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arXiv:1509.04395 · Physical Review B

Bosonic symmetry-protected topological phases with reflection symmetry

Tsuneya Yoshida, Takahiro Morimoto, Akira Furusaki

Published Dec 14, 2015• 18 citations
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Abstract

We study two-dimensional bosonic symmetry-protected topological (SPT) phases that are protected by reflection symmetry and local symmetry [${Z}_{N}\ensuremath{\rtimes}R, {Z}_{N}\ifmmode\times\else\texttimes\fi{}R$, U(1)$\ensuremath{\rtimes}R$, or U(1)$\ifmmode\times\else\texttimes\fi{}R$], in the search for two-dimensional bosonic analogs of topological crystalline insulators in integer-$S$ spin systems with reflection and spin-rotation symmetries. To classify them, we employ a Chern-Simons approach and examine the stability of edge states against perturbations that preserve the assumed symmetries. We find that SPT phases protected by ${Z}_{N}\ensuremath{\rtimes}R$ symmetry are classified as ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ for even $N$ and 0 (no SPT phase) for odd $N$, while those protected by U(1)$\ensuremath{\rtimes}R$ symmetry are ${\mathbb{Z}}_{2}$. We point out that the two-dimensional Affleck-Kennedy-Lieb-Tasaki state of $S=2$ spins on the square lattice is a ${\mathbb{Z}}_{2}$ SPT phase protected by reflection and $\ensuremath{\pi}$-rotation symmetries.

DOI
10.1103/physrevb.92.245122

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We study two-dimensional bosonic symmetry-protected topological (SPT) phases that are protected by reflection symmetry and local symmetry [ZN\ensuremath⋊R,ZN\ifmmode×\else\texttimes\fiR{Z}_{N}\ensuremath{\rtimes}R, {Z}_{N}\ifmmode\times\else\texttimes\fi{}RZN​\ensuremath⋊R,ZN​\ifmmode×\else\texttimes\fiR, U(1)\ensuremath⋊R\ensuremath{\rtimes}R\ensuremath⋊R, or U(1)\ifmmode×\else\texttimes\fiR\ifmmode\times\else\texttimes\fi{}R\ifmmode×\else\texttimes\fiR], in the search for two-dimensional bosonic analogs of topological crystalline insulators in integer-SSS spin systems with reflection and spin-rotation symmetries. To classify them, we employ a Chern-Simons approach and examine the stability of edge states against perturbations that preserve the assumed symmetries. We find that SPT phases protected by ZN\ensuremath⋊R{Z}_{N}\ensuremath{\rtimes}RZN​\ensuremath⋊R symmetry are classified as Z2\ifmmode×\else\texttimes\fiZ2{\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}Z2​\ifmmode×\else\texttimes\fiZ2​ for even NNN and 0 (no SPT phase) for odd NNN, while those protected by U(1)\ensuremath⋊R\ensuremath{\rtimes}R\ensuremath⋊R symmetry are Z2{\mathbb{Z}}_{2}Z2​. We point out that the two-dimensional Affleck-Kennedy-Lieb-Tasaki state of S=2S=2S=2 spins on the square lattice is a Z2{\mathbb{Z}}_{2}Z2​ SPT phase protected by reflection and \ensuremathπ\ensuremath{\pi}\ensuremathπ-rotation symmetries.
We study interaction effects on the topological crystalline insulators protected by time-reversal (T)(T)(T) and reflection symmetry (R)(R)(R) in two and three spatial dimensions. From the stability analysis of the edge states with bosonization, we find that the classification of the two-dimensional symmetry-protected topological (SPT) phases protected by ${Z}_{2}\ifmmode\times\else\texttimes\fi{}[\text{U(1)}\ensuremath{\rtimes}T]$ symmetry is reduced from Z\mathbb{Z}Z to Z4{\mathbb{Z}}_{4}Z4​ by interactions where the Z2{Z}_{2}Z2​ symmetry denotes the reflection whose mirror plane is the two-dimensional plane itself. By extending the approach recently proposed by Isobe and Fu, we show that the classification of the three-dimensional SPT phases (i.e., topological crystalline insulators) protected by $R\ifmmode\times\else\texttimes\fi{}[\text{U(1)}\ensuremath{\rtimes}T]$ symmetry is reduced from Z\mathbb{Z}Z to Z8{\mathbb{Z}}_{8}Z8​ by interactions.
It has been shown that the symmetry-protected topological (SPT) phases with finite Abelian symmetries can be described by Chern-Simons field theory. We propose a topological response theory to uniquely identify the SPT orders, which allows us to obtain a systematic scheme to classify bosonic SPT phases with any finite Abelian symmetry group. We point out that even for finite Abelian symmetry, there exist bosonic SPT phases beyond the current Chern-Simons theory framework. We also apply the theory to fermionic SPT phases with Zm symmetry and find the classification of SPT phases depends on the parity of m: for even m there are 2m classes, m out of which is intrinsically fermionic SPT phases and cannot be realized in any bosonic system. Finally we propose a classification scheme of fermionic SPT phases for any finite, Abelian symmetry.
Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry GGG. 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 SO(3)\mathit{SO}(3)SO(3) spin rotation symmetry. The topological insulator is another example of SPT phases which are protected by U(1)U(1)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 ddd-dimensional bosonic SPT phases with on-site symmetry GGG (which may contain antiunitary time-reversal symmetry) can be labeled by the elements in H1+d[G,UT(1)]{\mathcal{H}}^{1+d}[G,{U}_{T}(1)]H1+d[G,UT​(1)], the Borel (1+d)(1+d)(1+d)-group-cohomology classes of GGG over the GGG module UT(1){U}_{T}(1)UT​(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 GGG 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 Z2T{Z}_{2}^{T}Z2T​ 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 H1+d[Z2T,UT(1)]{\mathcal{H}}^{1+d}[{Z}_{2}^{T},{U}_{T}(1)]H1+d[Z2T​,UT​(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)U(1)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 GH{G}_{H}GH​ is the symmetry group of the Hamiltonian and ${G}_{\ensuremath{\Psi}}$ the symmetry group of the ground states.
We study one-dimensional systems of two-orbital SU(4) fermionic cold atoms. In particular, we focus on an SU(4) spin model [named SU(4) $e\text{\ensuremath{-}}g$ spin model] that is realized in a low-energy state in the Mott insulator phase at the filling ng=3,ne=1{n}_{g}=3,{n}_{e}=1ng​=3,ne​=1 (ng,ne{n}_{g},{n}_{e}ng​,ne​: numbers of atoms in ground and excited states, respectively). Our numerical study with the infinite-size density matrix renormalization group shows that the ground state of SU(4) $e\text{\ensuremath{-}}g$ spin model is a nontrivial symmetry protected topological (SPT) phase protected by ${Z}_{4}\ifmmode\times\else\texttimes\fi{}{Z}_{4}$ symmetry. Specifically, we find that the ground state belongs to an SPT phase with the topological index $2\ensuremath{\in}{\mathbb{Z}}_{4}$ and show sixfold degenerate edge states. This is topologically distinct from SPT phases with the index $1\ensuremath{\in}{\mathbb{Z}}_{4}$ that are realized in the SU(4) bilinear model and the SU(4) Affleck-Kennedy-Lieb-Tasaki (AKLT) model. We explore the phase diagram of SU(4) spin models including $e\text{\ensuremath{-}}g$ spin model, bilinear-biquadratic model, and AKLT model, and identify that antisymmetrization effect in neighboring spins (that we quantify with Casimir operators) is the driving force of the phase transitions. Furthermore, we demonstrate by using the matrix product state how the Z4{\mathbb{Z}}_{4}Z4​ SPT state with six edge states appears in the SU(4) $e\text{\ensuremath{-}}g$ spin model.
Symmetry protected topological (SPT) states are short-range entangled states with symmetry. The boundary of a SPT phases has either gapless excitations or degenerate ground states, around a gapped bulk. Recently, we proposed a systematic construction of SPT phases in interacting bosonic systems, however it is not very clear what is the form of the low-energy excitations on the gapless edge. In this paper, we answer this question for two-dimensional (2D) bosonic SPT phases with ZN{\mathbb{Z}}_{N}ZN​ and U(1)\text{U}(1)U(1) symmetry. We find that while the low-energy modes of the gapless edges are nonchiral, symmetry acts on them in a ``chiral'' way, i.e., acts on the right movers and the left movers differently. This special realization of symmetry protects the gaplessness of the otherwise unstable edge states by prohibiting a direct scattering between the left and right movers. Moreover, understanding of the low-energy effective theory leads to experimental predictions about the SPT phases. In particular, we find that all the 2D U(1)\text{U}(1)U(1) SPT phases have even integer quantized Hall conductance.
Symmetry-protected topological (SPT) phases are gapped quantum phases with a certain symmetry, which can all be smoothly connected to the same trivial product state if we break the symmetry. For noninteracting fermion systems with time reversal ($\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{T}$), charge conjugation ($\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{C}$), and/or U(1)U(1)U(1) ($\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{N}$) symmetries, the total symmetry group can depend on the relations between those symmetry operations, such as $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{T}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{N}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{T}}^{\ensuremath{-}1}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{N}$ or $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{T}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{N}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{T}}^{\ensuremath{-}1}=\ensuremath{-}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{N}$. As a result, the SPT phases of those fermion systems with different symmetry groups have different classifications. In this paper, we use Kitaev's K-theory approach to classify the gapped free-fermion phases for those possible symmetry groups. In particular, we can view the U(1)U(1)U(1) as a spin rotation. We find that superconductors with the Sz{S}_{z}Sz​ spin-rotation symmetry are classified by ZZZ in even dimensions, while superconductors with the time reversal plus the Sz{S}_{z}Sz​ spin-rotation symmetries are classified by ZZZ in odd dimensions. We show that all 10 classes of gapped free-fermion phases can be realized by electron systems with certain symmetries. We also point out that, to properly describe the symmetry of a fermionic system, we need to specify its full symmetry group that includes the fermion number parity transformation ${(\ensuremath{-})}^{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{N}}$. The full symmetry group is actually a projective symmetry group.