arXiv:1306.5671 · Physical Review B

Identifying symmetry-protected topological order by entanglement entropy

Wei Li, Andreas Weichselbaum, Jan von Delft

Published Dec 18, 2013• 25 citations

Abstract

According to the classification using projective representations of the SO(3) group, there exist two topologically distinct gapped phases in spin-1 chains. The symmetry-protected topological (SPT) phase possesses half-integer projective representations of the SO(3) group, while the trivial phase possesses integer linear representations. In the present work, we implement non-Abelian symmetries in the density matrix renormalization group (DMRG) method, allowing us to keep track of (and also control) the virtual bond representations and to readily distinguish the SPT phase from the trivial one by evaluating the multiplet entanglement spectrum. In particular, using the entropies ${S}^{I}$ (${S}^{H}$) of integer (half-integer) representations, we can define an entanglement gap $G={S}^{I}\ensuremath{-}{S}^{H}$, which equals 1 in the SPT phase, and $\ensuremath{-}1$ in the trivial phase. As an application of our proposal, we study the spin-1 models on various one-dimensional (1D) and quasi-1D lattices, including the bilinear-biquadratic model on a single chain, and the Heisenberg model on a two-leg ladder and a three-leg tube. Among others, we confirm the existence of a SPT phase in the spin-1 tube model and reveal that the phase transition between the SPT and the trivial phase is a continuous one. The transition point is found to be critical, with conformal central charge $c=3$ determined by fits to the block entanglement entropy.

DOI: 10.1103/physrevb.88.245121

Related papers

Symmetry-protected topological order identified via Gutzwiller-guided density matrix renormalization group: SO ( n ) spin chains

Pei-Yuan Cai, Hui-Ke Jin, Yi Zhou

arXiv:2504.10919Physical review. B./Physical review. BJul 7, 20253 citations

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

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

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

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

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

Jun 23, 201113 citations

Critical-entanglement spectrum of one-dimensional symmetry-protected topological phases

Wen-Jia Rao, Xin Wan, Guang-Ming Zhang

arXiv:1406.7113Physical Review BAug 28, 201425 citations

Density Matrix Renormalization Group studies of quantum antiferromagnets on an anisotropic triangular strip

裕一朗日髙

Mar 23, 20230 citations

Symmetry-twisted multi-entropies: Order parameters for 2D SPT phases

Ramanjit Sohal, Michael Levin, Ruben Verresen

cond-mat.str-elCross-listedarXiv:2607.12023Jul 15, 20260 citations

arXiv:1306.5671 · Physical Review B

Identifying symmetry-protected topological order by entanglement entropy

Wei Li, Andreas Weichselbaum, Jan von Delft

Published Dec 18, 2013• 25 citations

arXiv HTML PDF

Abstract

{S}^{I}

(

{S}^{H}

) of integer (half-integer) representations, we can define an entanglement gap

G={S}^{I}\ensuremath{-}{S}^{H}

, which equals 1 in the SPT phase, and

\ensuremath{-}1

in the trivial phase. As an application of our proposal, we study the spin-1 models on various one-dimensional (1D) and quasi-1D lattices, including the bilinear-biquadratic model on a single chain, and the Heisenberg model on a two-leg ladder and a three-leg tube. Among others, we confirm the existence of a SPT phase in the spin-1 tube model and reveal that the phase transition between the SPT and the trivial phase is a continuous one. The transition point is found to be critical, with conformal central charge

c=3

determined by fits to the block entanglement entropy.

DOI: 10.1103/physrevb.88.245121

Related papers

Symmetry-protected topological order identified via Gutzwiller-guided density matrix renormalization group: SO ( n ) spin chains

Pei-Yuan Cai, Hui-Ke Jin, Yi Zhou

We present a comprehensive study of topological phases in the SO( $n$ ) spin chains using a combination of analytical parton construction and numerical techniques. For even $n=2l$ , we identify a novel SPT $^2$ phase characterized by two distinct topological sectors, exhibiting exact degeneracy at the matrix product state (MPS) exactly solvable point. Through Gutzwiller-projected mean-field theory and density matrix renormalization group (DMRG) calculations, we demonstrate that these sectors remain topologically degenerate in close chains throughout the SPT $^2$ phase, with energy gaps decaying exponentially with system size. For odd $n=2l+1$ , we show that the ground state remains unique in close chains. We precisely characterize critical states using entanglement entropy scaling, confirming the central charges predicted by conformal field theories. Our results reveal fundamental differences between even and odd $n$ cases, provide numerical verification of topological protection, and establish reliable methods for studying high-symmetry quantum systems. The Gutzwiller-guided DMRG is demonstrated to be notably efficient in targeting specific topological sectors.We present a comprehensive study of topological phases in the SO($n$) spin chains using a combination of analytical parton construction and numerical techniques. For even $n=2l$, we identify a novel SPT$^2$ phase characterized by two distinct topological sectors, exhibiting exact degeneracy at the matrix product state (MPS) exactly solvable point. Through Gutzwiller-projected mean-field theory and density matrix renormalization group (DMRG) calculations, we demonstrate that these sectors remain topologically degenerate in close chains throughout the SPT$^2$ phase, with energy gaps decaying exponentially with system size. For odd $n=2l+1$, we show that the ground state remains unique in close chains. We precisely characterize critical states using entanglement entropy scaling, confirming the central charges predicted by conformal field theories. Our results reveal fundamental differences between even and odd $n$ cases, provide numerical verification of topological protection, and establish reliable methods for studying high-symmetry quantum systems. The Gutzwiller-guided DMRG is demonstrated to be notably efficient in targeting specific topological sectors.

arXiv:2504.10919Physical review. B./Physical review. BJul 7, 20253 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 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

Critical-entanglement spectrum of one-dimensional symmetry-protected topological phases

Wen-Jia Rao, Xin Wan, Guang-Ming Zhang

Under an appropriate symmetric extensive bipartition in a one-dimensional symmetry-protected topological (SPT) phase, a bulk critical-entanglement spectrum can be obtained, resembling the excitation spectrum of the critical point separating the SPT phase from the trivial (vacuum) state. Such a critical point is beyond the standard Landau-Ginzburg-Wilson paradigm for symmetry-breaking phase transitions. For the $S=1$ SPT (Haldane) phase with the Affleck-Kennedy-Lieb-Tasaki exact wave function, the resulting critical-entanglement spectrum has a residual entropy per lattice site ${s}_{r}=0.67602$ , showing a delocalized version of the edge excitations in the SPT phase. From the wave function corresponding to the lowest entanglement energy level, the central charge of the critical point can be extracted $c\ensuremath{\approx}1.01\ifmmode\pm\else\textpm\fi{}0.01$. The critical theory can be identified as the same effective field theory as the spin-1/2 antiferromagnetic Heisenberg chain or the spin-1/2 Haldane-Shastry model with inverse-square long-range interaction.Under an appropriate symmetric extensive bipartition in a one-dimensional symmetry-protected topological (SPT) phase, a bulk critical-entanglement spectrum can be obtained, resembling the excitation spectrum of the critical point separating the SPT phase from the trivial (vacuum) state. Such a critical point is beyond the standard Landau-Ginzburg-Wilson paradigm for symmetry-breaking phase transitions. For the $S=1$ SPT (Haldane) phase with the Affleck-Kennedy-Lieb-Tasaki exact wave function, the resulting critical-entanglement spectrum has a residual entropy per lattice site ${s}_{r}=0.67602$, showing a delocalized version of the edge excitations in the SPT phase. From the wave function corresponding to the lowest entanglement energy level, the central charge of the critical point can be extracted $c\ensuremath{\approx}1.01\ifmmode\pm\else\textpm\fi{}0.01$. The critical theory can be identified as the same effective field theory as the spin-1/2 antiferromagnetic Heisenberg chain or the spin-1/2 Haldane-Shastry model with inverse-square long-range interaction.

arXiv:1406.7113Physical Review BAug 28, 201425 citations

Density Matrix Renormalization Group studies of quantum antiferromagnets on an anisotropic triangular strip

裕一朗日髙

Many quantum phases are described by the Landau theory, which is based on the spontaneous symmetry breaking.However, the symmetry-protected topological (SPT) phases have been discovered as new quantum phases that cannot be described by the Landau theory.One of the examples is the Haldane phase in the one-dimensional spin-1 Heisenberg chain.This concept of the SPT phase is developed for gapped states.However, many researchers are attempting to extend this concept to gapless states.Such as SPT phase in a gapless system is called a gapless SPT phase.To calculate the ground state of a spin system, we perform the numerical calculation based on the density-matrix renormalization group (DMRG) method.All the quantum spin states on a lattice is written in the form of a matrix product state (MPS).The DMRG method is one of the variational methods by which we optimize the MPS.In addition, an MPS characterizes SPT phases.An entanglement spectrum is defined using singular values of an MPS and becomes even-fold degenerate when the state is in the Haldane state.We can also calculate an entanglement entropy that characterizes the gapless system.In this thesis, we investigate a three-leg spin-1/2 ladder with geometrically frustrated interleg interactions.We call this model an anisotropic triangular-strip (ATS) model.According to the quantum field theory and renormalization group theory, when the rung coupling is sufficiently small, the frustrated interchain interaction almost isolates the second chain in the middle from the other two chains.In addition, at the same time, the first and third chains effectively mediate the ferromagnetic interaction.As a result, the ground state of the ATS model is a gapless Tomonaga-Luttinger liquid weakly coupled to a spin-1 Haldane chain with irrelevant interactions.By calculating the entanglement spectrum and the entanglement entropy, we show that the ground state of the ATS model with weak rung coupling is the gapless SPT state.Moreover, the ground state of the ATS model with sufficiently large rung coupling becomes ferrimagnetic state.We also investigate the other phases of the ATS model by the DMRG method.Through this study, we found the gapless SPT state which is not trivially decoupled into the gapless system and the nontrivial gapped SPT system in the ATS model, and we determine the phase diagram of the ATS model.This doctoral thesis would not have been completed without the support of many people.I am grateful to Shunsuke C. Furuya for proposing the basic concept of this thesis and supporting my investigation.He also gives me much fruitful advice especially when I got stuck in my research.I also would like to thank Atsushi Ueda especially for teaching me how to use the supercomputer in ISSP.Thanks to Yasuhiro Tada for teaching me great advice which is valuable to my study and for polishing this thesis.This study is performed with their cooperation, and some results are published in the journal Physical Review B.I am grateful to my supervisor, Prof. Masaki Oshikawa, for supporting many things about my research.He introduced the gapless SPT phase to me.I had meaningful discussions with him and was given much fruitful advice for completing this doctoral thesis.Oshikawa Group members gave me great and meaningful advice when I presented in the group seminarsMany quantum phases are described by the Landau theory, which is based on the spontaneous symmetry breaking.However, the symmetry-protected topological (SPT) phases have been discovered as new quantum phases that cannot be described by the Landau theory.One of the examples is the Haldane phase in the one-dimensional spin-1 Heisenberg chain.This concept of the SPT phase is developed for gapped states.However, many researchers are attempting to extend this concept to gapless states.Such as SPT phase in a gapless system is called a gapless SPT phase.To calculate the ground state of a spin system, we perform the numerical calculation based on the density-matrix renormalization group (DMRG) method.All the quantum spin states on a lattice is written in the form of a matrix product state (MPS).The DMRG method is one of the variational methods by which we optimize the MPS.In addition, an MPS characterizes SPT phases.An entanglement spectrum is defined using singular values of an MPS and becomes even-fold degenerate when the state is in the Haldane state.We can also calculate an entanglement entropy that characterizes the gapless system.In this thesis, we investigate a three-leg spin-1/2 ladder with geometrically frustrated interleg interactions.We call this model an anisotropic triangular-strip (ATS) model.According to the quantum field theory and renormalization group theory, when the rung coupling is sufficiently small, the frustrated interchain interaction almost isolates the second chain in the middle from the other two chains.In addition, at the same time, the first and third chains effectively mediate the ferromagnetic interaction.As a result, the ground state of the ATS model is a gapless Tomonaga-Luttinger liquid weakly coupled to a spin-1 Haldane chain with irrelevant interactions.By calculating the entanglement spectrum and the entanglement entropy, we show that the ground state of the ATS model with weak rung coupling is the gapless SPT state.Moreover, the ground state of the ATS model with sufficiently large rung coupling becomes ferrimagnetic state.We also investigate the other phases of the ATS model by the DMRG method.Through this study, we found the gapless SPT state which is not trivially decoupled into the gapless system and the nontrivial gapped SPT system in the ATS model, and we determine the phase diagram of the ATS model.This doctoral thesis would not have been completed without the support of many people.I am grateful to Shunsuke C. Furuya for proposing the basic concept of this thesis and supporting my investigation.He also gives me much fruitful advice especially when I got stuck in my research.I also would like to thank Atsushi Ueda especially for teaching me how to use the supercomputer in ISSP.Thanks to Yasuhiro Tada for teaching me great advice which is valuable to my study and for polishing this thesis.This study is performed with their cooperation, and some results are published in the journal Physical Review B.I am grateful to my supervisor, Prof. Masaki Oshikawa, for supporting many things about my research.He introduced the gapless SPT phase to me.I had meaningful discussions with him and was given much fruitful advice for completing this doctoral thesis.Oshikawa Group members gave me great and meaningful advice when I presented in the group seminars

Mar 23, 20230 citations

Symmetry-twisted multi-entropies: Order parameters for 2D SPT phases

Ramanjit Sohal, Michael Levin, Ruben Verresen

Although symmetry-protected topological phases (SPTs) can be distinguished by their entanglement properties, it has been unclear how to extract this information directly from expectation values beyond the 1D case. Here, we close this gap and propose a pair of nonlocal order parameters that can detect and distinguish all bosonic SPTs in 2D protected by internal discrete, Abelian unitary symmetries. The desired topological invariants are extracted by these quantities by effectively simulating the SPT path integral on topologically non-trivial spacetime manifolds. Our order parameters are defined in terms of expectation values of partial symmetry and permutation operations acting on fixed numbers of replicas of the system in finite spatial regions. These expectation values correspond to symmetry-twisted versions of multipartite entanglement quantities known as multi-entropies. We show explicitly that our two order parameters detect symmetry-protected four-party and six-party entanglement, respectively, and we constrain possible "spurious" contributions. We analytically test our proposal in fixed-point lattice models. Our results suggest multipartite entanglement to be a defining feature of SPTs; indeed, we expect our methods to generalize to fermionic and higher-dimensional systems.Although symmetry-protected topological phases (SPTs) can be distinguished by their entanglement properties, it has been unclear how to extract this information directly from expectation values beyond the 1D case. Here, we close this gap and propose a pair of nonlocal order parameters that can detect and distinguish all bosonic SPTs in 2D protected by internal discrete, Abelian unitary symmetries. The desired topological invariants are extracted by these quantities by effectively simulating the SPT path integral on topologically non-trivial spacetime manifolds. Our order parameters are defined in terms of expectation values of partial symmetry and permutation operations acting on fixed numbers of replicas of the system in finite spatial regions. These expectation values correspond to symmetry-twisted versions of multipartite entanglement quantities known as multi-entropies. We show explicitly that our two order parameters detect symmetry-protected four-party and six-party entanglement, respectively, and we constrain possible "spurious" contributions. We analytically test our proposal in fixed-point lattice models. Our results suggest multipartite entanglement to be a defining feature of SPTs; indeed, we expect our methods to generalize to fermionic and higher-dimensional systems.

cond-mat.str-elCross-listedarXiv:2607.12023Jul 15, 20260 citations

{S}^{I}

(

{S}^{H}

) of integer (half-integer) representations, we can define an entanglement gap

G={S}^{I}\ensuremath{-}{S}^{H}

, which equals 1 in the SPT phase, and

\ensuremath{-}1

c=3

determined by fits to the block entanglement entropy.

We present a comprehensive study of topological phases in the SO(

n

) spin chains using a combination of analytical parton construction and numerical techniques. For even

n=2l

, we identify a novel SPT

^2

phase characterized by two distinct topological sectors, exhibiting exact degeneracy at the matrix product state (MPS) exactly solvable point. Through Gutzwiller-projected mean-field theory and density matrix renormalization group (DMRG) calculations, we demonstrate that these sectors remain topologically degenerate in close chains throughout the SPT

^2

phase, with energy gaps decaying exponentially with system size. For odd

n=2l+1

, we show that the ground state remains unique in close chains. We precisely characterize critical states using entanglement entropy scaling, confirming the central charges predicted by conformal field theories. Our results reveal fundamental differences between even and odd

n

cases, provide numerical verification of topological protection, and establish reliable methods for studying high-symmetry quantum systems. The Gutzwiller-guided DMRG is demonstrated to be notably efficient in targeting specific topological sectors.

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.

S=1

SPT (Haldane) phase with the Affleck-Kennedy-Lieb-Tasaki exact wave function, the resulting critical-entanglement spectrum has a residual entropy per lattice site

{s}_{r}=0.67602

, showing a delocalized version of the edge excitations in the SPT phase. From the wave function corresponding to the lowest entanglement energy level, the central charge of the critical point can be extracted $c\ensuremath{\approx}1.01\ifmmode\pm\else\textpm\fi{}0.01$. The critical theory can be identified as the same effective field theory as the spin-1/2 antiferromagnetic Heisenberg chain or the spin-1/2 Haldane-Shastry model with inverse-square long-range interaction.