Physical Review Letters

Construction of bosonic symmetry-protected-trivial states and their topological invariants via G x SO(∞) nonlinear σ

Xiao-Gang Wen

Published May 1, 2015• 4 citations

Abstract

It has been shown that the bosonic symmetry-protected-trivial (SPT) phases with pure gauge anomalous boundary can all be realized via nonlinear σ models (NLσMs) of the symmetry group G with various topological terms. Those SPT phases (called the pure SPT phases) can be classified by group cohomology H[superscript d](G,R/Z). But there are also SPT phases with mixed gauge-gravity anomalous boundary (which will be called the mixed SPT phases). Some of the mixed SPT states were also referred as the beyond-group-cohomology SPT states. In this paper, we show that those beyond-group-cohomology SPT states are actually within another type of group cohomology classification. More precisely, we show that both the pure and the mixed SPT phases can be realized by G × SO(∞) NLσMs with various topological terms. Through the group cohomology H[superscript d][G × SO(∞),R/Z], we find that the set of our constructed SPT phases in d-dimensional space-time are described by E[superscript d](G) ⋊ ⊕[d−1 over k=1]H[superscript k](G,iTO[d−k over L]) ⊕ H[superscript d](G,R/Z) where G may contain time reversal. Here iTO[d over L] is the set of the topologically ordered phases in d-dimensional space-time that have no topological excitations, and one has iTO[1 over L] = iTO[2 over L] = iTO[4 over L] = iTO[6 over L] = 0, iTO[3 over L] = Z, iTO[5 over L] = Z[subscript 2], iTO[7 over L] =2Z. For G=U(1) ⋊ Z[ T over 2] (charge conservation and time-reversal symmetry), we find that the mixed SPT phases beyond H[superscript d][U(1) ⋊ Z[T over 2],R/Z] are described by Z[subscript 2] in 3 + 1D, Z in 4 + 1D, 3Z[subscript 2] in 5 + 1D, and 4Z[subscript 2] in 6 + 1D. Our construction also gives us the topological invariants that fully characterize the corresponding SPT and iTO phases. Through several examples, we show how the universal physical properties of SPT phases can be obtained from those topological invariants.

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Construction of bosonic symmetry-protected-trivial states and their topological invariants via G×SO(∞) nonlinear σ models

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Symmetry protected topological orders and the group cohomology of their symmetry group

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Symmetry protected topological orders and the cohomology class of their symmetry group

Xie Chen, Zheng‐Cheng Gu, Zheng-Xin Liu +1 more

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Bosonic crystalline symmetry protected topological phases beyond the group cohomology proposal

Hao Song, Charles Zhaoxi Xiong, Sheng-Jie Huang

arXiv:1811.06558Physical review. B./Physical review. BApr 22, 202019 citations

Physical Review Letters

Construction of bosonic symmetry-protected-trivial states and their topological invariants via G x SO(∞) nonlinear σ

Xiao-Gang Wen

Published May 1, 2015• 4 citations

OpenAlex

Abstract

It has been shown that the bosonic symmetry-protected-trivial (SPT) phases with pure gauge anomalous boundary can all be realized via nonlinear σ models (NLσMs) of the symmetry group G with various topological terms. Those SPT phases (called the pure SPT phases) can be classified by group cohomology Hsuperscript d. But there are also SPT phases with mixed gauge-gravity anomalous boundary (which will be called the mixed SPT phases). Some of the mixed SPT states were also referred as the beyond-group-cohomology SPT states. In this paper, we show that those beyond-group-cohomology SPT states are actually within another type of group cohomology classification. More precisely, we show that both the pure and the mixed SPT phases can be realized by G × SO(∞) NLσMs with various topological terms. Through the group cohomology H[superscript d][G × SO(∞),R/Z], we find that the set of our constructed SPT phases in d-dimensional space-time are described by Esuperscript d ⋊ ⊕[d−1 over k=1]H[superscript k](G,iTO[d−k over L]) ⊕ Hsuperscript d where G may contain time reversal. Here iTO[d over L] is the set of the topologically ordered phases in d-dimensional space-time that have no topological excitations, and one has iTO[1 over L] = iTO[2 over L] = iTO[4 over L] = iTO[6 over L] = 0, iTO[3 over L] = Z, iTO[5 over L] = Z[subscript 2], iTO[7 over L] =2Z. For G=U(1) ⋊ Z[ T over 2] (charge conservation and time-reversal symmetry), we find that the mixed SPT phases beyond H[superscript d][U(1) ⋊ Z[T over 2],R/Z] are described by Z[subscript 2] in 3 + 1D, Z in 4 + 1D, 3Z[subscript 2] in 5 + 1D, and 4Z[subscript 2] in 6 + 1D. Our construction also gives us the topological invariants that fully characterize the corresponding SPT and iTO phases. Through several examples, we show how the universal physical properties of SPT phases can be obtained from those topological invariants.

Related papers

Construction of bosonic symmetry-protected-trivial states and their topological invariants via G×SO(∞) nonlinear σ models

Xiao-Gang Wen

It has been shown that the bosonic symmetry-protected-trivial (SPT) phases with pure gauge anomalous boundary can all be realized via nonlinear $\ensuremath{\sigma}$ models $(\mathrm{NL}\ensuremath{\sigma}\mathrm{Ms})$ of the symmetry group $G$ with various topological terms. Those SPT phases (called the pure SPT phases) can be classified by group cohomology ${\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})$ . But there are also SPT phases with mixed gauge-gravity anomalous boundary (which will be called the mixed SPT phases). Some of the mixed SPT states were also referred as the beyond-group-cohomology SPT states. In this paper, we show that those beyond-group-cohomology SPT states are actually within another type of group cohomology classification. More precisely, we show that both the pure and the mixed SPT phases can be realized by $G\ifmmode\times\else\texttimes\fi{}SO(\ensuremath{\infty}) \mathrm{NL}\ensuremath{\sigma}\mathrm{Ms}$ with various topological terms. Through the group cohomology ${\mathcal{H}}^{d}[G\ifmmode\times\else\texttimes\fi{}SO(\ensuremath{\infty}),\mathbb{R}/\mathbb{Z}]$, we find that the set of our constructed SPT phases in $d$ -dimensional space-time are described by ${E}^{d}(G)\ensuremath{\rtimes}{\ensuremath{\bigoplus}}_{k=1}^{d\ensuremath{-}1}{\mathcal{H}}^{k}(G,{\text{iTO}}_{L}^{d\ensuremath{-}k})\ensuremath{\bigoplus}{\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})$ where $G$ may contain time reversal. Here ${\text{iTO}}_{L}^{d}$ is the set of the topologically ordered phases in $d$ -dimensional space-time that have no topological excitations, and one has ${\text{iTO}}_{L}^{1}={\text{iTO}}_{L}^{2}={\text{iTO}}_{L}^{4}={\text{iTO}}_{L}^{6}=0,\phantom{\rule{0.16em}{0ex}}{\text{iTO}}_{L}^{3}=\mathbb{Z},\phantom{\rule{0.16em}{0ex}}{\text{iTO}}_{L}^{5}={\mathbb{Z}}_{2},\phantom{\rule{0.16em}{0ex}}{\text{iTO}}_{L}^{7}=2\mathbb{Z}$ . For $G=U(1)\ensuremath{\rtimes}{Z}_{2}^{T}$ (charge conservation and time-reversal symmetry), we find that the mixed SPT phases beyond ${\mathcal{H}}^{d}[U(1)\ensuremath{\rtimes}{Z}_{2}^{T},\mathbb{R}/\mathbb{Z}]$ are described by ${\mathbb{Z}}_{2}$ in 3 + 1D, $\mathbb{Z}$ in 4 + 1D, $3{\mathbb{Z}}_{2}$ in 5 + 1D, and $4{\mathbb{Z}}_{2}$ in 6 + 1D. Our construction also gives us the topological invariants that fully characterize the corresponding SPT and iTO phases. Through several examples, we show how the universal physical properties of SPT phases can be obtained from those topological invariants.It has been shown that the bosonic symmetry-protected-trivial (SPT) phases with pure gauge anomalous boundary can all be realized via nonlinear $\ensuremath{\sigma}$ models $(\mathrm{NL}\ensuremath{\sigma}\mathrm{Ms})$ of the symmetry group $G$ with various topological terms. Those SPT phases (called the pure SPT phases) can be classified by group cohomology ${\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})$. But there are also SPT phases with mixed gauge-gravity anomalous boundary (which will be called the mixed SPT phases). Some of the mixed SPT states were also referred as the beyond-group-cohomology SPT states. In this paper, we show that those beyond-group-cohomology SPT states are actually within another type of group cohomology classification. More precisely, we show that both the pure and the mixed SPT phases can be realized by $G\ifmmode\times\else\texttimes\fi{}SO(\ensuremath{\infty}) \mathrm{NL}\ensuremath{\sigma}\mathrm{Ms}$ with various topological terms. Through the group cohomology ${\mathcal{H}}^{d}[G\ifmmode\times\else\texttimes\fi{}SO(\ensuremath{\infty}),\mathbb{R}/\mathbb{Z}]$, we find that the set of our constructed SPT phases in $d$-dimensional space-time are described by ${E}^{d}(G)\ensuremath{\rtimes}{\ensuremath{\bigoplus}}_{k=1}^{d\ensuremath{-}1}{\mathcal{H}}^{k}(G,{\text{iTO}}_{L}^{d\ensuremath{-}k})\ensuremath{\bigoplus}{\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})$ where $G$ may contain time reversal. Here ${\text{iTO}}_{L}^{d}$ is the set of the topologically ordered phases in $d$-dimensional space-time that have no topological excitations, and one has ${\text{iTO}}_{L}^{1}={\text{iTO}}_{L}^{2}={\text{iTO}}_{L}^{4}={\text{iTO}}_{L}^{6}=0,\phantom{\rule{0.16em}{0ex}}{\text{iTO}}_{L}^{3}=\mathbb{Z},\phantom{\rule{0.16em}{0ex}}{\text{iTO}}_{L}^{5}={\mathbb{Z}}_{2},\phantom{\rule{0.16em}{0ex}}{\text{iTO}}_{L}^{7}=2\mathbb{Z}$. For $G=U(1)\ensuremath{\rtimes}{Z}_{2}^{T}$ (charge conservation and time-reversal symmetry), we find that the mixed SPT phases beyond ${\mathcal{H}}^{d}[U(1)\ensuremath{\rtimes}{Z}_{2}^{T},\mathbb{R}/\mathbb{Z}]$ are described by ${\mathbb{Z}}_{2}$ in 3 + 1D, $\mathbb{Z}$ in 4 + 1D, $3{\mathbb{Z}}_{2}$ in 5 + 1D, and $4{\mathbb{Z}}_{2}$ in 6 + 1D. Our construction also gives us the topological invariants that fully characterize the corresponding SPT and iTO phases. Through several examples, we show how the universal physical properties of SPT phases can be obtained from those topological invariants.

arXiv:1410.8477Physical Review BMay 4, 201577 citations

Symmetry protected topological orders and the group cohomology of their symmetry group

Xie Chen, Zheng‐Cheng Gu, Zheng-Xin Liu +1 more

Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry $G$ . They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phases which is protected by $\mathit{SO}(3)$ spin rotation symmetry. The topological insulator is another example of SPT phases which are protected by $U(1)$ and time-reversal symmetries. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: Distinct $d$ -dimensional bosonic SPT phases with on-site symmetry $G$ (which may contain antiunitary time-reversal symmetry) can be labeled by the elements in ${\mathcal{H}}^{1+d}[G,{U}_{T}(1)]$ , the Borel $(1+d)$ -group-cohomology classes of $G$ over the $G$ module ${U}_{T}(1)$ . Our theory, which leads to explicit ground-state wave functions and commuting projector Hamiltonians, is based on a new type of topological term that generalizes the topological $\ensuremath{\theta}$ term in continuous nonlinear $\ensuremath{\sigma}$ models to lattice nonlinear $\ensuremath{\sigma}$ models. The boundary excitations of the nontrivial SPT phases are described by lattice nonlinear $\ensuremath{\sigma}$ models with a nonlocal Lagrangian term that generalizes the Wess-Zumino-Witten term for continuous nonlinear $\ensuremath{\sigma}$ models. As a result, the symmetry $G$ must be realized as a non-on-site symmetry for the low-energy boundary excitations, and those boundary states must be gapless or degenerate. As an application of our result, we can use ${\mathcal{H}}^{1+d}[U(1)\ensuremath{\rtimes}{Z}_{2}^{T},{U}_{T}(1)]$ to obtain interacting bosonic topological insulators (protected by time reversal ${Z}_{2}^{T}$ and boson number conservation), which contain one nontrivial phase in one-dimensional (1D) or 2D and three in 3D. We also obtain interacting bosonic topological superconductors (protected by time-reversal symmetry only), in term of ${\mathcal{H}}^{1+d}[{Z}_{2}^{T},{U}_{T}(1)]$ , which contain one nontrivial phase in odd spatial dimensions and none for even dimensions. Our result is much more general than the above two examples, since it is for any symmetry group. For example, we can use ${\mathcal{H}}^{1+d}[U(1)\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{T},{U}_{T}(1)]$ to construct the SPT phases of integer spin systems with time-reversal and $U(1)$ spin rotation symmetry, which contain three nontrivial SPT phases in 1D, none in 2D, and seven in 3D. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: $({G}_{H},{G}_{\ensuremath{\Psi}},{\mathcal{H}}^{1+d}[{G}_{\ensuremath{\Psi}},{U}_{T}(1)])$, where ${G}_{H}$ is the symmetry group of the Hamiltonian and ${G}_{\ensuremath{\Psi}}$ the symmetry group of the ground states.Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry $G$. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phases which is protected by $\mathit{SO}(3)$ spin rotation symmetry. The topological insulator is another example of SPT phases which are protected by $U(1)$ and time-reversal symmetries. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: Distinct $d$-dimensional bosonic SPT phases with on-site symmetry $G$ (which may contain antiunitary time-reversal symmetry) can be labeled by the elements in ${\mathcal{H}}^{1+d}[G,{U}_{T}(1)]$, the Borel $(1+d)$-group-cohomology classes of $G$ over the $G$ module ${U}_{T}(1)$. Our theory, which leads to explicit ground-state wave functions and commuting projector Hamiltonians, is based on a new type of topological term that generalizes the topological $\ensuremath{\theta}$ term in continuous nonlinear $\ensuremath{\sigma}$ models to lattice nonlinear $\ensuremath{\sigma}$ models. The boundary excitations of the nontrivial SPT phases are described by lattice nonlinear $\ensuremath{\sigma}$ models with a nonlocal Lagrangian term that generalizes the Wess-Zumino-Witten term for continuous nonlinear $\ensuremath{\sigma}$ models. As a result, the symmetry $G$ must be realized as a non-on-site symmetry for the low-energy boundary excitations, and those boundary states must be gapless or degenerate. As an application of our result, we can use ${\mathcal{H}}^{1+d}[U(1)\ensuremath{\rtimes}{Z}_{2}^{T},{U}_{T}(1)]$ to obtain interacting bosonic topological insulators (protected by time reversal ${Z}_{2}^{T}$ and boson number conservation), which contain one nontrivial phase in one-dimensional (1D) or 2D and three in 3D. We also obtain interacting bosonic topological superconductors (protected by time-reversal symmetry only), in term of ${\mathcal{H}}^{1+d}[{Z}_{2}^{T},{U}_{T}(1)]$, which contain one nontrivial phase in odd spatial dimensions and none for even dimensions. Our result is much more general than the above two examples, since it is for any symmetry group. For example, we can use ${\mathcal{H}}^{1+d}[U(1)\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{T},{U}_{T}(1)]$ to construct the SPT phases of integer spin systems with time-reversal and $U(1)$ spin rotation symmetry, which contain three nontrivial SPT phases in 1D, none in 2D, and seven in 3D. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: $({G}_{H},{G}_{\ensuremath{\Psi}},{\mathcal{H}}^{1+d}[{G}_{\ensuremath{\Psi}},{U}_{T}(1)])$, where ${G}_{H}$ is the symmetry group of the Hamiltonian and ${G}_{\ensuremath{\Psi}}$ the symmetry group of the ground states.

arXiv:1106.4772Physical Review BApr 4, 20131,308 citations

Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory

Zheng‐Cheng Gu, Xiao-Gang Wen

Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry $G$ , which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of supercocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (i.e., non-onsite) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric boundary must be gapless for (1+1)-dimensional [(1+1)D] boundary, and must be gapless or topologically ordered beyond (1+1)D. As an application of this general result, we construct a new SPT phase in three dimensions, for interacting fermionic superconductors with coplanar spin order (which have ${T}^{2}=1$ time-reversal ${Z}_{2}^{T}$ and fermion-number-parity ${Z}_{2}^{f}$ symmetries described by a full symmetry group ${Z}_{2}^{T}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion pairs), and thus are not included in the $K$ -theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in two dimensions (2D) with a full symmetry group ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$. Those 2D fermionic SPT phases all have central-charge $c=1$ gapless edge excitations, if the symmetry is not broken.Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry $G$, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of supercocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (i.e., non-onsite) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric boundary must be gapless for (1+1)-dimensional [(1+1)D] boundary, and must be gapless or topologically ordered beyond (1+1)D. As an application of this general result, we construct a new SPT phase in three dimensions, for interacting fermionic superconductors with coplanar spin order (which have ${T}^{2}=1$ time-reversal ${Z}_{2}^{T}$ and fermion-number-parity ${Z}_{2}^{f}$ symmetries described by a full symmetry group ${Z}_{2}^{T}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion pairs), and thus are not included in the $K$-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in two dimensions (2D) with a full symmetry group ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$. Those 2D fermionic SPT phases all have central-charge $c=1$ gapless edge excitations, if the symmetry is not broken.

arXiv:1201.2648Physical Review BSep 23, 2014306 citations

Classification and description of bosonic symmetry protected topological phases with semiclassical nonlinear sigma models

Zhen Bi, Alex Rasmussen, Kevin Slagle +1 more

In this paper we systematically classify and describe bosonic symmetry protected topological (SPT) phases in all physical spatial dimensions using semiclassical nonlinear Sigma model (NLSM) field theories. All the SPT phases on a $d-$ dimensional lattice discussed in this paper can be described by the same NLSM, which is an O(d+2) NLSM in $(d+1)-$ dimensional space-time, with a topological $Θ-$ term. The field in the NLSM is a semiclassical Landau order parameter with a unit length constraint. The classification of SPT phases discussed in this paper based on their NLSMs is consistent with the more mathematical classification based on group cohomology. Besides the classification, the formalism used in this paper also allows us to explicitly discuss the physics at the boundary of the SPT phases, and it reveals the relation between SPT phases with different symmetries. For example, it gives many of these SPT states a natural "decorated defect" construction.In this paper we systematically classify and describe bosonic symmetry protected topological (SPT) phases in all physical spatial dimensions using semiclassical nonlinear Sigma model (NLSM) field theories. All the SPT phases on a $d-$dimensional lattice discussed in this paper can be described by the same NLSM, which is an O(d+2) NLSM in $(d+1)-$dimensional space-time, with a topological $Θ-$term. The field in the NLSM is a semiclassical Landau order parameter with a unit length constraint. The classification of SPT phases discussed in this paper based on their NLSMs is consistent with the more mathematical classification based on group cohomology. Besides the classification, the formalism used in this paper also allows us to explicitly discuss the physics at the boundary of the SPT phases, and it reveals the relation between SPT phases with different symmetries. For example, it gives many of these SPT states a natural "decorated defect" construction.

arXiv:1309.0515Physical Review BApr 6, 2015108 citations

Symmetry protected topological orders and the cohomology class of their symmetry group

Xie Chen, Zheng‐Cheng Gu, Zheng-Xin Liu +1 more

Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phase which is protected by SO(3) spin rotation symmetry. The topological insulator is another exam- ple of SPT phase which is protected by U(1) and time reversal symmetries. It has been shown that free fermion SPT phases can be systematically described by the K-theory. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: distinct d-dimensional bosonic SPT phases with on-site symmetry G (which may contain anti-unitary time reversal symmetry) can be labeled by the elements in H^{1+d}[G, U_T(1)] - the Borel (1 + d)-group-cohomology classes of G over the G-module U_T(1). The boundary excitations of the non-trivial SPT phases are gapless or degenerate. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: (G_H, G_{\Psi}, H^{1+d}[G_{\Psi}, U_T(1)], where G_H is the symmetry group of the Hamiltonian and G_{\Psi} the symmetry group of the ground states.Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phase which is protected by SO(3) spin rotation symmetry. The topological insulator is another exam- ple of SPT phase which is protected by U(1) and time reversal symmetries. It has been shown that free fermion SPT phases can be systematically described by the K-theory. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: distinct d-dimensional bosonic SPT phases with on-site symmetry G (which may contain anti-unitary time reversal symmetry) can be labeled by the elements in H^{1+d}[G, U_T(1)] - the Borel (1 + d)-group-cohomology classes of G over the G-module U_T(1). The boundary excitations of the non-trivial SPT phases are gapless or degenerate. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: (G_H, G_{\Psi}, H^{1+d}[G_{\Psi}, U_T(1)], where G_H is the symmetry group of the Hamiltonian and G_{\Psi} the symmetry group of the ground states.

Jun 23, 201113 citations

Bosonic crystalline symmetry protected topological phases beyond the group cohomology proposal

Hao Song, Charles Zhaoxi Xiong, Sheng-Jie Huang

It is demonstrated by explicit construction that three-dimensional bosonic crystalline symmetry protected topological (cSPT) phases are classified by ${H}_{\ensuremath{\phi}}^{5}(G;\mathbb{Z})\ensuremath{\bigoplus}{H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ for all 230 space groups $G$ , where ${H}_{\ensuremath{\phi}}^{n}(G;\mathbb{Z})$ denotes the $n\mathrm{th}$ twisted group cohomology of $G$ with $\mathbb{Z}$ coefficients, and $\ensuremath{\phi}$ indicates that $g\ensuremath{\in}G$ acts nontrivially on coefficients by sending them to their inverses if $g$ reverses space-time orientation and acts trivially otherwise. The previously known summand ${H}_{\ensuremath{\phi}}^{5}(G;\mathbb{Z})$ corresponds only to crystalline phases built without the ${E}_{8}$ state or its multiples on 2-cells of space. It is the crystalline analog of the ``group cohomology proposal'' for classifying bosonic symmetry protected topological (SPT) phases, which takes the form ${H}_{\ensuremath{\phi}}^{d+2}(G;\mathbb{Z})\ensuremath{\cong}{H}_{\ensuremath{\phi}}^{d+1}(G;\text{U}(1))$ for finite internal symmetry groups in $d$ spatial dimensions. The new summand ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ classifies possible configurations of ${E}_{8}$ states on 2-cells that can be used to build crystalline phases beyond the group cohomology proposal. The completeness of our classification and the physical meaning of ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ are established through a combination of dimensional reduction, surface topological order, and explicit cellular construction. The value of ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ can be easily read off from the international symbol for $G$ . Our classification agrees with the prediction of the ``generalized cohomology hypothesis,'' which concerns the general structure of the classification of SPT phases, and therefore provides strong evidence for the validity of the said hypothesis in the realm of crystalline symmetries.It is demonstrated by explicit construction that three-dimensional bosonic crystalline symmetry protected topological (cSPT) phases are classified by ${H}_{\ensuremath{\phi}}^{5}(G;\mathbb{Z})\ensuremath{\bigoplus}{H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ for all 230 space groups $G$, where ${H}_{\ensuremath{\phi}}^{n}(G;\mathbb{Z})$ denotes the $n\mathrm{th}$ twisted group cohomology of $G$ with $\mathbb{Z}$ coefficients, and $\ensuremath{\phi}$ indicates that $g\ensuremath{\in}G$ acts nontrivially on coefficients by sending them to their inverses if $g$ reverses space-time orientation and acts trivially otherwise. The previously known summand ${H}_{\ensuremath{\phi}}^{5}(G;\mathbb{Z})$ corresponds only to crystalline phases built without the ${E}_{8}$ state or its multiples on 2-cells of space. It is the crystalline analog of the ``group cohomology proposal'' for classifying bosonic symmetry protected topological (SPT) phases, which takes the form ${H}_{\ensuremath{\phi}}^{d+2}(G;\mathbb{Z})\ensuremath{\cong}{H}_{\ensuremath{\phi}}^{d+1}(G;\text{U}(1))$ for finite internal symmetry groups in $d$ spatial dimensions. The new summand ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ classifies possible configurations of ${E}_{8}$ states on 2-cells that can be used to build crystalline phases beyond the group cohomology proposal. The completeness of our classification and the physical meaning of ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ are established through a combination of dimensional reduction, surface topological order, and explicit cellular construction. The value of ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ can be easily read off from the international symbol for $G$. Our classification agrees with the prediction of the ``generalized cohomology hypothesis,'' which concerns the general structure of the classification of SPT phases, and therefore provides strong evidence for the validity of the said hypothesis in the realm of crystalline symmetries.

arXiv:1811.06558Physical review. B./Physical review. BApr 22, 202019 citations

G

with various topological terms. Those SPT phases (called the pure SPT phases) can be classified by group cohomology

{\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})

. But there are also SPT phases with mixed gauge-gravity anomalous boundary (which will be called the mixed SPT phases). Some of the mixed SPT states were also referred as the beyond-group-cohomology SPT states. In this paper, we show that those beyond-group-cohomology SPT states are actually within another type of group cohomology classification. More precisely, we show that both the pure and the mixed SPT phases can be realized by $G\ifmmode\times\else\texttimes\fi{}SO(\ensuremath{\infty}) \mathrm{NL}\ensuremath{\sigma}\mathrm{Ms}$ with various topological terms. Through the group cohomology ${\mathcal{H}}^{d}[G\ifmmode\times\else\texttimes\fi{}SO(\ensuremath{\infty}),\mathbb{R}/\mathbb{Z}]$, we find that the set of our constructed SPT phases in

d

-dimensional space-time are described by ${E}^{d}(G)\ensuremath{\rtimes}{\ensuremath{\bigoplus}}_{k=1}^{d\ensuremath{-}1}{\mathcal{H}}^{k}(G,{\text{iTO}}_{L}^{d\ensuremath{-}k})\ensuremath{\bigoplus}{\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})$ where

G

may contain time reversal. Here

{\text{iTO}}_{L}^{d}

is the set of the topologically ordered phases in

d

-dimensional space-time that have no topological excitations, and one has

{\text{iTO}}_{L}^{1}={\text{iTO}}_{L}^{2}={\text{iTO}}_{L}^{4}={\text{iTO}}_{L}^{6}=0,\phantom{\rule{0.16em}{0ex}}{\text{iTO}}_{L}^{3}=\mathbb{Z},\phantom{\rule{0.16em}{0ex}}{\text{iTO}}_{L}^{5}={\mathbb{Z}}_{2},\phantom{\rule{0.16em}{0ex}}{\text{iTO}}_{L}^{7}=2\mathbb{Z}

. For $G=U(1)\ensuremath{\rtimes}{Z}_{2}^{T}$ (charge conservation and time-reversal symmetry), we find that the mixed SPT phases beyond ${\mathcal{H}}^{d}[U(1)\ensuremath{\rtimes}{Z}_{2}^{T},\mathbb{R}/\mathbb{Z}]$ are described by

{\mathbb{Z}}_{2}

in 3 + 1D,

\mathbb{Z}

in 4 + 1D,

3{\mathbb{Z}}_{2}

in 5 + 1D, and

4{\mathbb{Z}}_{2}

in 6 + 1D. Our construction also gives us the topological invariants that fully characterize the corresponding SPT and iTO phases. Through several examples, we show how the universal physical properties of SPT phases can be obtained from those topological invariants.

Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry

G

. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phases which is protected by

\mathit{SO}(3)

spin rotation symmetry. The topological insulator is another example of SPT phases which are protected by

U(1)

and time-reversal symmetries. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: Distinct

d

-dimensional bosonic SPT phases with on-site symmetry

G

(which may contain antiunitary time-reversal symmetry) can be labeled by the elements in

{\mathcal{H}}^{1+d}[G,{U}_{T}(1)]

, the Borel

(1+d)

-group-cohomology classes of

G

over the

G

module

{U}_{T}(1)

. Our theory, which leads to explicit ground-state wave functions and commuting projector Hamiltonians, is based on a new type of topological term that generalizes the topological $\ensuremath{\theta}$ term in continuous nonlinear $\ensuremath{\sigma}$ models to lattice nonlinear $\ensuremath{\sigma}$ models. The boundary excitations of the nontrivial SPT phases are described by lattice nonlinear $\ensuremath{\sigma}$ models with a nonlocal Lagrangian term that generalizes the Wess-Zumino-Witten term for continuous nonlinear $\ensuremath{\sigma}$ models. As a result, the symmetry

G

must be realized as a non-on-site symmetry for the low-energy boundary excitations, and those boundary states must be gapless or degenerate. As an application of our result, we can use ${\mathcal{H}}^{1+d}[U(1)\ensuremath{\rtimes}{Z}_{2}^{T},{U}_{T}(1)]$ to obtain interacting bosonic topological insulators (protected by time reversal

{Z}_{2}^{T}

and boson number conservation), which contain one nontrivial phase in one-dimensional (1D) or 2D and three in 3D. We also obtain interacting bosonic topological superconductors (protected by time-reversal symmetry only), in term of

{\mathcal{H}}^{1+d}[{Z}_{2}^{T},{U}_{T}(1)]

, which contain one nontrivial phase in odd spatial dimensions and none for even dimensions. Our result is much more general than the above two examples, since it is for any symmetry group. For example, we can use ${\mathcal{H}}^{1+d}[U(1)\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{T},{U}_{T}(1)]$ to construct the SPT phases of integer spin systems with time-reversal and

U(1)

spin rotation symmetry, which contain three nontrivial SPT phases in 1D, none in 2D, and seven in 3D. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: $({G}_{H},{G}_{\ensuremath{\Psi}},{\mathcal{H}}^{1+d}[{G}_{\ensuremath{\Psi}},{U}_{T}(1)])$, where

{G}_{H}

is the symmetry group of the Hamiltonian and ${G}_{\ensuremath{\Psi}}$ the symmetry group of the ground states.

Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry

G

, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of supercocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (i.e., non-onsite) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric boundary must be gapless for (1+1)-dimensional [(1+1)D] boundary, and must be gapless or topologically ordered beyond (1+1)D. As an application of this general result, we construct a new SPT phase in three dimensions, for interacting fermionic superconductors with coplanar spin order (which have

{T}^{2}=1

time-reversal

{Z}_{2}^{T}

and fermion-number-parity

{Z}_{2}^{f}

symmetries described by a full symmetry group ${Z}_{2}^{T}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion pairs), and thus are not included in the

K

-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in two dimensions (2D) with a full symmetry group ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$. Those 2D fermionic SPT phases all have central-charge

c=1

gapless edge excitations, if the symmetry is not broken.

d-

dimensional lattice discussed in this paper can be described by the same NLSM, which is an O(d+2) NLSM in

(d+1)-

dimensional space-time, with a topological

Θ-

term. The field in the NLSM is a semiclassical Landau order parameter with a unit length constraint. The classification of SPT phases discussed in this paper based on their NLSMs is consistent with the more mathematical classification based on group cohomology. Besides the classification, the formalism used in this paper also allows us to explicitly discuss the physics at the boundary of the SPT phases, and it reveals the relation between SPT phases with different symmetries. For example, it gives many of these SPT states a natural "decorated defect" construction.

G

, where ${H}_{\ensuremath{\phi}}^{n}(G;\mathbb{Z})$ denotes the

n\mathrm{th}

twisted group cohomology of

G

with

\mathbb{Z}

coefficients, and $\ensuremath{\phi}$ indicates that $g\ensuremath{\in}G$ acts nontrivially on coefficients by sending them to their inverses if

g

reverses space-time orientation and acts trivially otherwise. The previously known summand ${H}_{\ensuremath{\phi}}^{5}(G;\mathbb{Z})$ corresponds only to crystalline phases built without the

{E}_{8}

state or its multiples on 2-cells of space. It is the crystalline analog of the ``group cohomology proposal'' for classifying bosonic symmetry protected topological (SPT) phases, which takes the form ${H}_{\ensuremath{\phi}}^{d+2}(G;\mathbb{Z})\ensuremath{\cong}{H}_{\ensuremath{\phi}}^{d+1}(G;\text{U}(1))$ for finite internal symmetry groups in

d

spatial dimensions. The new summand ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ classifies possible configurations of

{E}_{8}

states on 2-cells that can be used to build crystalline phases beyond the group cohomology proposal. The completeness of our classification and the physical meaning of ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ are established through a combination of dimensional reduction, surface topological order, and explicit cellular construction. The value of ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ can be easily read off from the international symbol for

G

. Our classification agrees with the prediction of the ``generalized cohomology hypothesis,'' which concerns the general structure of the classification of SPT phases, and therefore provides strong evidence for the validity of the said hypothesis in the realm of crystalline symmetries.