{n}^{I}} & = & {{\rm{e}}}^{{\rm{i}}(\tfrac{{\phi }_{n}^{4}}{2}-{\phi }_{n}^{7})},m\sim {{\rm{e}}}^{\tfrac{{\rm{i}}}{2}(\displaystyle \sum _{I=2}^{4}{\tilde{\phi }}_{n}^{I}-{\tilde{\phi }}_{n}^{1})}={{\rm{e}}}^{{\rm{i}}(\tfrac{{\phi }_{n}^{4}-{\phi }_{n}^{7}}{2}-{\phi }_{n}^{1})},\\ \epsilon \sim {{\rm{e}}}^{{\rm{i}}{\tilde{\phi }}_{I}^{4}} & = & {{\rm{e}}}^{{\rm{i}}({\phi }_{n}^{4}-{\phi }_{n}^{3}-\tfrac{{\phi }_{n}^{7}}{2})}.\end{eqnarray} \tag{ 42 }$$

These Abelian anyons can also be obtained from the gapless edge states of the STO, by cutting an edge between A_n and B_n layer. All three anyons obey fermi self-statistics and satisfy the following ${Z}_{2}\times {Z}_{2}$ fusion rule:

$$\begin{eqnarray}&&e\times e=m\times m=1,\end{eqnarray} \tag{ 43 }$$

$$\begin{eqnarray}&&\epsilon =e\times m.\end{eqnarray} \tag{ 44 }$$

They also obey mutual semion statistics, i.e. ${{\rm{e}}}^{{\rm{i}}\pi }=-1$ phase when braiding e around m once. Such a STO is known as Z₂ topological order 'efmf' [6, 8], described by the following ${\bf{K}}$ matrix (Cartan matrix of Lie group SO(8))

$$\begin{eqnarray}{{\bf{K}}}_{{SO}(8)}=\left(\begin{array}{cccc}2 & -1 & -1 & -1\\ -1 & 2 & 0 & 0\\ -1 & 0 & 2 & 0\\ -1 & 0 & 0 & 2\end{array}\right)\end{eqnarray} \tag{ 45 }$$

in the Abelian Chern–Simons theory framework. A pure 2D efmf phase have chiral central charge ${c}_{-}=4$ and necessarily breaks orientation-reversing glide (and PT) symmetry. Therefore our glide-invariant efmf STO is anomalous and can only be realized on the surface of 3D GSPT phase ${| {E}_{8}\rangle }_{{{ \mathcal G }}_{y}}$. Therefore we have justified that ${| {E}_{8}\rangle }_{{{ \mathcal G }}_{y}}$ is a nontrivial GSPT phase, and hence the ${{\mathbb{Z}}}_{2}$ classification of GSPT phase with no on-site symmetry.

With on-site time reversal symmetry, E₈ states are forbidden and 2D bosonic SRE phases have a trivial classification [15] ${H}^{3}({Z}_{2}^{{ \mathcal T }},U(1))={{\mathbb{Z}}}_{1}$. With no nontrivial 2D root phases, the classification of 3D GSPT phases with ${G}_{0}={Z}_{2}^{{ \mathcal T }}$ is hence also trivial, as shown in table 1.

In the presence of on-site U(1) charge (or spin) symmetry, 2D SRE boson phases have a ${\mathbb{Z}}\times {\mathbb{Z}}$ classification [43], labeled by a pair of integers $\{\nu \in {\mathbb{Z}},q\in {\mathbb{Z}}\}$. Physically the pair of integers correspond to chiral central charge and Hall conductance of a 2D SRE phase:

$$\begin{eqnarray}&&{c}_{-}=8\nu ,\,\,\,\,\,{\sigma }_{{xy}}=2q\displaystyle \frac{{Q}_{0}^{2}}{h},\end{eqnarray} \tag{ 46 }$$

where Q₀ is the fundamental charge carried by one boson. Physically a state $\{\nu ,q\}$ can be obtained by stacking ν copies of E₈ states of charge-neutral bosonic excitons, and q copies of bosonic integer quantum Hall (BIQH) states [43, 48] of charge-Q₀ bosons. These 2D SRE phases labeled as $[\nu ,q]$ satisfy the following addition rule:

$$\begin{eqnarray}&&[{\nu }_{1},{q}_{1}]\oplus [{\nu }_{2},{q}_{2}]=[{\nu }_{1}+{\nu }_{2},{q}_{1}+{q}_{2}].\end{eqnarray} \tag{ 47 }$$

Two generators of these 2D U(1)-symmetric SRE phases are the neutral E₈ state $[\nu =1,q=0]$ and the BIQH state $[\nu =0,q=1]$. Their associated 3D GSPT phases are hence generated by ${| \mathrm{1,}0\rangle }_{{{ \mathcal G }}_{y}}={| {E}_{8}\rangle }_{{{ \mathcal G }}_{y}}$ and ${| \mathrm{0,1}\rangle }_{{{ \mathcal G }}_{y}}={| \mathrm{BIQH}\rangle }_{{{ \mathcal G }}_{y}}$, satisfying ${Z}_{2}\times {Z}_{2}$ fusion rule (14) i.e.

$$\begin{eqnarray}&&{| \nu \in 2{\mathbb{Z}},q\rangle }_{{{ \mathcal G }}_{y}}\simeq {| \nu ,q\in 2{\mathbb{Z}}\rangle }_{{{ \mathcal G }}_{y}}\simeq {| 0\rangle }_{{{ \mathcal G }}_{y}}.\end{eqnarray} \tag{ 48 }$$

Previously in section 3.1 we have shown that ${| {E}_{8}\rangle }_{{{ \mathcal G }}_{y}}={| \mathrm{1,0}\rangle }_{{{ \mathcal G }}_{y}}$ is a nontrivial GSPT phase with anomalous STOs like efmf. Below we show that ${| \mathrm{BIQH}\rangle }_{{{ \mathcal G }}_{y}}={| \mathrm{0,1}\rangle }_{{{ \mathcal G }}_{y}}$ is also a nontrivial GSPT phase by constructing its anomalous STO, and therefore establish the ${{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2}$ classification of 3D GSPT phases with on-site U(1) symmetry.

The edge states of 2D BIQH root state $| \mathrm{BIQH}\rangle =| 0,1\rangle$ is described by [43] chiral bosons $\{{\phi }^{I}| I=1,2\}$

$$\begin{eqnarray}&&{{ \mathcal L }}_{{\rm{edge}}}=\displaystyle \frac{1}{4\pi }\displaystyle \sum _{I,J=1,2}{\partial }_{t}{\phi }^{I}{{\bf{K}}}_{I,J}{\partial }_{y}{\phi }^{J}\,+\,\cdots ,\end{eqnarray} \tag{ 49 }$$

with ${\bf{K}}$ matrix and charge vector ${\bf{q}}$

$$\begin{eqnarray}{\bf{K}}=\left(\begin{array}{cc}0 & 1\\ 1 & 0\end{array}\right),\,\,\,{{\bf{q}}}_{\mathrm{BIQH}}=\left(\begin{array}{c}1\\ 1\end{array}\right).\end{eqnarray} \tag{ 50 }$$

The chiral bosons transform under U(1) charge/spin rotation (by angle α) as

$$\begin{eqnarray}&&\vec{\phi }\equiv \left(\begin{array}{c}{\phi }^{1}\\ {\phi }^{2}\end{array}\right)\mathop{\longrightarrow }\limits^{{{\rm{e}}}^{{\rm{i}}\alpha \hat{Q}}}\vec{\phi }+\alpha \cdot {{\bf{K}}}^{-1}{{\bf{q}}}_{\mathrm{BIQH}}=\left(\begin{array}{c}{\phi }^{1}+\alpha \\ {\phi }^{2}+\alpha \end{array}\right).\end{eqnarray} \tag{ 51 }$$

Now we consider the glide-invariant [001] side surface of 3D GSPT phase ${| \mathrm{BIQH}\rangle }_{{{ \mathcal G }}_{y}}={| \nu =0,q=1\rangle }_{{{ \mathcal G }}_{y}}$. Gapless edge states of BIQH states will appear at the intersection of each 2D plane $\{{A}_{n},{B}_{n}\}$ in (10) and the side surface. We label these gapless chiral bosons from A_n layer by $\{{\phi }_{n}^{I}| n\in {\mathbb{Z}}\}$, and those from B_n layer by $\{{\phi }_{n+\tfrac{1}{2}}^{I}| n\in {\mathbb{Z}}\}$, illustrated by the 'edge wires' in figure 4. These gapless surface states are described by the following glide-invariant Lagrangian density

$$\begin{eqnarray}&&{{ \mathcal L }}_{0}=\displaystyle \sum _{n\in {\mathbb{Z}}/2}\displaystyle \frac{{(-1)}^{2n}}{4\pi }\displaystyle \sum _{I,J}{\partial }_{t}{\phi }_{n}^{I}{{\bf{K}}}_{I,J}{\partial }_{y}{\phi }_{n}^{J}+\cdots \end{eqnarray} \tag{ 52 }$$

with commutation relation

$$\begin{eqnarray}&&[{\phi }_{m}^{I}(x),{\phi }_{n}^{J}(y)]={(-1)}^{2n}\pi \hspace{1pt}{\rm{i}}\hspace{1pt}{{\bf{K}}}_{I,J}^{-1}{\delta }_{m,n}\mathrm{Sign}(x-y).\end{eqnarray} \tag{ 53 }$$

The anomalous STO on side surface of ${| \mathrm{BIQH}\rangle }_{{{ \mathcal G }}_{y}}$ is realized by the following CWC:

$$\begin{eqnarray}&&{{ \mathcal L }}_{{\rm{STO}}}={{ \mathcal L }}_{0}+{{ \mathcal H }}_{{\rm{int}}},\end{eqnarray} \tag{ 54 }$$

where ${{ \mathcal L }}_{0}$ is given in (52), and the inter-wire coupling terms are given by

$$\begin{eqnarray}&&{{ \mathcal H }}_{{\rm{int}}}={C}_{0}\displaystyle \sum _{n\in \tfrac{{\mathbb{Z}}}{2}}\cos ({\hat{L}}_{n}),\,\,\,{\hat{L}}_{n}={\phi }_{n-\tfrac{1}{2}}^{1}-2{\phi }_{n}^{2}+{\phi }_{n+\tfrac{1}{2}}^{1},\end{eqnarray} \tag{ 55 }$$

where C₀ is a real constant. They preserve both on-site U(1) and glide symmetries. These inter-wire terms are illustrated in figure 3, where every three neighboring wires are coupled together. Again all cosine terms commute with each other and hence can be minimized simultaneously.

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What type of STO is developed out of inter-wire couplings (55)? This issue can be addressed in two approaches. One way is to identify the 2D 'bulk' anyons in the above coupled-wire model and their hopping operators, as outlined in the appendix. One can easily identify three types of inequivalent anyons:

$$\begin{eqnarray}{e}\sim {{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{{\rm{e}}}} & = & {{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{1}/2},\,\,\,m\sim {{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{m}}={{\rm{e}}}^{{\rm{i}}(\tfrac{1}{2}{\phi }_{n+\tfrac{1}{2}}^{1}-{\phi }_{n}^{2})},\\ \epsilon & = & {e}\times m\sim {{\rm{e}}}^{{\rm{i}}({\phi }_{n}^{{\rm{e}}}+{\phi }_{n}^{m})}={{{\rm{e}}}^{{\rm{i}}}}^{(\tfrac{{\phi }_{n+\tfrac{1}{2}}^{1}+{\phi }_{n}^{1}}{2}-{\phi }_{n}^{2})},\,\,n\in {\mathbb{Z}}\end{eqnarray} \tag{ 56 }$$

that can freely hop in the STO. From the U(1) symmetry transformation rules (51), both e and m anyons carry half-integer charges of U(1) symmetry. It is also straightforward to show that they satisfy the ${Z}_{2}\times {Z}_{2}$ fusion rules in (44), and mutual semion statistics between two different anyons. Both e and m obey bose self-statistics, while obeys fermi self-statistics. Another approach to identify the STO is to consider a fictitious (since the surface of a 3D system is not 'edgable') open edge of the 2D STO, by cutting a boundary between any two wires. For example for the boundary between wires ${\phi }_{n-\tfrac{1}{2}}^{1,2}$ and ${\phi }_{n}^{1,2}$, the gapless edge modes on the right edge are given by $\{{\phi }_{n}^{e},{\phi }_{n}^{m}\}$ in (56). The ${\bf{K}}$ matrix and charge vector ${\bf{q}}$ for the STO are given by

$$\begin{eqnarray}{{\bf{K}}}_{{\rm{STO}}}=-\left(\begin{array}{cc}0 & 2\\ 2 & 0\end{array}\right),\,\,\,{{\bf{q}}}_{{\rm{STO}}}=\left(\begin{array}{c}1\\ -1\end{array}\right)\mathrm{mod}2.\end{eqnarray} \tag{ 57 }$$

The charge vector is defined only $\mathrm{mod}2$, since one can always redefine the anyon e (or m) by combining a local boson with it, which changes the anyon charge by 1 without affecting fractional statistics.

Therefore the STO on top of GSPT phase ${| \mathrm{QHE}\rangle }_{{{ \mathcal G }}_{y}}$ is a Z₂ topological order of toric code type [49]. The anomaly of this toric code STO lies in its non-vanishing Hall conductance:

$$\begin{eqnarray}&&{\sigma }_{{xy}}^{{\rm{STO}}}={{\bf{q}}}_{{\rm{STO}}}^{T}{{\bf{K}}}_{{\rm{STO}}}^{-1}{{\bf{q}}}_{{\rm{STO}}}=1\mathrm{mod}2\ne 0.\end{eqnarray} \tag{ 58 }$$

Such a PT-breaking nonzero Hall conductance contradicts the orientation-reversing glide symmetry, and hence cannot be realized in a pure 2D system. It characterizes the anomalous glide symmetry implementation on the [001] side surface of GSPT phase ${| \mathrm{BIQH}\rangle }_{{{ \mathcal G }}_{y}}$. This anomalous STO is known as eCmC [6, 8] in the context of 3D SPT phase with $U(1)\rtimes {Z}_{2}^{{ \mathcal T }}$ symmetry.

As mentioned in section 2.3, indeed the construction of STO (55) is similar to a pure 2D Z₂ gauge theory (23), except that the microscopic building blocks are replaced from 1D spin chains (19) for usual Z₂ gauge theory to the 2D U(1)-SPT edge states (52). This difference leads to the anomaly of the eCmC STO here.

After identifying the anomalous STO of 3D GSPT phase ${| \mathrm{BIQH}\rangle }_{{{ \mathcal G }}_{y}}$, we have demonstrated that ${| \mathrm{BIQH}\rangle }_{{{ \mathcal G }}_{y}}$ is a nontrivial 3D GSPT phase with on-site U(1) symmetry. Therefore we established the ${{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2}$ classification of 3D GSPT phase with on-site symmetry group ${G}_{0}=U(1)$, as summarized in table 1. These are bosonic analogs of glide-protected TIs [21, 23], generated by two root phases ${| {E}_{8}\rangle }_{{{ \mathcal G }}_{y}}$ and ${| \mathrm{BIQH}\rangle }_{{{ \mathcal G }}_{y}}$, characterized by anomalous STOs efmf and eCmC respectively.

With both U(1) charge and ${{\mathbb{Z}}}_{2}^{{ \mathcal T }}$ time reversal symmetries, the 2D SRE phases are classified by ${H}^{3}(U(1)\rtimes {Z}_{2}^{{ \mathcal T }},U(1))={{\mathbb{Z}}}_{2}$. Note that E₈ states with chiral edge modes necessarily break time reversal and hence are forbidden here. The 2D root state for this ${{\mathbb{Z}}}_{2}$ classification is the bosonic quantum spin Hall (BQSH) state [43, 50] $| \mathrm{BQSH}\rangle$. Its protected edge states are still described by Lagrangian (49), with the same ${\bf{K}}$ matrix but a different charge vector

$$\begin{eqnarray}{\bf{K}}=\left(\begin{array}{cc}0 & 1\\ 1 & 0\end{array}\right),\,\,\,{{\bf{q}}}_{\mathrm{BQSH}}=\left(\begin{array}{c}1\\ 0\end{array}\right).\end{eqnarray} \tag{ 59 }$$

The edge chiral bosons $\{{\phi }^{\mathrm{1,2}}\}$ transform under symmetries as

$$\begin{eqnarray}&&\left(\begin{array}{c}{\phi }^{1}\\ {\phi }^{2}\end{array}\right)\mathop{\longrightarrow }\limits^{{{\rm{e}}}^{{\rm{i}}\alpha \hat{Q}}}\left(\begin{array}{c}{\phi }^{1}\\ {\phi }^{2}+\alpha \end{array}\right),\end{eqnarray} \tag{ 60 }$$

$$\begin{eqnarray}&&\left(\begin{array}{c}{\phi }^{1}\\ {\phi }^{2}\end{array}\right)\mathop{\longrightarrow }\limits^{{ \mathcal T }}\left(\begin{array}{c}{\phi }^{1}+\pi \\ -{\phi }^{2}\end{array}\right).\end{eqnarray} \tag{ 61 }$$

From general arguments in section 2, the 3D GSPT phases with on-site ${G}_{0}=U(1)\rtimes {Z}_{2}^{{ \mathcal T }}$ symmetry have at most a ${{\mathbb{Z}}}_{2}$ classification from its 2D root state $| \mathrm{BQSH}\rangle$. The question is, is the 3D state ${| \mathrm{BQSH}\rangle }_{{{ \mathcal G }}_{y}}$ out of the coupled layer construction a nontrivial GSPT or not? The answer is yes. Below we construct and analyze the anomalous STO on [001] side surface of ${| \mathrm{BQSH}\rangle }_{{{ \mathcal G }}_{y}}$ state, therefore establishing the ${{\mathbb{Z}}}_{2}$ classification of 3D GSPT phases with onsite $U(1)\rtimes { \mathcal T }$ symmetry.

We construct the gapped symmetric STO by depositing a 1D quantum wire, i.e. 1D spin chains described by (20) and (21), between every two neighboring layers (see figure 4), and coupling all gapless modes on the side surface in proper ways. In particular as illustrated in figure 4, we label these bosonic modes in each spin chain by $({\varphi }_{n},{\theta }_{n})$ with Lagrangian density

$$\begin{eqnarray}&&{{ \mathcal L }}_{1}=\displaystyle \frac{1}{2\pi }\displaystyle \sum _{n\in {\mathbb{Z}}/2}{\partial }_{t}{\varphi }_{n}{\partial }_{y}{\theta }_{n}+\cdots \end{eqnarray} \tag{ 62 }$$

and commutation relation

$$\begin{eqnarray}&&[{\varphi }_{m}(x),{\theta }_{n}(y)]=[{\theta }_{n}(x),{\varphi }_{m}(y)]=\pi \hspace{1pt}{\rm{i}}\hspace{1pt}{\delta }_{m,n}\mathrm{Sign}(x-y).\,\,\,\end{eqnarray} \tag{ 63 }$$

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After depositing the spin chains between two neighboring layers, the whole surface states are described by

$$\begin{eqnarray}&&{{ \mathcal L }}_{{\rm{STO}}}={{ \mathcal L }}_{0}+{{ \mathcal L }}_{1}+{{ \mathcal H }}_{{\rm{int}}},\end{eqnarray} \tag{ 64 }$$

where ${{ \mathcal L }}_{0}$ given in (52) describes the edge states of BQSH layers. Under U(1) and time reversal symmetries, chiral boson fields of the BQSH edge states ${\phi }_{n}^{1,2}$ and deposited spin chains $({\varphi }_{n},{\theta }_{n})$ transform as

$$\begin{eqnarray}&&\left(\begin{array}{c}{\phi }_{n}^{1}\\ {\phi }_{n}^{2}\\ {\varphi }_{n}\\ {\theta }^{n}\end{array}\right)\mathop{\longrightarrow }\limits^{{{\rm{e}}}^{{\rm{i}}\alpha \hat{Q}}}\left(\begin{array}{c}{\phi }_{n}^{1}\\ {\phi }_{n}^{2}+\alpha \\ {\varphi }_{n}\\ {\theta }^{n}\end{array}\right),\,\,\,\left(\begin{array}{c}{\phi }_{n}^{1}\\ {\phi }_{n}^{2}\\ {\varphi }_{n}\\ {\theta }^{n}\end{array}\right)\mathop{\longrightarrow }\limits^{{ \mathcal T }}\left(\begin{array}{c}{\phi }_{n}^{1}+\pi \\ -{\phi }_{n}^{2}\\ {\varphi }_{n}+\pi \\ -{\theta }^{n}\end{array}\right).\end{eqnarray} \tag{ 65 }$$

Under glide symmetry operation, these chiral boson fields transform as

$$\begin{eqnarray}&&\left(\begin{array}{c}{\phi }_{n}^{1}\\ {\phi }_{n}^{2}\\ {\varphi }_{n}\\ {\theta }_{n}\end{array}\right)\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}\left(\begin{array}{c}{\phi }_{n+\tfrac{1}{2}}^{1}\\ {\phi }_{n+\tfrac{1}{2}}^{2}\\ {\varphi }_{n+\tfrac{1}{2}}\\ -{\theta }_{n+\tfrac{1}{2}}\end{array}\right).\end{eqnarray} \tag{ 66 }$$

As shown in figure 4, we consider the following interwire terms that couple three neighboring 1D Luttinger liquids together:

$$\begin{eqnarray}{{ \mathcal H }}_{{\rm{int}}} & = & \displaystyle \sum _{n\in {\mathbb{Z}}/2}{C}_{0}{\rm{\cos }}{\hat{L}}_{n}^{0}+{C}_{1}{\rm{\cos }}{\hat{L}}_{n}^{1},{\hat{L}}_{n}^{0}={\varphi }_{n}-k{\phi }_{n}^{1}+{\varphi }_{n+\tfrac{1}{2}},\\ {\hat{L}}_{n}^{1} & = & {\phi }_{n}^{2}-k{(-1)}^{2n+1}{\theta }_{n+\tfrac{1}{2}}-{\phi }_{n+\tfrac{1}{2}}^{2}.\end{eqnarray} \tag{ 67 }$$

Clearly the above interwire couplings preserve glide and all onsite symmetries, if we choose the integer k = even.

The inter-wire couplings (67) again stablize a toric-code-type Z_k gauge theory on the GSPT surface. The anyons (generated by gauge charge e and gauge flux m) in this STO are given by

$$\begin{eqnarray}&&{e}\sim {{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{e}},\,\,\,{\phi }_{n}^{e}=\displaystyle \frac{1}{k}{\varphi }_{n},\end{eqnarray} \tag{ 68 }$$

$$\begin{eqnarray}&&m\sim {{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{m}},\,\,\,{\phi }_{n}^{m}=-{\theta }_{n}-\displaystyle \frac{1}{k}{\phi }_{n}^{2},\,\,\,n\in {\mathbb{Z}}.\end{eqnarray} \tag{ 69 }$$

These anyons obey ${Z}_{k}\times {Z}_{k}$ fusion rules

$$\begin{eqnarray}&&{e}^{k}\sim {m}^{k}\sim 1\end{eqnarray} \tag{ 70 }$$

as in a 2D Z_k gauge theory described by ${{\bf{K}}}_{{Z}_{k}}=-\left(\begin{array}{cc}0 & k\\ k & 0\end{array}\right)$. How do these anyons transform under various symmetries? First of all, $m\sim {{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{m}}$ carries fractional ($-1/k$) charge of the U(1) symmetry while ${e}\sim {{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{e}}$ is charge neutral:

$$\begin{eqnarray}&&\left(\begin{array}{c}{\phi }_{n}^{e}\\ {\phi }_{n}^{m}\end{array}\right)\mathop{\longrightarrow }\limits^{{{\rm{e}}}^{{\rm{i}}\alpha \hat{Q}}}\left(\begin{array}{c}{\phi }_{n}^{e}\\ {\phi }_{n}^{m}-\displaystyle \frac{\alpha }{k}\end{array}\right)\end{eqnarray} \tag{ 71 }$$

While m is invariant under time reversal symmetry ${ \mathcal T }$, e transforms as a 'Kramers doublet' of ${ \mathcal T }$ (for k = even):

$$\begin{eqnarray}&&\left(\begin{array}{c}{\phi }_{n}^{e}\\ {\phi }_{n}^{m}\end{array}\right)\mathop{\longrightarrow }\limits^{{ \mathcal T }}\left(\begin{array}{c}{\phi }_{n}^{e}+\displaystyle \frac{\pi }{k}\\ -{\phi }_{n}^{m}\end{array}\right)\Longrightarrow {{ \mathcal T }}^{k}{{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{e}}{{ \mathcal T }}^{-k}=-{{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{e}}.\end{eqnarray} \tag{ 72 }$$

These symmetry operations can all be realized in a pure 2D system, e.g. by gauging fermion parity in a quantum spin Hall (QSH) insulator [51, 52] for simplest k = 2 case. The anomaly of this STO lies in its glide symmetry implementation [53], which we reveal below.

Since glide operation changes spatial locations, we need to be able to write down anyon operators associated with one certain anyon type at different spatial locations. As discussed in the appendix, this is determined by anyon hopping operators: two anyons of the same type can hop into each other, via a string of local operators invariant under any global symmetry. In particular the string takes the form of ${\hat{O}}_{{n}_{1},{n}_{2}}^{a}={{\rm{e}}}^{{\rm{i}}{\hat{T}}_{{n}_{1},{n}_{2}}^{a}}$, where a labels the anyon type and

$$\begin{eqnarray}&&{T}_{{n}_{1},{n}_{2}}^{a}={\phi }_{{n}_{2}}^{a}-{\phi }_{{n}_{1}}^{a}+\displaystyle \sum _{I,n}{t}_{n}^{I}{L}_{n}^{I},\,\,\,{t}_{n}^{I}\in {\mathbb{R}}.\end{eqnarray} \tag{ 73 }$$

Here ${\hat{L}}_{n}^{I}$ are the arguments of cosine terms in inter-wire tunneling terms (55), and ${{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{a}}$ creates an anyon a at location n. In our STO case, knowing the anyon operators (56) at location n, we can use hopping operators to identify anyon operator at location $n+\tfrac{1}{2}$. It is straightforward to work out their hopping operators (73):

$$\begin{eqnarray}&&{T}_{n,n+\tfrac{1}{2}}^{e}=0,\,\,{t}_{n}^{0}=\displaystyle \frac{1}{k}\Longrightarrow {\phi }_{n+\tfrac{1}{2}}^{e}={\phi }_{n}^{1}-\displaystyle \frac{1}{k}{\varphi }_{n+\tfrac{1}{2}},\,\,\end{eqnarray} \tag{ 74 }$$

$$\begin{eqnarray}&&{T}_{n,n+\tfrac{1}{2}}^{m}={\theta }_{n},\,\,{t}_{n}^{1}=-\displaystyle \frac{1}{k}\Longrightarrow {\phi }_{n+\tfrac{1}{2}}^{m}={\theta }_{n+\tfrac{1}{2}}-\displaystyle \frac{1}{k}{\phi }_{n+\tfrac{1}{2}}^{2}.\,\,\end{eqnarray} \tag{ 75 }$$

It is straightforward to see the anyon hopping operators ${{\rm{e}}}^{{\rm{i}}{T}_{n,n+\tfrac{1}{2}}^{e/m}}$ remain invariant under U(1) charge and time reversal symmetries. Notice that for a fixed anyon operator ${\phi }_{n}^{a}$ on link n, one can always redefine the anyon hopping operator ${T}_{n,n+\tfrac{1}{2}}^{a}$ and anyon ${\phi }_{n+\tfrac{1}{2}}^{a}$ on link $n+\tfrac{1}{2}$ as

$$\begin{eqnarray}&&{\phi }_{n+\tfrac{1}{2}}^{a}\to {\phi }_{n+\tfrac{1}{2}}^{a}+2{M}_{a}{\phi }_{n}^{1},\,\,\,{M}_{a}\in {\mathbb{Z}},\\ &&\quad {T}_{n,n+\tfrac{1}{2}}^{a}\to {T}_{n,n+\tfrac{1}{2}}^{a}+2{M}_{a}{\phi }_{n}^{1}\end{eqnarray} \tag{ 76 }$$

so that all symmetry requirements for anyon hopping operators remain satisfied. Therefore under a generic glide symmetry operation, the anyons transform as

$$\begin{eqnarray}&&{{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{e}}\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}{{\rm{e}}}^{{\rm{i}}(\tfrac{1}{k}{\varphi }_{n+\tfrac{1}{2}}+2{M}_{e}{\phi }_{n}^{1})}={{\rm{e}}}^{{\rm{i}}(2{M}_{e}+1){\phi }_{n}^{1}}{{\rm{e}}}^{-{\rm{i}}{\phi }_{n+\tfrac{1}{2}}^{e}},\\ &&\quad {{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{m}}\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}{{\rm{e}}}^{{\rm{i}}2{M}_{m}{\phi }_{n}^{1}}{{\rm{e}}}^{{\rm{i}}{\phi }_{n+\tfrac{1}{2}}^{m}}.\end{eqnarray} \tag{ 77 }$$

While m anyon can simply be relocated spatially under glide operation, e anyon is attached to a local boson ${{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{1}}$ after glide operation. Notice that this local boson transforms nontrivially under time reversal operation ${ \mathcal T }$

$$\begin{eqnarray}&&{b}_{n}\equiv {{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{1}}\mathop{\longrightarrow }\limits^{{ \mathcal T }}{{\rm{e}}}^{-{\rm{i}}({\phi }_{n}^{1}+\pi )}=-{b}_{n}^{\dagger }.\end{eqnarray} \tag{ 78 }$$

To fully describe the Abelian STO we must also take this boson into account. Therefore the minimal description of the symmetric STO is a 4 × 4 ${\bf{K}}$ matrix and charge vector

$$\begin{eqnarray}{{\bf{K}}}_{{\rm{STO}}}=\left(\begin{array}{cccc}0 & -k & 0 & 0\\ -k & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\end{array}\right),\,\,{{\bf{q}}}_{{\rm{STO}}}=\left(\begin{array}{c}0\\ 1\\ 1\\ 0\end{array}\right),\,\,k=\mathrm{even}.\end{eqnarray} \tag{ 79 }$$

Under U(1) rotation, the chiral boson fields transform as

$$\begin{eqnarray}&&{\vec{\phi }}_{n}\equiv \left(\begin{array}{c}{\phi }_{n}^{m}\\ {\phi }_{n}^{e}\\ {(-1)}^{2n-1}{\phi }_{n-\tfrac{1}{2}}^{1}\\ {\phi }_{n-\tfrac{1}{2}}^{2}\end{array}\right)\mathop{\longrightarrow }\limits^{{{\rm{e}}}^{{\rm{i}}\alpha \hat{Q}}}\left(\begin{array}{c}{\phi }_{n}^{m}-\displaystyle \frac{\alpha }{k}\\ {\phi }_{n}^{e}\\ {(-1)}^{2n-1}{\phi }_{n-\tfrac{1}{2}}^{1}\\ {\phi }_{n-\tfrac{1}{2}}^{2}+\alpha \end{array}\right)={\vec{\phi }}_{n}+\alpha {{\bf{K}}}_{{\rm{STO}}}^{-1}{{\bf{q}}}_{{\rm{STO}}}.\end{eqnarray} \tag{ 80 }$$

Under time reversal ${ \mathcal T }$ they transform as

$$\begin{eqnarray}&&{\vec{\phi }}_{n}\mathop{\longrightarrow }\limits^{{ \mathcal T }}\left(\begin{array}{c}-{\phi }_{n}^{m}\\ {\phi }_{n}^{e}+\displaystyle \frac{\pi }{k}\\ {(-1)}^{2n-1}{\phi }_{n-\tfrac{1}{2}}^{1}+\pi \\ -{\phi }_{n-\tfrac{1}{2}}^{2}\end{array}\right)={{\bf{W}}}_{{ \mathcal T }}{\vec{\phi }}_{n}+\left(\begin{array}{c}0\\ \displaystyle \frac{\pi }{k}\\ \pi \\ 0\end{array}\right),{{\bf{W}}}_{{ \mathcal T }}=\left(\begin{array}{cccc}-1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & -1\end{array}\right).\end{eqnarray} \tag{ 81 }$$

Though not explicit, the above time reversal symmetry plays a crucial role in establishing the following anomalous glide symmetry operation:

$$\begin{eqnarray}&&{\vec{\phi }}_{n}\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}\left(\begin{array}{c}{\phi }_{n+\tfrac{1}{2}}^{m}+2{M}_{m}{\phi }_{n}^{1}\\ -{\phi }_{n+\tfrac{1}{2}}^{e}+(2{M}_{e}+1){\phi }_{n}^{1}\\ -{(-1)}^{2n}{\phi }_{n}^{1}\\ {\phi }_{n}^{2}\end{array}\right)={{\bf{W}}}_{{{ \mathcal G }}_{y}}{\vec{\phi }}_{n+\tfrac{1}{2}},{{\bf{W}}}_{{{ \mathcal G }}_{y}}=\left(\begin{array}{cccc}1 & 0 & 2{M}_{m} & 0\\ 0 & -1 & 2{M}_{e}+1 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 1\end{array}\right),\,\,\,{M}_{e,m}\in {\mathbb{Z}}.\end{eqnarray} \tag{ 82 }$$

In the absence of time reversal symmetry (81), on the other hand, ${b}_{n}={{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{1}}$ is simply a charge-neutral boson operator, and hence we can redefine anyon and hopping operators as

$$\begin{eqnarray*}&&{\phi }_{n+\tfrac{1}{2}}^{a}\to {\phi }_{n+\tfrac{1}{2}}^{a}+{M}_{a}{\phi }_{n}^{1},\,\,\,{M}_{a}\in {\mathbb{Z}},\quad {T}_{n,n+\tfrac{1}{2}}^{a}\to {T}_{n,n+\tfrac{1}{2}}^{a}+{M}_{a}{\phi }_{n}^{1}.\end{eqnarray*}$$

Consequently by choosing ${M}_{e}=-1$ and M_m = 0, the glide symmetry is implemented simply as

$$\begin{eqnarray*}&&{{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{e}}\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}{{\rm{e}}}^{-{\rm{i}}{\phi }_{n+\tfrac{1}{2}}^{e}},\,\,\,{{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{m}}\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}{{\rm{e}}}^{{\rm{i}}{\phi }_{n+\tfrac{1}{2}}^{m}}.\end{eqnarray*}$$

This is nothing but a toric-code-type Abelian topological order with ${\bf{K}}=\left(\begin{array}{cc}0 & -k\\ -k & 0\end{array}\right)$, where m particles carry fractional $\left(\tfrac{1}{k}\right)$ charge and e particles become its anti-particle under glide operation. It can be easily realized in a pure 2D boson system, by proliferating vorticity-k vortices in a superfluid. Therefore there will be no anomaly in the STO once time reversal symmetry ${ \mathcal T }$ is absent. A similar analysis applies to fermon systems in section 4.6.

Although the STO (79) has zero Hall conductance and zero central charge compatible with glide and time reversal symmetries, the anomaly shows up in a more subtle way. To manifest the anomaly, we gauge a discrete Z_N subgroup of the on-site U(1) symmetry, generated by ${\hat{R}}_{N}\equiv \exp (\hspace{1pt}{\rm{i}}\hspace{1pt}\tfrac{2\pi }{N}\hat{Q})$. For a pure 2D topological order, gauging any discrete on-site symmetry should lead to a consistent 2D topological order. Therefore inconsistencies in the gauged topological order can serve as a fingerprint of symmetry anomalies in STOs.

Under the discrete Z_N charge rotation, the anyons in STO transform as

$$\begin{eqnarray*}&&{\vec{\phi }}_{n}\mathop{\longrightarrow }\limits^{{\hat{R}}_{N}}{\vec{\phi }}_{n}+\displaystyle \frac{2\pi }{N}\left(\begin{array}{c}-1/k\\ 0\\ 0\\ 1\end{array}\right).\end{eqnarray*}$$

After gauging the discrete symmetry ${\hat{R}}_{N}$, the Z_N fluxes ('symmetry defects') ${{ \mathcal F }}_{{Z}_{N}}$ become dynamical excitations in the gauged topological order. This brings a new type of anyons [54] into the original topological order (79)

$$\begin{eqnarray}&&{{ \mathcal F }}_{{Z}_{N}}\sim {{\rm{e}}}^{{\rm{i}}{{\rm{\Phi }}}_{n}^{{Z}_{N}}},\,\,\,{{\rm{\Phi }}}_{n}^{{Z}_{N}}=\left(0,\displaystyle \frac{1}{N},\displaystyle \frac{1}{N},0\right)\cdot {\vec{\phi }}_{n}.\end{eqnarray} \tag{ 83 }$$

In Abelian topological order (79), an arbitrary anyon a can be represented as

$$\begin{eqnarray}&&a\sim {{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{a}},\,\,\,{\phi }_{n}^{a}={\vec{l}}_{a}\cdot {\vec{\phi }}_{n},\,\,\,{\vec{l}}_{a}\in {{\mathbb{Z}}}^{4}.\end{eqnarray} \tag{ 84 }$$

Geometrically all Abelian anyons can be viewed as living on a d_K-dimensional (for ${d}_{K}\times {d}_{K}{\bf{K}}$ matrix) integer lattice [47] ${\vec{l}}_{a}\in {\rm{\Lambda }}={{\mathbb{Z}}}^{{d}_{K}}$. The 'primitive Bravais vectors' of this lattice can be chosen as $\{{({e}^{I})}_{J}={\delta }_{I,J}| 1\leqslant I\leqslant {d}_{K}\}$, corresponding to anyons m, e and bosons ${{\rm{e}}}^{{\rm{i}}{\phi }^{\mathrm{1,2}}}$. Since gauging the Z_N symmetry will bring in a new anyon (83), it introduces a new primitive vector

$$\begin{eqnarray}&&{\vec{l}}_{{{ \mathcal F }}_{{Z}_{N}}}={\left(0,\displaystyle \frac{1}{N},\displaystyle \frac{1}{N},0\right)}^{T}.\end{eqnarray} \tag{ 85 }$$

for the anyon lattice of the gauged topological order. After gauging the symmetry, the new topological order has a anyon lattice expanded by all integer vectors and (85). The basis of the new anyon lattice ${{\rm{\Lambda }}}_{{Z}_{N}}$ can e.g. be chosen as

$$\begin{eqnarray}{\vec{l}}_{m} & = & {(\mathrm{1,}0,\mathrm{0,}0)}^{T},\,\,\,{\vec{l}}_{e}={(\mathrm{0,}1,\mathrm{0,}0)}^{T},\\ {\vec{l}}_{{\phi }^{2}} & = & {(\mathrm{0,}0,\mathrm{0,}1)}^{T},\,\,\,{\vec{l}}_{{{ \mathcal F }}_{{Z}_{N}}}={\left(0,\displaystyle \frac{1}{N},\displaystyle \frac{1}{N},0\right)}^{T}.\end{eqnarray} \tag{ 86 }$$

For an arbitrary Abelian anyon labeled by vector ${\vec{l}}_{a}\in {\rm{\Lambda }}$, it transforms under time reversal and glide as

$$\begin{eqnarray}&&\vec{l}\mathop{\longrightarrow }\limits^{{ \mathcal T }}{{\bf{W}}}_{{ \mathcal T }}^{T}\vec{l},\,\,\,\,\,\vec{l}\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}{{\bf{W}}}_{{{ \mathcal G }}_{y}}^{T}\vec{l}.\end{eqnarray} \tag{ 87 }$$

For a symmetric 2D Abelian topological order, its anyon lattice must be invariant under all symmetry operations, i.e. any 'lattice site' (corresponding to one anyon) must be mapped to another lattice site on the same lattice Λ. In our case of STO (79), apparently any integer vectors is mapped to another integer vector by (86). However, the anyon ('Z_N symmetry defect') labeled by a fractional vector ${\vec{l}}_{{{ \mathcal F }}_{{Z}_{N}}}$ in the gauged topological order is mapped to a new vector

$$\begin{eqnarray}&&{{\bf{W}}}_{{{ \mathcal G }}_{y}}^{T}{\vec{l}}_{{{ \mathcal F }}_{{Z}_{N}}}={\left(\mathrm{0,}0,\displaystyle \frac{2{M}_{e}+1}{N},0\right)}^{T}-{\left(0,\displaystyle \frac{1}{N},\displaystyle \frac{1}{N},0\right)}^{T},\,\,{M}_{e}\in {\mathbb{Z}},\\ &&\quad {{\bf{W}}}_{{{ \mathcal G }}_{y}}^{T}{\vec{l}}_{{{ \mathcal F }}_{{Z}_{N}}}\notin {{\rm{\Lambda }}}_{{Z}_{N}},\,\,\,\,\forall \,N=0\mathrm{mod}2.\end{eqnarray} \tag{ 88 }$$

For any odd integer $N=1\mathrm{mod}2$, one can always find an integer ${M}_{e}\in {\mathbb{Z}}$ so that $\tfrac{2{M}_{e}+1}{N}\in {\mathbb{Z}}$ and the new vector ${{\bf{W}}}_{{{ \mathcal G }}_{y}}^{T}{\vec{l}}_{{{ \mathcal F }}_{{Z}_{N}}}$ still belong to the same anyon lattice. For even integer $N=0\mathrm{mod}2$, on the other hand, glide symmetry maps the original anyon lattice (86) of the gauged topological order into a different lattice [13] as shown in (88). Therefore glide symmetry is broken in the new topological order from gauging Z_N symmetry in STO (79). As discussed earlier, time reversal symmetry is crucial for the anomaly. Without time reversal symmetry, 2M_e can be replaced by an arbitrary integer M_e. For any integer N, one can always find an integer such that $\tfrac{{M}_{e}+1}{N}\in {\mathbb{Z}}$. The anomalous STO here needs all three symmetries to be preserved. We coin this anomalous STO as ${\rm{e}}{{ \mathcal G }}_{y}{ \mathcal T }$ mC. The anomaly of STO ${\rm{e}}{{ \mathcal G }}_{y}{ \mathcal T }$ mC on the glide-preserving surface of ${| \mathrm{BQSH}\rangle }_{{{ \mathcal G }}_{y}}$ therefore demonstrates that ${| \mathrm{BQSH}\rangle }_{{{ \mathcal G }}_{y}}$ is a nontrivial 3D GSPT phase with $U(1)\rtimes {Z}_{2}^{{ \mathcal T }}$ global symmetries.

Physically, unlike PT-breaking thermal/charge Hall response of efmf/eCmC STOs on the surface of ${| {E}_{8}\rangle }_{{{ \mathcal G }}_{y}}$ and ${| \mathrm{BIQH}\rangle }_{{{ \mathcal G }}_{y}}$ states, here the anomaly of STO ${\rm{e}}{{ \mathcal G }}_{y}{ \mathcal T }$ mC on the surface of ${| \mathrm{BQSH}\rangle }_{{{ \mathcal G }}_{y}}$ state is manifested in the glide symmetry implementations on $\tfrac{\pi }{N/2}$ flux of U(1) symmetry for any even integer N. In the simplest case, a π flux (N = 2) would preserve both time reversal and glide symmetries in a 2D symmetric topological order. In our anomalous STO, however, π-flux associated with anyon vector ${\vec{l}}_{\pi }={\left(0,,,\tfrac{1}{2},,,\tfrac{1}{2},,,0\right)}^{T}$ is transformed into a different vector ${\vec{l}}_{\pi }^{\prime }={{\bf{W}}}_{{{ \mathcal G }}_{y}}^{T}{\vec{l}}_{\pi }={\left(0,,,-,\tfrac{1}{2},,,{M}_{e},\in ,{\mathbb{Z}},,,0\right)}^{T}$ under glide symmetry operation. As shown in (88), the two vectors ${\vec{l}}_{\pi }$ and ${\vec{l}}_{\pi }^{\prime }$ differ by a fractional vector $(0,0,{M}_{e}+\tfrac{1}{2},0)$, which transforms as a Kramers doublet under time reversal symmetry [55]. In other words, the glide operation (82) attaches a Kramers doublet to each π flux, which changes the time reversal symmetry character of a π flux in the STO. This means the glide symmetry operation (82) in the STO is incompatible with [13] time reversal symmetry ${ \mathcal T }$, and thus secretly breaks it. Such an anomalous symmetry operation is impossible in a pure 2D system, and can only be realized in the anomalous STO on the surface of GSPT phases. Establishing ${| \mathrm{BQSH}\rangle }_{{{ \mathcal G }}_{y}}$ as a nontrivial 3D GSPT, we achieve the ${{\mathbb{Z}}}_{2}$ classification of GSPT phases with onsite $U(1)\rtimes {Z}_{2}^{{ \mathcal T }}$ symmetry, as summarized in table 1.

Unlike the previous case considered in section 3.4, U(1) spin rotational symmetry and time reversal corresponds to a different symmetry group ${G}_{0}=U(1)\times {Z}_{2}^{{ \mathcal T }}$ and a different classification. In this case, the 2D SRE boson phases are classified by ${H}^{3}(U(1)\times {Z}_{2}^{{ \mathcal T }},U(1))={{\mathbb{Z}}}_{1}$, without any nontrivial SPT phases. Therefore the associated 3D GSPT classification is also trivial, as shown in table 1.

In this section we consider discrete symmetries, by breaking the U(1) symmetry discussed previously into its Z_N subgroup. The results are summarized in the three rows at the bottom of table 1.

We start with on-site symmetry group ${G}_{0}={Z}_{N}$, generated by discrete Z_N rotation ${\hat{R}}_{N}$ satisfying ${({\hat{R}}_{N})}^{N}\,=\,1$. The situation is quite similar to ${G}_{0}=U(1)$ case discussed in section 3.3. The 2D SRE phases have a ${{\mathbb{Z}}}_{N}\times {\mathbb{Z}}$ classification. The root phase associated with the integer index $\nu \in {\mathbb{Z}}$ is again the chiral E₈ state $| {E}_{8}\rangle$, leading to 3D GSPT phase ${| {E}_{8}\rangle }_{{{ \mathcal G }}_{y}}$ featured by anomalous STO efmf (see section 3.1). The other root state for cyclic $q\in {{\mathbb{Z}}}_{N}$ index is the Z_N-SPT phase $| q=1\mathrm{mod}N\rangle$, whose edge states (49) transform under Z_N rotation as

$$\begin{eqnarray}&&\left(\begin{array}{c}{\phi }^{1}\\ {\phi }^{2}\end{array}\right)\mathop{\longrightarrow }\limits^{{\hat{R}}_{N}}\left(\begin{array}{c}{\phi }^{1}+\displaystyle \frac{2\pi }{N}\\ {\phi }^{2}+\displaystyle \frac{2\pi }{N}q\end{array}\right).\end{eqnarray} \tag{ 89 }$$

These 2D Z_N SPT phases obey the following ${{\mathbb{Z}}}_{N}$ addition rule

$$\begin{eqnarray}&&| {q}_{1}\rangle \oplus | {q}_{2}\rangle =| {q}_{1}+{q}_{2}\mathrm{mod}N\rangle .\end{eqnarray} \tag{ 90 }$$

Therefore for any odd integer N, any Z_N-SPT phase can be viewed as the sum of two identical Z_N-SPT phases:

$$\begin{eqnarray}| q\rangle =\left|\displaystyle \frac{N+q}{2}\right\rangle \oplus \left|\displaystyle \frac{N+q}{2}\right\rangle ,\,\,\,q & = & \mathrm{odd},\\ | q\rangle =\left|\displaystyle \frac{q}{2}\right\rangle \oplus \left|\displaystyle \frac{q}{2}\right\rangle ,\,\,\,q & = & \mathrm{even}.\end{eqnarray} \tag{ 91 }$$

As a result according to Z₂ addition rule (14) of 3D GSPT phases, none of these 2D Z_N-SPT phases will lead to a nontrivial 3D GSPT phase if N = odd. On the other hand, this argument stops working for the root Z_N-SPT phase $| q=1\mathrm{mod}N\rangle$ when N = even. In this case, the surface states (52) on [001] side surface can be symmetrically gapped out by same interwire couplings (55) as the ${G}_{0}=U(1)$ case. The resulting STO is again toric-code-type Z₂ topological order eCmC, where both e and m anyons carry projective representation of Z_N symmetry. Unlike in ${G}_{0}=U(1)$ case, the Hall conductance of symmetric STO is not well defined here due to lack of continuous U(1) symmetry.

To diagnose the STO anomaly for N = even case, we gauge the discrete Z_N symmetry generated by ${\hat{R}}_{N}$. In particular, discrete ${\hat{R}}_{N}$ flux ('symmetry defects')

$$\begin{eqnarray}&&{{ \mathcal F }}_{{Z}_{N}}\sim {{\rm{e}}}^{{\rm{i}}({\phi }^{e}-{\phi }^{m})/N}\end{eqnarray} \tag{ 92 }$$

become new emergent anyons in the gauged topological order. Among them, the π flux

$$\begin{eqnarray}&&{{ \mathcal F }}_{\pi }\sim {{\rm{e}}}^{{\rm{i}}\tfrac{{\phi }^{{\rm{e}}}-{\phi }^{m}}{2}}\end{eqnarray} \tag{ 93 }$$

corresponds to a bound state of $N/2$ elementary ${\hat{R}}_{N}$ fluxes and preserves glide symmetry. The statistical angle of π flux (93) is ${\theta }_{{{ \mathcal F }}_{\pi }}=\pi /4\mathrm{mod}\pi$. Since this new anyon ${{ \mathcal F }}_{\pi }$ corresponding to π flux is invariant under glide operation which reverses the statistical angle, we must have

$$\begin{eqnarray}&&{{ \mathcal F }}_{\pi }\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}{{ \mathcal F }}_{\pi }\Longrightarrow {\theta }_{{{ \mathcal F }}_{\pi }}=-{\theta }_{{{ \mathcal F }}_{\pi }}\mathrm{mod}2\pi .\end{eqnarray} \tag{ 94 }$$

Hence the self statistics ${\theta }_{{{ \mathcal F }}_{\pi }}=\pi /4\mathrm{mod}\pi$ and the gauged STO would violate the orientation-reversing glide symmetry in a pure 2D system. This dictates the anomalous glide symmetry implementation in the Z_N-symmetric STO of N = even GSPT phase ${| q=1\mathrm{mod}N\rangle }_{{{ \mathcal G }}_{y}}$. Therefore we have shown that 3D GSPT phase ${| q=1\mathrm{mod}N\rangle }_{{{ \mathcal G }}_{y}}$ is nontrivial only when N = even, hence arriving at the ${{\mathbb{Z}}}_{(N,2)}\times {{\mathbb{Z}}}_{2}$ classification of ${G}_{0}={Z}_{N}$ 3D GSPT phases.

Next we discuss ${G}_{0}={Z}_{N}\rtimes {Z}_{2}^{{ \mathcal T }}$ case with discrete Z_N charge conservation (generated by ${\hat{R}}_{N}$) and time reversal, as summarized in the second row from the bottom of table 1. The 2D SRE phases have a ${{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2}$ classification for integer N = even, or a trivial classification for N = odd [43]. For nontrivial N = even case, there are two SRE root phases with gapless edge modes protected by ${Z}_{N}\rtimes {Z}_{2}^{{ \mathcal T }}$ symmetry. One is an analog of BQSH state with discrete Z_N symmetry, coined as $| {Z}_{N}-\mathrm{BQSH}\rangle$, whose edge states (49) transform as

$$\begin{eqnarray}&&\left(\begin{array}{c}{\phi }^{1}\\ {\phi }^{2}\end{array}\right)\mathop{\longrightarrow }\limits^{{\hat{R}}_{N}}\left(\begin{array}{c}{\phi }^{1}\\ {\phi }^{2}+\displaystyle \frac{2\pi }{N}\end{array}\right),\,\,\,\left(\begin{array}{c}{\phi }^{1}\\ {\phi }^{2}\end{array}\right)\mathop{\longrightarrow }\limits^{{ \mathcal T }}\left(\begin{array}{c}{\phi }^{1}+\pi \\ -{\phi }^{2}\end{array}\right).\end{eqnarray} \tag{ 95 }$$

These edge states need the protection of both ${\hat{R}}_{N}$ and time reversal symmetry. In comparison, the other root phase is closer to BIQH state with discrete Z_N symmetry, whose edge states (49) transform as

$$\begin{eqnarray}&&\left(\begin{array}{c}{\phi }^{1}\\ {\phi }^{2}\end{array}\right)\mathop{\longrightarrow }\limits^{{\hat{R}}_{N}}\left(\begin{array}{c}{\phi }^{1}+\pi \\ {\phi }^{2}+\displaystyle \frac{2\pi }{N}\end{array}\right),\,\,\,\left(\begin{array}{c}{\phi }^{1}\\ {\phi }^{2}\end{array}\right)\mathop{\longrightarrow }\limits^{{ \mathcal T }}\left(\begin{array}{c}{\phi }^{1}\\ -{\phi }^{2}\end{array}\right).\end{eqnarray} \tag{ 96 }$$

Unlike the previous case, these gapless edge modes remain stable even if time reversal is broken. In both cases the construction of STO is exactly the same as (67) for ${| \mathrm{BQSH}\rangle }_{{{ \mathcal G }}_{y}}$ state, which lead to a Z_k (k = even) gauge theory on the surface. The anomaly of their STOs again lies in the glide action on π flux (i.e. bound state of $N/2$ symmetry defects ${{ \mathcal F }}_{{Z}_{N}}$) of the STO: the π flux secretly breaks glide symmetry in the sense that anyon lattice is not invariant under glide operation after gauging Z_N symmetry. The first case (95) is exactly the same as ${\rm{e}}{{ \mathcal G }}_{y}{ \mathcal T }$ mC STO in GSPT state ${| {Z}_{N}-\mathrm{BQSH}\rangle }_{{{ \mathcal G }}_{y}}$. For the latter case (96), the anyons ${\vec{\phi }}_{n}\equiv {\left({\phi }_{n}^{m},,,{\phi }_{n}^{e},,,{(-1)}^{2n-1},{\phi }_{n-\tfrac{1}{2}}^{1},,,{\phi }_{n-\tfrac{1}{2}}^{2}\right)}^{T}$ in the anomalous STO transform as

$$\begin{eqnarray}&&{\vec{\phi }}_{n}\mathop{\longrightarrow }\limits^{{\hat{R}}_{N}}{\vec{\phi }}_{n}+\displaystyle \frac{2\pi }{N}\left(\begin{array}{c}-\tfrac{1}{k}\\ 0\\ \tfrac{N}{2}\\ 1\end{array}\right),\,\,\,{\vec{\phi }}_{n}\mathop{\longrightarrow }\limits^{{ \mathcal T }}{{\bf{W}}}_{{ \mathcal T }}{\vec{\phi }}_{n},\end{eqnarray} \tag{ 97 }$$

where matrix ${{\bf{W}}}_{{ \mathcal T }}$ is defined in (81). The glide symmetry operation is the same as in (82). Again gauging ${\hat{R}}_{N}$ symmetry leads to new anyon (symmetry defect):

$$\begin{eqnarray}&&{{ \mathcal F }}_{{Z}_{N}}=\left(0,\displaystyle \frac{1}{N},\displaystyle \frac{1}{N},\displaystyle \frac{1}{2}\right)\cdot {\vec{\phi }}_{n}\end{eqnarray} \tag{ 98 }$$

and it is straightforward to verify that the anyon lattice of gauged topological order is not invariant under glide operation (82). Since after glide operation each e particle is dressed with a boson ${{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{1}}$ that transforms nontrivially under Z_N symmetry, we coin this anomalous STO as ${\rm{e}}{{ \mathcal G }}_{y}C$ mC. Therefore we establish the ${{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2}$ classification and two root phases of 3D GSPT phases with ${G}_{0}={Z}_{n}\rtimes {Z}_{2}^{{ \mathcal T }}$, as summarized in table 1.

Finally we discuss ${G}_{0}={Z}_{N}\times {Z}_{2}^{{ \mathcal T }}$ case, with discrete Z_N spin rotation ${\hat{R}}_{N}$ and time reversal symmetry. Again the 2D SRE phases have a ${{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2}$ classification if N = even, or a trivial one if N = odd. When N = even, one root SPT phase has edge states (49) that transform under symmetries as

$$\begin{eqnarray}&&\left(\begin{array}{c}{\phi }^{1}\\ {\phi }^{2}\end{array}\right)\mathop{\longrightarrow }\limits^{{\hat{R}}_{N}}\left(\begin{array}{c}{\phi }^{1}\\ {\phi }^{2}+\pi \end{array}\right),\,\,\,\left(\begin{array}{c}{\phi }^{1}\\ {\phi }^{2}\end{array}\right)\mathop{\longrightarrow }\limits^{{ \mathcal T }}\left(\begin{array}{c}{\phi }^{1}+\pi \\ -{\phi }^{2}\end{array}\right)\end{eqnarray} \tag{ 99 }$$

while the other root phase has

$$\begin{eqnarray}&&\left(\begin{array}{c}{\phi }^{1}\\ {\phi }^{2}\end{array}\right)\mathop{\longrightarrow }\limits^{{\hat{R}}_{N}}\left(\begin{array}{c}{\phi }^{1}+\displaystyle \frac{2\pi }{N}\\ {\phi }^{2}+\pi \end{array}\right),\,\,\,\left(\begin{array}{c}{\phi }^{1}\\ {\phi }^{2}\end{array}\right)\mathop{\longrightarrow }\limits^{{ \mathcal T }}\left(\begin{array}{c}{\phi }^{1}\\ -{\phi }^{2}\end{array}\right).\end{eqnarray} \tag{ 100 }$$

In their corresponding 3D GSPT phases, the anomalous STOs are completely similar to those previously discussed in ${G}_{0}={Z}_{N}\rtimes {Z}_{2}^{{ \mathcal T }}$ case. In a CWC in parallel to (67), we can obtain the anomalous STOs: ${\rm{e}}{{ \mathcal G }}_{y}{ \mathcal T }$ mC for (99) case and ${\rm{e}}{{ \mathcal G }}_{y}C$ mC for (100) case. Therefore we establish the ${{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2}$ classification for 3D GSPT phases with ${G}_{0}={Z}_{N}\times {Z}_{b}^{{ \mathcal T }}$ onsite symmetry when N = even, as summarized in the bottom row of table 1.

Any local fermion Hamiltonian always preserves the fermion parity ${{\bf{P}}}_{{\rm{f}}}={(-1)}^{\hat{F}}$ as a global (on-site) symmetry, since each term in a local Hamiltonian must contain an even number of fermion creation/annihilation operators. Without any other on-site symmetry, the fermion system belongs to symmetry class D in Altland-Zirnbauer's 10-fold way language [45].

The 2D SRE fermion phases in class D have an integer (${\mathbb{Z}}$) classification, labeled by an integer-valued index $\nu \in {\mathbb{Z}}$. The $\nu =1$ root phase is the ${p}_{x}+\hspace{1pt}{\rm{i}}\hspace{1pt}{p}_{y}$ chiral superconductor $| {p}_{x}+\hspace{1pt}{\rm{i}}\hspace{1pt}{p}_{y}\rangle$ of spinless fermions [40], featured by chiral Majorana edge modes with ${c}_{-}=\nu /2=\tfrac{1}{2}$. From the coupled layer construction and Z₂ addition rule (14) of 3D GSPT phases, ${| {p}_{x}+{\rm{i}}{p}_{y}\rangle }_{{{ \mathcal G }}_{y}}$ is the only possible nontrivial GSPT in symmetry class D (i.e. with onsite symmetry ${G}_{0}={Z}_{2}^{{{\bf{P}}}_{{\rm{f}}}}$). Below we show that ${| {p}_{x}+{\rm{i}}{p}_{y}\rangle }_{{{ \mathcal G }}_{y}}$ is indeed a nontrivial GSPT hosting anomalous STOs on its glide-invariant surface, therefore establishing ${{\mathbb{Z}}}_{2}$ classification of fermion GSPT phases in class D.

In the coupled layer construction of ${| {p}_{x}+{\rm{i}}{p}_{y}\rangle }_{{{ \mathcal G }}_{y}}$ phase, its gapless surface states on glide-invariant [001] side surface are described by

$$\begin{eqnarray}&&{{ \mathcal H }}_{0}=\hspace{1pt}{\rm{i}}\hspace{1pt}{v}_{F}\displaystyle \sum _{n\in {\mathbb{Z}}/2}{(-1)}^{2n}{\chi }_{n}{\partial }_{y}{\chi }_{n},\end{eqnarray} \tag{ 101 }$$

where ${\chi }_{n}$ (${\chi }_{n+\tfrac{1}{2}}$) is the chiral Majorana mode of layer A_n (B_n) for $n\in {\mathbb{Z}}$, and v_F is the fermi velocity. Now let us deposit two quantum wires of spinless free fermion gas between each two neighboring layers (${\chi }_{n}$ and ${\chi }_{n+\tfrac{1}{2}}$) of opposite chirality, described by

$$\begin{eqnarray}&&{{ \mathcal H }}_{1}=\hspace{1pt}{\rm{i}}\hspace{1pt}{v}_{{\rm{F}}}\displaystyle \sum _{n\in {\mathbb{Z}}/2}\displaystyle \sum _{f=1,2}\displaystyle \sum _{\alpha =\pm }{(-1)}^{\alpha }{\psi }_{n,f,\alpha }^{\dagger }{\partial }_{y}{\psi }_{n,f,\alpha },\end{eqnarray} \tag{ 102 }$$

where f = 1, 2 is the 'flavor' index for deposited quantum wires, and $\alpha =\pm$ is the chirality ('handedness') index for right/left movers of each quantum wire. These Dirac fermion modes can also be expressed in terms of Majorana modes

$$\begin{eqnarray}&&{\chi }_{n,f,\alpha }^{1}\equiv \displaystyle \frac{{\psi }_{n,f,\alpha }^{\dagger }+{\psi }_{n,f,\alpha }}{2},\,\,{\chi }_{n,f,\alpha }^{2}\equiv \displaystyle \frac{{\psi }_{n,f,\alpha }^{\dagger }-{\psi }_{n,f,\alpha }}{2\hspace{1pt}{\rm{i}}\hspace{1pt}}.\end{eqnarray} \tag{ 103 }$$

With the deposited array of quantum wires, one can define the following nine branches of chiral Majorana modes associated with each layer:

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{n}\equiv {({\chi }_{n},{\chi }_{n-\tfrac{1}{2},f,{(-1)}^{2n}}^{1,2},{\chi }_{n,f,{(-1)}^{2n}}^{1,2})}^{T}\end{eqnarray} \tag{ 104 }$$

and they transform under glide symmetry as

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{n}\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}{{\rm{\Psi }}}_{n+\tfrac{1}{2}}.\end{eqnarray} \tag{ 105 }$$

This is exactly the same surface states in the AFM TSC considered in [36], where a symmetric STO is established by certain gapping terms that couple the nine branches of Majorana wires. The consequent STO is a non-Abelian topological order, whose edge states are described by ${SO}{(3)}_{3}$ Wess–Zumino–Witten model with chiral central charge ${c}_{-}=9/4$. Such a nonzero chiral central charge is incompatible with orientation-reversing glide operation in a pure 2D topological order, and manifests the anomaly of the glide-symmetric non-Abelian STO. Therefore ${| {p}_{x}+{\rm{i}}{p}_{y}\rangle }_{{{ \mathcal G }}_{y}}$ is indeed a nontrivial 3D GSPT phase with anomalous STOs, and 3D fermion GSTP phases in AZ symmetry class D is classified by ${{\mathbb{Z}}}_{2}$ as summarized in table 2.

Now let's consider time reversal symmetry ${ \mathcal T }$ satisfying ${{ \mathcal T }}^{2}=1$, corresponding to on-site symmetry group ${G}_{0}={Z}_{2}^{{ \mathcal T }}\times {Z}_{2}^{{{\bf{P}}}_{{\rm{f}}}}$ in a fermion system. Such a 'spinless' time reversal symmetry can be realized in a magnetic superconductor where the combination of time reversal and π spin rotation is preserved. In AZ's 10-fold way this corresponds to symmetry class BDI.

2D SRE fermion phases in class BDI has a trivial ${{\mathbb{Z}}}_{1}$ classification, i.e. there is no nontrivial fermion 2D SPT phases. As a result, the 3D GSPT phases in class BDI also has a trivial ${{\mathbb{Z}}}_{1}$ classification as shown in table 2.

Symmetry class DIII corresponds to spin-1/2 fermion system with time reversal symmetry ${ \mathcal T }$, satisfying ${{ \mathcal T }}^{2}={{\bf{P}}}_{f}$ where fermions transform as Kramers doublets. The on-site symmetry group is hence ${G}_{0}={Z}_{4}^{{ \mathcal T }}$ since ${{ \mathcal T }}^{4}=1$. The 2D fermion SRE phases have a ${{\mathbb{Z}}}_{2}$ classification [43], where the only nontrivial root phase is the helical TSC $| {({p}_{x}+{\rm{i}}{p}_{y})}_{\uparrow }\otimes {({p}_{x}-{\rm{i}}{p}_{y})}_{\downarrow }\rangle$ with a pair of counter-propagating Majorana fermion edge modes. Below we briefly show this 2D root phase leads to a nontrivial 3D GSPT phase in class DIII with anomalous STOs. This establishes the ${{\mathbb{Z}}}_{2}$ classification of 3D fermion GSPT phases in class DIII.

Clearly the 2D TSC in class DIII can be simply viewed as a tensor product of $\nu =1$ TSC in class D for spin-$\uparrow$ fermions and its time reversal partner. Its helical edge states indeed consist of a spin-$\uparrow$ right-moving Majorana mode and a spin-$-\downarrow$ left-moving Majorana mode. Therefore we can simply double the surface degrees of freedom by including both spin-$\uparrow$ and spin-$-\downarrow$, and follow the same construction for ${| {p}_{x}+{\rm{i}}{p}_{y}\rangle }_{{{ \mathcal G }}_{y}}$ state discussed in section 4.1. Naturally the STO of ${| {({p}_{x}+{\rm{i}}{p}_{y})}_{\uparrow }\otimes {({p}_{x}-{\rm{i}}{p}_{y})}_{\downarrow }\rangle }_{{{ \mathcal G }}_{y}}$ state is given by ${[{SO}{(3)}_{3}]}_{\uparrow }\times {[{SO}{(3)}_{-3}]}_{\downarrow }$. This non-chiral STO is obtained by stacking two chiral STOs (${[{SO}{(3)}_{3}]}_{\uparrow }$ and ${[{SO}{(3)}_{-3}]}_{\downarrow }$) on top of each other, and time reversal operation transforms one chiral STO (with chiral central charge ${c}_{-,\uparrow /\downarrow }=\pm 9/4$) into the other. Most crucially, each chiral STO (e.g. ${[{SO}{(3)}_{-3}]}_{\downarrow }$) is invariant under the glide symmetry operation, which is impossible for a pure 2D topological order. Therefore ${| {({p}_{x}+{\rm{i}}{p}_{y})}_{\uparrow }\otimes {({p}_{x}-{\rm{i}}{p}_{y})}_{\downarrow }\rangle }_{{{ \mathcal G }}_{y}}$ hosts an anomalous STO, and belongs to a nontrivial GSPT phase in class DIII. This dictates the ${{\mathbb{Z}}}_{2}$ classification of 3D fermion GSPT phase in class DIII, as summarized in table 2.

A continuous U(1) symmetry leads to symmetry class A of fermion systems. In 2D, SRE fermion phases in class A have a ${\mathbb{Z}}\times {\mathbb{Z}}$ classification, labeled by two integer indices $(\nu ,q)$ similar to the 2D boson case. Physically a SRE fermion phase $[\nu ,q]$ in class A is characterized by the following chiral central charge (thermal response) and Hall conductance (charge response)

$$\begin{eqnarray}&&{c}_{-}=8\nu +q,\,\,\,{\sigma }_{{xy}}=q\displaystyle \frac{{e}^{2}}{h},\,\,\,\nu ,q\in {\mathbb{Z}},\end{eqnarray} \tag{ 106 }$$

where e is the fundamental charge carried by each fermion. There are two SRE root phases: (i) the IQH state $| \mathrm{IQH}\rangle$ of fermions with $[\nu =0,q=1];$ (ii) the E₈ state $| {E}_{8}\rangle$ of neutral bosons with $[\nu =1,q=0]$, built from particle-hole excitations of fermions. The 3D GSPT state ${| {E}_{8}\rangle }_{{{ \mathcal G }}_{y}}$ associated with 2D E₈ state of neutral bosonic particle-hole excitations is clearly a nontrivial GSPT phase, which features anomalous Z₂ STO 'efmf' as discussed in section 3.1.

Now let's look into 3D GSPT state ${| \mathrm{IQH}\rangle }_{{{ \mathcal G }}_{y}}$ constructed by stacking ${\sigma }_{{xy}}={e}^{2}/h$ IQH layers of fermions. Its gapless surface states on glide-invariant [001] side surface is described by

$$\begin{eqnarray}&&{{ \mathcal H }}_{0}=\hspace{1pt}{\rm{i}}\hspace{1pt}{v}_{F}\displaystyle \sum _{n\in {\mathbb{Z}}/2}{(-1)}^{2n}{\psi }_{n}^{\dagger }{\partial }_{y}{\psi }_{n},\end{eqnarray} \tag{ 107 }$$

where glide symmetry is implemented as

$$\begin{eqnarray}&&{\psi }_{n}\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}{\psi }_{n+\tfrac{1}{2}}.\end{eqnarray} \tag{ 108 }$$

Such an array of staggered 1D chiral fermion wires is exactly the same as the surface states of the 3D AFM-TI, whose symmetric STO is explicitly constructed by proper inter-wire couplings in [35, 36]. This leads to a non-Abelian STO coined 'T-Pfaffian' [12], which has a chiral central charge ${c}_{-}=1/2$ and Hall conductance ${\sigma }_{{xy}}={e}^{2}/2h$. Both the thermal and charge Hall responses are contradictory to the glide symmetry in a pure 2D system, demonstrating the anomaly of the glide-symmetric STO. Therefore ${| \mathrm{IQH}\rangle }_{{{ \mathcal G }}_{y}}$ is also a nontrivial fermion GSPT phase in class A.

As a result, due to Z₂ addition rule (14) we achieve the ${{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2}$ classification of 3D fermion GSPT phass in symmetry class A, as summarized in table 2. The associated root phases are ${| {E}_{8}\rangle }_{{{ \mathcal G }}_{y}}$ and ${| \mathrm{IQH}\rangle }_{{{ \mathcal G }}_{y}}$ states.

Class AI corresponds to on-site symmetry group ${G}_{0}=U(1)\rtimes {Z}_{2}^{{ \mathcal T }}$ where time reversal symmetry ${ \mathcal T }$ satisfies ${{ \mathcal T }}^{2}=1$. Such a 'non-Kramers' time reversal symmetry can be realized in magnetic insulators, where only the combination of Kramers time reversal and π spin rotation is preserved. The 2D SRE fermion phases in class AI have a ${{\mathbb{Z}}}_{2}$ classification [43], where the nontrivial root phase is the BQSH state $| 2e-\mathrm{BQSH}\rangle$ of bosonic charge-$2e$ Cooper pairs. The corresponding 3D GSPT state ${| 2e-\mathrm{BQSH}\rangle }_{{{ \mathcal G }}_{y}}$ from the coupled layer construction is a nontrivial GSPT phase, characterized by an anomalous STO detailed in section 3.4. Specifically, the π flux of charge-$2e$ Cooper pairs is not invariant under glide symmetry operation, manifesting the anomaly of the glide-symmetric STO. Note that a π flux for charge-$2e$ Cooper pairs is a $\pm \tfrac{\pi }{2}$ flux for charge-e fundamental fermions. Therefore in the STO of ${| 2e-\mathrm{BQSH}\rangle }_{{{ \mathcal G }}_{y}}$ state, the anomalous glide symmetry operation will not transform a $\pi /2$ flux into a $-\pi /2$ flux, a phenomena impossible in a pure 2D system. In summary, 3D fermion GSPT phases in class AI have a ${{\mathbb{Z}}}_{2}$ classification, whose root phase is the ${| 2e-\mathrm{BQSH}\rangle }_{{{ \mathcal G }}_{y}}$ state.

In a spin-1/2 fermion system with time reversal ${{ \mathcal T }}^{2}={{\bf{P}}}_{{\rm{f}}}$ and U(1) charge symmetries, the on-site symmetry group is ${G}_{0}=U(1)\rtimes {Z}_{4}^{{ \mathcal T }}$, corresponding to symmetry class AII. Most TI materials belong to this symmetry class. The 2D fermion SRE phases in class AII have a ${{\mathbb{Z}}}_{2}$ classification, whose root phase is the 2D QSH insulator $| \mathrm{QSH}\rangle$ of spin-1/2 fermions. Stacking these 2D QSH layers [26] give rise to the 'hourglass fermion' surface states of the 3D non-symmorphic free-fermion TI [23, 24]. Is such a 3D state ${| \mathrm{QSH}\rangle }_{{{ \mathcal G }}_{y}}$ a nontrivial GSPT phase in the presence of strong electronic interactiosn? If yes, what are the properties of its anomalous STO? Below we answer these questions by explict construction of the symmetric STO on top of ${| \mathrm{QSH}\rangle }_{{{ \mathcal G }}_{y}}$ state.

Since each QSH layer intersects with the glide-symmetric [001] side surface by its gapless helical edge, the gapless [001] surface states of ${| \mathrm{QSH}\rangle }_{{{ \mathcal G }}_{y}}$ phase is described by a 2D array of 1D helical edge modes:

$$\begin{eqnarray}&&{{ \mathcal H }}_{0}=-\hspace{1pt}{\rm{i}}\hspace{1pt}{v}_{F}\displaystyle \sum _{n\in {\mathbb{Z}}}({\psi }_{n,R}^{\dagger }{\partial }_{y}{\psi }_{n,R}-{\psi }_{n,L}^{\dagger }{\partial }_{y}{\psi }_{n,L}).\end{eqnarray} \tag{ 109 }$$

Under glide and time reversal operations, the chiral fermion modes transform as

$$\begin{eqnarray}&&\left(\begin{array}{c}{\psi }_{n,R}\\ {\psi }_{n,L}\end{array}\right)\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}\left(\begin{array}{c}{\psi }_{n+\tfrac{1}{2},L}\\ -{\psi }_{n+\tfrac{1}{2},R}\end{array}\right),\end{eqnarray} \tag{ 110 }$$

$$\begin{eqnarray}&&\left(\begin{array}{c}{\psi }_{n,R}\\ {\psi }_{n,L}\end{array}\right)\mathop{\longrightarrow }\limits^{{ \mathcal T }}\left(\begin{array}{c}{\psi }_{n,L}\\ -{\psi }_{n,R}\end{array}\right).\end{eqnarray} \tag{ 111 }$$

Meanwhile all chiral fermions carry charge-e each. To study interaction effects on the surface states, we bosonize the chiral fermion and obtain the following chiral boson action:

$$\begin{eqnarray}&&{{ \mathcal L }}_{0}=\displaystyle \sum _{n\in {\mathbb{Z}}/2}({\partial }_{t}{\phi }_{n}^{R}{\partial }_{y}{\phi }_{n}^{R}-{\partial }_{t}{\phi }_{n}^{L}{\partial }_{y}{\phi }_{n}^{L})+\cdots \end{eqnarray} \tag{ 112 }$$

with commutation relations

$$\begin{eqnarray}&&[{\phi }_{m}^{{\chi }_{1}}({y}_{1}),{\phi }_{n}^{{\chi }_{2}}({y}_{2})]=\hspace{1pt}{\rm{i}}\hspace{1pt}\pi {(-1)}^{{\chi }_{1}}\mathrm{Sign}({y}_{1}-{y}_{2}){\delta }_{m,n}{\delta }_{{\chi }_{1},{\chi }_{2}}.\end{eqnarray} \tag{ 113 }$$

The chiral fermions are related to chiral boson fields by

$$\begin{eqnarray}&&{\psi }_{n,R}\sim {{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{R}},\,\,\,{\psi }_{n,L}\sim {{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{L}}.\end{eqnarray} \tag{ 114 }$$

Hence the chiral bosons transform under symmetries as

$$\begin{eqnarray}&&{\phi }_{n}^{R/L}\mathop{\longrightarrow }\limits^{{{\rm{e}}}^{{\rm{i}}\alpha \hat{Q}}}{\phi }_{n}^{R/L}+\alpha ,\end{eqnarray} \tag{ 115 }$$

$$\begin{eqnarray}&&\left(\begin{array}{c}{\phi }_{n}^{R}\\ {\phi }_{n}^{L}\end{array}\right)\mathop{\longrightarrow }\limits^{{ \mathcal T }}\left(\begin{array}{c}-{\phi }_{n}^{L}\\ \pi -{\phi }_{n}^{R}\end{array}\right),\end{eqnarray} \tag{ 116 }$$

$$\begin{eqnarray}&&\left(\begin{array}{c}{\phi }_{n}^{R}\\ {\phi }_{n}^{L}\end{array}\right)\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}\left(\begin{array}{c}{\phi }_{n+\tfrac{1}{2}}^{L}\\ {\phi }_{n+\tfrac{1}{2}}^{R}+\pi \end{array}\right).\end{eqnarray} \tag{ 117 }$$

To reveal the connection to ${| \mathrm{BQSH}\rangle }_{{{ \mathcal G }}_{y}}$ case studied in section 3.4, we reorganize the chiral boson fields ${\phi }_{n}^{R/L}$ into a different basis:

$$\begin{eqnarray}&&{\phi }_{n}^{1}\equiv {(-1)}^{2n}\displaystyle \frac{{\phi }_{n}^{R}-{\phi }_{n}^{L}}{2},\,\,\,{\phi }_{n}^{2}\equiv {\phi }_{n}^{R}+{\phi }_{n}^{L}.\end{eqnarray} \tag{ 118 }$$

And it is straightforward to verify their commutation relation (53) with ${\bf{K}}=\left(\begin{array}{cc}0 & 1\\ 1 & 0\end{array}\right)$, as well as symmetry transformation rules:

$$\begin{eqnarray}&&\left(\begin{array}{c}{\phi }_{n}^{1}\\ {\phi }_{n}^{2}\end{array}\right)\mathop{\longrightarrow }\limits^{{{\rm{e}}}^{{\rm{i}}\alpha \hat{Q}}}\left(\begin{array}{c}{\phi }_{n}^{1}\\ {\phi }_{n}^{2}+2\alpha \end{array}\right),\end{eqnarray} \tag{ 119 }$$

$$\begin{eqnarray}&&\left(\begin{array}{c}{\phi }_{n}^{1}\\ {\phi }_{n}^{2}\end{array}\right)\mathop{\longrightarrow }\limits^{{ \mathcal T }}\left(\begin{array}{c}{\phi }_{n}^{1}-{(-1)}^{2n}\displaystyle \frac{\pi }{2}\\ \pi -{\phi }_{n}^{2}\end{array}\right),\end{eqnarray} \tag{ 120 }$$

$$\begin{eqnarray}&&\left(\begin{array}{c}{\phi }_{n}^{1}\\ {\phi }_{n}^{2}\end{array}\right)\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}\left(\begin{array}{c}{\phi }_{n+\tfrac{1}{2}}^{1}+\displaystyle \frac{\pi }{2}\\ {\phi }_{n+\tfrac{1}{2}}^{2}+\pi \end{array}\right).\end{eqnarray} \tag{ 121 }$$

It is straightforward to see that by depositing 1 spin chain (62) and (63) between two neighboring QSH layers, we can obtain a symmetric STO using the same interwire coupling terms (67) as the ${| \mathrm{BQSH}\rangle }_{{{ \mathcal G }}_{y}}$ case. The only difference is, to preserve time reversal symmetry (120), we must have

$$\begin{eqnarray}&&k=0\mathrm{mod}4\end{eqnarray} \tag{ 122 }$$

for interwire couplings (67) and resultant Z_k gauge theory on the side surface. In terms of original chiral boson fields $\{{\phi }_{n}^{R/L}\}$, the interwire couplings are written as

$$\begin{eqnarray}{{ \mathcal H }}_{{\rm{int}}} & = & \displaystyle \sum _{n\in \tfrac{{\mathbb{Z}}}{2}}{C}_{0}{\rm{\cos }}{\hat{L}}_{n}^{0}+{C}_{1}{\rm{\cos }}{\hat{L}}_{n}^{1},{L}_{n}^{0}={\varphi }_{n}-{(-1)}^{2n}\displaystyle \frac{k}{2}({\phi }_{n}^{R}-{\phi }_{n}^{L})+{\varphi }_{n+\tfrac{1}{2}},\\ {L}_{n}^{1} & = & {\phi }_{n}^{R}+{\phi }_{n}^{L}+k{(-1)}^{2n}{\theta }_{n+\tfrac{1}{2}}-{\phi }_{n+\tfrac{1}{2}}^{R}-{\phi }_{n+\tfrac{1}{2}}^{L}.\end{eqnarray} \tag{ 123 }$$

Following the same calculations for the ${| \mathrm{BQSH}\rangle }_{{{ \mathcal G }}_{y}}$ STO, we can identify the bulk anyons and their symmetry transformations ($\forall \,n\in {\mathbb{Z}}$)

$$\begin{eqnarray}&&\left(\begin{array}{c}{\phi }_{n}^{e}=\displaystyle \frac{1}{k}{\varphi }_{n}\\ {\phi }_{n}^{m}=-{\theta }_{n}-\displaystyle \frac{{\phi }_{n}^{R}+{\phi }_{n}^{L}}{k}\end{array}\right)\mathop{\longrightarrow }\limits^{{{\rm{e}}}^{{\rm{i}}\alpha \hat{Q}}}\left(\begin{array}{c}{\phi }_{n}^{e}\\ {\phi }_{n}^{m}-\displaystyle \frac{2\alpha }{k}\end{array}\right),\end{eqnarray} \tag{ 124 }$$

$$\begin{eqnarray}&&\left(\begin{array}{c}{\phi }_{n}^{e}\\ {\phi }_{n}^{m}\end{array}\right)\mathop{\longrightarrow }\limits^{{ \mathcal T }}\left(\begin{array}{c}{\phi }_{n}^{e}+\displaystyle \frac{\pi }{k}\\ -{\phi }_{n}^{m}-\displaystyle \frac{\pi }{k}\end{array}\right).\end{eqnarray} \tag{ 125 }$$

Due to non-locality of electron operator ${{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{R/L}}$, here the anyon hopping operators are different from ${| \mathrm{BQSH}\rangle }_{{{ \mathcal G }}_{y}}$ case:

$$\begin{eqnarray*}{T}_{n,n+\tfrac{1}{2}}^{e} & = & 0,\,{t}_{n}^{0}=\displaystyle \frac{1}{k},\,{t}_{n}^{1}=-\displaystyle \frac{1}{2}\Longrightarrow {\phi }_{n+\tfrac{1}{2}}^{e}=-\displaystyle \frac{1}{k}{\varphi }_{n+\tfrac{1}{2}}+{\phi }_{n}^{R}+\displaystyle \frac{k}{2}({\theta }_{n+\tfrac{1}{2}}-\displaystyle \frac{{\phi }_{n+\tfrac{1}{2}}^{R}+{\phi }_{n+\tfrac{1}{2}}^{L}}{k}),\,\,\\ {T}_{n,n+\tfrac{1}{2}}^{m} & = & {\theta }_{n},\,{t}_{n}^{1}=-\displaystyle \frac{1}{k}\Longrightarrow {\phi }_{n+\tfrac{1}{2}}^{m}={\theta }_{n+\tfrac{1}{2}}-\displaystyle \frac{{\phi }_{n+\tfrac{1}{2}}^{R}+{\phi }_{n+\tfrac{1}{2}}^{L}}{k}.\,\,\,\end{eqnarray*}$$

Again notice that we can redefine the anyon a and their hopping operators ${T}_{{n}_{1},{n}_{2}}^{a}$ by local bosonic operators that transform trivially under all on-site symmetries:

$$\begin{eqnarray}&&{\phi }_{n}^{a}\longrightarrow {\phi }_{n}^{a}+2{M}_{a}({\phi }_{n}^{R}-{\phi }_{n}^{L}),\,\,\,{M}_{a}\in {\mathbb{Z}}.\end{eqnarray} \tag{ 126 }$$

From this we can identify the generic glide symmetry operation on anyons:

$$\begin{eqnarray}&&\left(\begin{array}{c}{\phi }_{n}^{e}\\ {\phi }_{n}^{m}\end{array}\right)\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}\left(\begin{array}{c}\tfrac{k}{2}{\phi }_{n+\tfrac{1}{2}}^{m}-{\phi }_{n+\tfrac{1}{2}}^{e}\\ {\phi }_{n+\tfrac{1}{2}}^{m}\end{array}\right)+\left(\begin{array}{c}{\phi }_{n}^{R}\\ 0\end{array}\right)+2({\phi }_{n}^{R}-{\phi }_{n}^{L})\left(\begin{array}{c}{M}_{e}\\ {M}_{m}\end{array}\right).\end{eqnarray} \tag{ 127 }$$

Therefore to fully describe the symmetric STO, we need the following 4 × 4 ${\bf{K}}$ matrix and charge vector ${\bf{q}}$

$$\begin{eqnarray}{{\bf{K}}}_{{\rm{STO}}}=\left(\begin{array}{cccc}0 & -k & 0 & 0\\ -k & 0 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & -1\end{array}\right),\,\,\,k=0\mathrm{mod}4.\end{eqnarray} \tag{ 128 }$$

The associated excitations of the STO transform under global symmetries as

$$\begin{eqnarray}&&{\vec{\phi }}_{n}\equiv \left(\begin{array}{c}{\phi }_{n}^{e}\\ {\phi }_{n}^{m}\\ {\phi }_{n-\tfrac{1}{2}}^{R}\\ {\phi }_{n-\tfrac{1}{2}}^{L}\end{array}\right)\mathop{\longrightarrow }\limits^{{{\rm{e}}}^{{\rm{i}}\alpha \hat{Q}}}\left(\begin{array}{c}{\phi }_{n}^{e}\\ {\phi }_{n}^{m}-\displaystyle \frac{2\alpha }{k}\\ {\phi }_{n}^{R}+\alpha \\ {\phi }_{n}^{L}+\alpha \end{array}\right),\end{eqnarray} \tag{ 129 }$$

$$\begin{eqnarray}&&{\vec{\phi }}_{n}\mathop{\longrightarrow }\limits^{{ \mathcal T }}{{\bf{W}}}_{{ \mathcal T }}{\vec{\phi }}_{n}+\left(\begin{array}{c}\tfrac{\pi }{k}\\ -\tfrac{\pi }{k}\\ 0\\ \pi \end{array}\right),\,\,{{\bf{W}}}_{{ \mathcal T }}=\left(\begin{array}{cccc}1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 0 & -1\\ 0 & 0 & -1 & 0\end{array}\right).\,\,\end{eqnarray} \tag{ 130 }$$

Under glide operation, gauge charge e is dressed by an extra electron ${\psi }_{n,R}\sim {{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{R}}$:

$$\begin{eqnarray}&&\vec{\phi }\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}{{\bf{W}}}_{{{ \mathcal G }}_{y}}{\vec{\phi }}_{n+\tfrac{1}{2}}+\left(\begin{array}{c}0\\ 0\\ 0\\ \pi \end{array}\right),\\ {{\bf{W}}}_{{{ \mathcal G }}_{y}}=\left(\begin{array}{cccc}-1 & k/2 & 1+2{M}_{e} & -2{M}_{e}\\ 0 & 1 & 2{M}_{m} & -2{M}_{m}\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\end{array}\right),\,{M}_{e,m}\in {\mathbb{Z}}.\,\,\,\end{eqnarray} \tag{ 131 }$$

Similar to the bosonic case in section 3.4, the above glide operation doesn't preserve the form of ${{\bf{K}}}_{{\rm{STO}}}$ matrix: it is an 'anyonic symmetry' [13] that keeps the fractional statistics (S and T matrices) and all bulk data of the STO invariant.

To demonstrate the anomaly of the above glide symmetry operation on STO, we again gauge a discrete Z_N subgroup of the U(1) charge symmetry, generated by ${\hat{R}}_{N}\equiv \exp (\hspace{1pt}{\rm{i}}\hspace{1pt}\tfrac{2\pi }{N}\hat{Q})$. Under this discrete Z_N charge rotation, the chiral bosons $\vec{\phi }$ of STO transform as

$$\begin{eqnarray}&&\vec{\phi }\mathop{\longrightarrow }\limits^{{\hat{R}}_{N}}\left(\begin{array}{c}{\phi }^{e}\\ {\phi }^{m}-\tfrac{4\pi }{{kN}}\\ {\phi }^{R}+\tfrac{2\pi }{N}\\ {\phi }^{L}+\tfrac{2\pi }{N}\end{array}\right).\end{eqnarray} \tag{ 132 }$$

After gauging this discrete Z_N symmetry, the Z_N gauge flux

$$\begin{eqnarray}&&{{ \mathcal F }}_{{Z}_{N}}\sim {{\rm{e}}}^{{\rm{i}}{{\rm{\Phi }}}_{{Z}_{N}}},\,\,\,{{\rm{\Phi }}}_{n}^{{Z}_{N}}=\left(\displaystyle \frac{2}{N},0,\displaystyle \frac{1}{N},-\displaystyle \frac{1}{N}\right)\cdot {\vec{\phi }}_{n}\end{eqnarray} \tag{ 133 }$$

becomes a new deconfined anyon excitation. Consequently the anyon lattice of the gauged new topological order is expanded by four primitive vectors:

$$\begin{eqnarray}{\vec{l}}_{m} & = & {(\mathrm{1,}0,\mathrm{0,}0)}^{T},\,\,\,{\vec{l}}_{e}={(\mathrm{0,1},\mathrm{0,}0)}^{T},\\ {\vec{l}}_{{\psi }_{R}} & = & {(\mathrm{0,}0,\mathrm{1,}0)}^{T},\,\,\,{\vec{l}}_{{{ \mathcal F }}_{{Z}_{N}}}=\displaystyle \frac{1}{N}{(\mathrm{2,}\mathrm{0,}1,-1)}^{T}.\end{eqnarray} \tag{ 134 }$$

Under time reversal and glide operations, an arbitrary Abelian anyon a labeled by vector ${\vec{l}}_{a}$ is mapped to another vector

$$\begin{eqnarray}&&{\vec{l}}_{a}\mathop{\longrightarrow }\limits^{{ \mathcal T }}{{\bf{W}}}_{{ \mathcal T }}^{T}{\vec{l}}_{a},\,\,\,{\vec{l}}_{a}\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}{{\bf{W}}}_{{{ \mathcal G }}_{y}}^{T}{\vec{l}}_{a}.\end{eqnarray} \tag{ 135 }$$

The anyon lattice is preserved under time reversal, but under glide operation the new vector ${\vec{l}}_{{{ \mathcal F }}_{{Z}_{N}}}$ is mapped into

$$\begin{eqnarray}{{\bf{W}}}_{{{ \mathcal G }}_{y}}^{T}{\vec{l}}_{{{ \mathcal F }}_{{Z}_{N}}} & = & \displaystyle \frac{1}{N}{(-2,k,4{M}_{e}+\mathrm{1,1}-4{M}_{e})}^{T}\\ & = & -{\vec{l}}_{{{ \mathcal F }}_{{Z}_{N}}}+\displaystyle \frac{{(0,k,4{M}_{e}+2,-4{M}_{e})}^{T}}{N}\end{eqnarray} \tag{ 136 }$$

Therefore for any even integer $N\geqslant 4$ (equivalent to odd $N\geqslant 3$ due to conserved fermion parity), glide symmetry ${{\bf{W}}}_{{{ \mathcal G }}_{y}}$ does not preserve the anyon lattice structure and is anomalous. Physically, this implies that glide symmetry will not transform a $\pi /N$ flux into a $-\pi /N$ flux for $N\geqslant 3$ in the STO, a contradiction for any glide-symmetric topological order in a pure 2D system.

Similar to boson case in section 3.4, time reversal symmetry is crucial for the anomalous STO. If time reversal is absent, we can arbitrarily dress anyon and hopping operators by charge-neutral boson operators such as ${{\rm{e}}}^{{\rm{i}}({\phi }_{n}^{R}-{\phi }_{n}^{L})}$, ${{\rm{e}}}^{{\rm{i}}k{\phi }_{n}^{{\rm{e}}}}$ and ${{\rm{e}}}^{{\rm{i}}(2{\phi }_{n}^{R/L}+k{\phi }_{n}^{m})}$. Therefore one can redefine the glide action on fermion operator ${{\rm{e}}}^{{\rm{i}}{\phi }^{R}}$ as follows:

$$\begin{eqnarray}&&{{\rm{e}}}^{{\rm{i}}{\phi }_{n}^{R}}\mathop{\longrightarrow }\limits^{{{ \mathcal G }}_{y}}{{\rm{e}}}^{{\rm{i}}{\phi }_{n+\tfrac{1}{2}}^{L}}{{\rm{e}}}^{{\rm{i}}(2{\phi }_{n+\tfrac{1}{2}}^{R}+k{\phi }_{n+\tfrac{1}{2}}^{m})}.\end{eqnarray} \tag{ 137 }$$

Meanwhile the even integers 2M_a in (126) can be replaced by an arbitrary integer ${M}_{a}\in {\mathbb{Z}}$. The new glide symmetry operation can be written as following

$$\begin{eqnarray}{{\bf{W}}}_{{{ \mathcal G }}_{y}}=\left(\begin{array}{cccc}-1 & k/2 & 1+{M}_{e} & -{M}_{e}\\ 0 & 1 & {M}_{m} & -{M}_{m}\\ 0 & k & 2 & 1\\ 0 & 0 & 1 & 0\end{array}\right),\,{M}_{e,m}\in {\mathbb{Z}}.,\end{eqnarray} \tag{ 138 }$$

Therefore we have

$$\begin{eqnarray}{{\bf{W}}}_{{{ \mathcal G }}_{y}}^{T}{\vec{l}}_{{{ \mathcal F }}_{{Z}_{N}}} & = & \displaystyle \frac{1}{N}{(-\mathrm{2,}\mathrm{0,}2{M}_{e}-\mathrm{1,1}-2{M}_{e})}^{T}\\ & = & -{\vec{l}}_{{{ \mathcal F }}_{{Z}_{N}}}+\displaystyle \frac{{(\mathrm{0,0,2}{M}_{e},-2{M}_{e})}^{T}}{N}.\end{eqnarray} \tag{ 139 }$$

By choosing M_e = 0 the anyon lattice remains invariant under glide operation, and hence there is no anomaly without time reversal symmetry. Therefore we have shown that time reversal symmetry ${ \mathcal T }$ is necessary for the anomalous STO.

So far, we have shown that the gapless 'hourglass fermion' surface states on glide-invariant [001] side surface of 3D non-symmorphic TI can be gapped out without breaking any symmetry, through strong electronic interactions described in (123). The consequent gapped STO preserves all symmetries, and in particular the glide symmetry is implemented in an anomalous way that is impossible in any pure 2D system. This establishes a bulk-boundary correspondence for fermion GSPT phases in the strong-interacting limit. As a result the 3D fermion GSPT phases in class AII have a ${{\mathbb{Z}}}_{2}$ classification (see table 2), where the nontrivial GSPT phase ${| \mathrm{QSH}\rangle }_{{{ \mathcal G }}_{y}}$ hosts the hourglass fermion surface states in the weakly-interacting limit.

Finally we consider U(1) spin rotational symmetry and time reversal associated with onsite symmetry group ${G}_{0}=U(1)\times {Z}_{2}^{{ \mathcal T }}$, which corresponds to symmetry class AIII. The associated 2D SRE fermion phases have a trivial ${{\mathbb{Z}}}_{1}$ classification, without any nontrivial bosonic/fermionic SPT phases [43]. Therefore the 3D fermion GSPT phases in class AIII also have a trivial ${{\mathbb{Z}}}_{1}$ classification, see table 2.