arXiv:1209.4399 · Physical Review B

Three-dimensional symmetry-protected topological phase close to antiferromagnetic Néel order

Cenke Xu

Published Apr 25, 2013• 39 citations

Abstract

It is well known that the Haldane phase of a one-dimensional spin-1 chain is a symmetry-protected topological (SPT) phase, which is described by a nonlinear sigma model (NLSM) with a $\ensuremath{\Theta}$ term at $\ensuremath{\Theta}=2\ensuremath{\pi}$. In this work we study a three-dimensional (3d) SPT phase of a SU(2$N$) antiferromagnetic spin system with a self-conjugate representation on every site. The spin-ordered N\'eel phase of this system has a ground state manifold $\mathcal{M}=\frac{\mathrm{U}(2N)}{\mathrm{U}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(N)}$, and this system is described by a NLSM defined with manifold $\mathcal{M}$. Since the homotopy group ${\ensuremath{\pi}}_{4}[\mathcal{M}]=\mathbb{Z}$ for $N>1$, this NLSM can naturally have a $\ensuremath{\Theta}$ term. We will argue that when $\ensuremath{\Theta}=2\ensuremath{\pi}$ this NLSM describes a SPT phase. This SPT phase is protected by the SU(2$N$) spin symmetry, or its subgroup SU($N$)$\ifmmode\times\else\texttimes\fi{}$SU$(N)\ensuremath{\rtimes}{Z}_{2}$, without assuming any other discrete symmetry. We will also construct a trial SU(2$N$) spin state on a 3d lattice; we argue that the long-wavelength physics of this state is precisely described by the aforementioned NLSM with $\ensuremath{\Theta}=2\ensuremath{\pi}$.

DOI: 10.1103/physrevb.87.144421

Related papers

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 phases in a two-leg SU(N) spin ladder with unequal spins

Sylvain Capponi, Pierre Fromholz, P. Lecheminant +1 more

arXiv:1909.13553Physical review. B./Physical review. BMay 12, 202013 citations

Z3 symmetry-protected topological phases in the SU(3) AKLT model

Takahiro Morimoto, Hiroshi Ueda, Tsutomu Momoi +1 more

arXiv:1409.1659Physical Review BDec 4, 201454 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

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

Zheng‐Cheng Gu, Xiao-Gang Wen

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

arXiv:1309.0515Physical Review BApr 6, 2015108 citations

arXiv:1209.4399 · Physical Review B

Three-dimensional symmetry-protected topological phase close to antiferromagnetic Néel order

Cenke Xu

Published Apr 25, 2013• 39 citations

arXiv HTML PDF

Abstract

It is well known that the Haldane phase of a one-dimensional spin-1 chain is a symmetry-protected topological (SPT) phase, which is described by a nonlinear sigma model (NLSM) with a

\ensuremath{\Theta}

term at

\ensuremath{\Theta}=2\ensuremath{\pi}

. In this work we study a three-dimensional (3d) SPT phase of a SU(2

N

) antiferromagnetic spin system with a self-conjugate representation on every site. The spin-ordered N'eel phase of this system has a ground state manifold

\mathcal{M}=\frac{\mathrm{U}(2N)}{\mathrm{U}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(N)}

, and this system is described by a NLSM defined with manifold

\mathcal{M}

. Since the homotopy group

{\ensuremath{\pi}}_{4}[\mathcal{M}]=\mathbb{Z}

for

N>1

, this NLSM can naturally have a

\ensuremath{\Theta}

term. We will argue that when

\ensuremath{\Theta}=2\ensuremath{\pi}

this NLSM describes a SPT phase. This SPT phase is protected by the SU(2

N

) spin symmetry, or its subgroup SU(

N

)

\ifmmode\times\else\texttimes\fi{}

(N)\ensuremath{\rtimes}{Z}_{2}

, without assuming any other discrete symmetry. We will also construct a trial SU(2

N

) spin state on a 3d lattice; we argue that the long-wavelength physics of this state is precisely described by the aforementioned NLSM with

\ensuremath{\Theta}=2\ensuremath{\pi}

DOI: 10.1103/physrevb.87.144421

Related papers

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 phases in a two-leg SU(N) spin ladder with unequal spins

Sylvain Capponi, Pierre Fromholz, P. Lecheminant +1 more

Chiral Haldane phases are examples of one-dimensional topological states of matter which are protected by the projective $\mathrm{SU}(N)$ group (or its subgroup ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N})$ with $N>2$ . The unique feature of these symmetry-protected topological (SPT) phases is that they are accompanied by inversion-symmetry breaking and the emergence of different left and right edge states which transform, for instance, respectively in the fundamental $(\mathbit{N})$ and antifundamental $(\overline{\mathbit{N}})$ representations of $\mathrm{SU}(N)$ . We show, by means of complementary analytical and numerical approaches, that these chiral SPT phases as well as the nonchiral ones are realized as the ground states of a generalized two-leg $\mathrm{SU}(N)$ spin ladder in which the spins in the first chain transform in $\mathbit{N}$ and the second in $\overline{\mathbit{N}}$. In particular, we map out the phase diagram for $N=3$ and 4 to show that all the possible symmetry-protected topological phases with projective $\mathrm{SU}(N)$ symmetry appear in this simple ladder model.Chiral Haldane phases are examples of one-dimensional topological states of matter which are protected by the projective $\mathrm{SU}(N)$ group (or its subgroup ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N})$ with $N>2$. The unique feature of these symmetry-protected topological (SPT) phases is that they are accompanied by inversion-symmetry breaking and the emergence of different left and right edge states which transform, for instance, respectively in the fundamental $(\mathbit{N})$ and antifundamental $(\overline{\mathbit{N}})$ representations of $\mathrm{SU}(N)$. We show, by means of complementary analytical and numerical approaches, that these chiral SPT phases as well as the nonchiral ones are realized as the ground states of a generalized two-leg $\mathrm{SU}(N)$ spin ladder in which the spins in the first chain transform in $\mathbit{N}$ and the second in $\overline{\mathbit{N}}$. In particular, we map out the phase diagram for $N=3$ and 4 to show that all the possible symmetry-protected topological phases with projective $\mathrm{SU}(N)$ symmetry appear in this simple ladder model.

arXiv:1909.13553Physical review. B./Physical review. BMay 12, 202013 citations

Z3 symmetry-protected topological phases in the SU(3) AKLT model

Takahiro Morimoto, Hiroshi Ueda, Tsutomu Momoi +1 more

We study ${\mathbb{Z}}_{3}$ symmetry-protected topological (SPT) phases in one-dimensional spin systems with ${Z}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3}$ symmetry. We construct ground-state wave functions of the matrix product form for nontrivial ${\mathbb{Z}}_{3}$ phases and their parent Hamiltonian from a cocycle of the group cohomology ${H}^{2}({Z}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3},U(1))$. The Hamiltonian is an SU(3) version of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, consisting of bilinear and biquadratic terms of su(3) generators in the adjoint representation. A generalization to the $\mathrm{SU}(N)$ case, the $\mathrm{SU}(N)$ AKLT Hamiltonian, is also presented, which realizes nontrivial ${\mathbb{Z}}_{N}$ SPT phases. We use the infinite-size variant of the density matrix renormalization group (iDMRG) method to determine the ground-state phase diagram of the SU(3) bilinear-biquadratic model as a function of the parameter $\ensuremath{\theta}$ controlling the ratio of the bilinear and biquadratic coupling constants. The nontrivial ${\mathbb{Z}}_{3}$ SPT phase is found for a range of the parameter $\ensuremath{\theta}$ including the point of vanishing biquadratic term $(\ensuremath{\theta}=0)$ as well as the SU(3) AKLT point [$\ensuremath{\theta}=arctan(2/9)$]. A continuous phase transition to the SU(3) dimer phase takes place at $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.027\ensuremath{\pi}$, with a central charge $c\ensuremath{\approx}3.2$. For SU(3) symmetric cases, we define string order parameters for the ${\mathbb{Z}}_{3}$ SPT phases in a similar way to the conventional Haldane phase. We propose simple spin models that effectively realize the SU(3) and SU(4) AKLT models.We study ${\mathbb{Z}}_{3}$ symmetry-protected topological (SPT) phases in one-dimensional spin systems with ${Z}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3}$ symmetry. We construct ground-state wave functions of the matrix product form for nontrivial ${\mathbb{Z}}_{3}$ phases and their parent Hamiltonian from a cocycle of the group cohomology ${H}^{2}({Z}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3},U(1))$. The Hamiltonian is an SU(3) version of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, consisting of bilinear and biquadratic terms of su(3) generators in the adjoint representation. A generalization to the $\mathrm{SU}(N)$ case, the $\mathrm{SU}(N)$ AKLT Hamiltonian, is also presented, which realizes nontrivial ${\mathbb{Z}}_{N}$ SPT phases. We use the infinite-size variant of the density matrix renormalization group (iDMRG) method to determine the ground-state phase diagram of the SU(3) bilinear-biquadratic model as a function of the parameter $\ensuremath{\theta}$ controlling the ratio of the bilinear and biquadratic coupling constants. The nontrivial ${\mathbb{Z}}_{3}$ SPT phase is found for a range of the parameter $\ensuremath{\theta}$ including the point of vanishing biquadratic term $(\ensuremath{\theta}=0)$ as well as the SU(3) AKLT point [$\ensuremath{\theta}=arctan(2/9)$]. A continuous phase transition to the SU(3) dimer phase takes place at $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.027\ensuremath{\pi}$, with a central charge $c\ensuremath{\approx}3.2$. For SU(3) symmetric cases, we define string order parameters for the ${\mathbb{Z}}_{3}$ SPT phases in a similar way to the conventional Haldane phase. We propose simple spin models that effectively realize the SU(3) and SU(4) AKLT models.

arXiv:1409.1659Physical Review BDec 4, 201454 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

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

It is well known that the Haldane phase of a one-dimensional spin-1 chain is a symmetry-protected topological (SPT) phase, which is described by a nonlinear sigma model (NLSM) with a

\ensuremath{\Theta}

term at

\ensuremath{\Theta}=2\ensuremath{\pi}

. In this work we study a three-dimensional (3d) SPT phase of a SU(2

N

) antiferromagnetic spin system with a self-conjugate representation on every site. The spin-ordered N'eel phase of this system has a ground state manifold

\mathcal{M}=\frac{\mathrm{U}(2N)}{\mathrm{U}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(N)}

, and this system is described by a NLSM defined with manifold

\mathcal{M}

. Since the homotopy group

{\ensuremath{\pi}}_{4}[\mathcal{M}]=\mathbb{Z}

for

N>1

, this NLSM can naturally have a

\ensuremath{\Theta}

term. We will argue that when

\ensuremath{\Theta}=2\ensuremath{\pi}

this NLSM describes a SPT phase. This SPT phase is protected by the SU(2

N

) spin symmetry, or its subgroup SU(

N

)

\ifmmode\times\else\texttimes\fi{}

(N)\ensuremath{\rtimes}{Z}_{2}

, without assuming any other discrete symmetry. We will also construct a trial SU(2

N

) spin state on a 3d lattice; we argue that the long-wavelength physics of this state is precisely described by the aforementioned NLSM with

\ensuremath{\Theta}=2\ensuremath{\pi}

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.

Chiral Haldane phases are examples of one-dimensional topological states of matter which are protected by the projective

\mathrm{SU}(N)

group (or its subgroup ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N})$ with

N>2

. The unique feature of these symmetry-protected topological (SPT) phases is that they are accompanied by inversion-symmetry breaking and the emergence of different left and right edge states which transform, for instance, respectively in the fundamental $(\mathbit{N})$ and antifundamental $(\overline{\mathbit{N}})$ representations of

\mathrm{SU}(N)

. We show, by means of complementary analytical and numerical approaches, that these chiral SPT phases as well as the nonchiral ones are realized as the ground states of a generalized two-leg

\mathrm{SU}(N)

spin ladder in which the spins in the first chain transform in $\mathbit{N}$ and the second in $\overline{\mathbit{N}}$. In particular, we map out the phase diagram for

N=3

and 4 to show that all the possible symmetry-protected topological phases with projective

\mathrm{SU}(N)

symmetry appear in this simple ladder model.

We study

{\mathbb{Z}}_{3}

symmetry-protected topological (SPT) phases in one-dimensional spin systems with ${Z}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3}$ symmetry. We construct ground-state wave functions of the matrix product form for nontrivial

{\mathbb{Z}}_{3}

phases and their parent Hamiltonian from a cocycle of the group cohomology ${H}^{2}({Z}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3},U(1))$. The Hamiltonian is an SU(3) version of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, consisting of bilinear and biquadratic terms of su(3) generators in the adjoint representation. A generalization to the

\mathrm{SU}(N)

case, the

\mathrm{SU}(N)

AKLT Hamiltonian, is also presented, which realizes nontrivial

{\mathbb{Z}}_{N}

SPT phases. We use the infinite-size variant of the density matrix renormalization group (iDMRG) method to determine the ground-state phase diagram of the SU(3) bilinear-biquadratic model as a function of the parameter $\ensuremath{\theta}$ controlling the ratio of the bilinear and biquadratic coupling constants. The nontrivial

{\mathbb{Z}}_{3}

SPT phase is found for a range of the parameter $\ensuremath{\theta}$ including the point of vanishing biquadratic term $(\ensuremath{\theta}=0)$ as well as the SU(3) AKLT point [$\ensuremath{\theta}=arctan(2/9)$]. A continuous phase transition to the SU(3) dimer phase takes place at $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.027\ensuremath{\pi}$, with a central charge $c\ensuremath{\approx}3.2$. For SU(3) symmetric cases, we define string order parameters for the

{\mathbb{Z}}_{3}

SPT phases in a similar way to the conventional Haldane phase. We propose simple spin models that effectively realize the SU(3) and SU(4) AKLT models.

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.