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arXiv:1508.05689 · Physical review. B./Physical review. B

Topological quantum field theory of three-dimensional bosonic Abelian-symmetry-protected topological phases

Peng Ye, Zheng‐Cheng Gu

Published May 31, 2016• 54 citations
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Abstract

Symmetry-protected topological phases (SPT) are short-range entangled gapped states protected by global symmetry. Nontrivial SPT phases cannot be adiabatically connected to the trivial disordered state (or atomic insulator) as long as certain global symmetry $G$ is unbroken. At low energies, most of the two-dimensional SPTs with Abelian symmetry can be described by topological quantum field theory (TQFT) of the multicomponent Chern-Simons type. However, in contrast to the fractional quantum Hall effect where TQFT can give rise to interesting bulk anyons, TQFT for SPTs only supports trivial bulk excitations. The essential question in TQFT descriptions for SPTs is to understand how the global symmetry is implemented in the partition function. In this paper, we systematically study TQFT of three-dimensional SPTs with unitary Abelian symmetry (e.g., ${\mathbb{Z}}_{{N}_{1}}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{N}_{2}}\ifmmode\times\else\texttimes\fi{}...$). In addition to the usual multicomponent $BF$ topological term at level-1, we find that there are new topological terms with quantized coefficients (e.g., ${a}^{1}\ensuremath{\wedge}{a}^{2}\ensuremath{\wedge}d{a}^{2}$ and ${a}^{1}\ensuremath{\wedge}{a}^{2}\ensuremath{\wedge}{a}^{3}\ensuremath{\wedge}{a}^{4}$) in TQFT actions, where ${a}^{1},{a}^{2},...$ are 1-form U(1) gauge fields. These additional topological terms cannot be adiabatically turned off as long as $G$ is unbroken. By investigating symmetry transformations for the TQFT partition function, we end up with the classification of SPTs that is consistent with the well-known group cohomology approach. We also discuss how to gauge the global symmetry and possible TQFT descriptions of Dijkgraaf-Witten gauge theory.

DOI
10.1103/physrevb.93.205157

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arXiv:2101.03359•Jan 9, 2021•0 citations
Symmetry-protected topological phases (SPT) are short-range entangled gapped states protected by global symmetry. Nontrivial SPT phases cannot be adiabatically connected to the trivial disordered state (or atomic insulator) as long as certain global symmetry GGG is unbroken. At low energies, most of the two-dimensional SPTs with Abelian symmetry can be described by topological quantum field theory (TQFT) of the multicomponent Chern-Simons type. However, in contrast to the fractional quantum Hall effect where TQFT can give rise to interesting bulk anyons, TQFT for SPTs only supports trivial bulk excitations. The essential question in TQFT descriptions for SPTs is to understand how the global symmetry is implemented in the partition function. In this paper, we systematically study TQFT of three-dimensional SPTs with unitary Abelian symmetry (e.g., ZN1\ifmmode×\else\texttimes\fiZN2\ifmmode×\else\texttimes\fi...{\mathbb{Z}}_{{N}_{1}}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{N}_{2}}\ifmmode\times\else\texttimes\fi{}...ZN1​​\ifmmode×\else\texttimes\fiZN2​​\ifmmode×\else\texttimes\fi...). In addition to the usual multicomponent BFBFBF topological term at level-1, we find that there are new topological terms with quantized coefficients (e.g., a1\ensuremath∧a2\ensuremath∧da2{a}^{1}\ensuremath{\wedge}{a}^{2}\ensuremath{\wedge}d{a}^{2}a1\ensuremath∧a2\ensuremath∧da2 and a1\ensuremath∧a2\ensuremath∧a3\ensuremath∧a4{a}^{1}\ensuremath{\wedge}{a}^{2}\ensuremath{\wedge}{a}^{3}\ensuremath{\wedge}{a}^{4}a1\ensuremath∧a2\ensuremath∧a3\ensuremath∧a4) in TQFT actions, where a1,a2,...{a}^{1},{a}^{2},...a1,a2,... are 1-form U(1) gauge fields. These additional topological terms cannot be adiabatically turned off as long as GGG is unbroken. By investigating symmetry transformations for the TQFT partition function, we end up with the classification of SPTs that is consistent with the well-known group cohomology approach. We also discuss how to gauge the global symmetry and possible TQFT descriptions of Dijkgraaf-Witten gauge theory.
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.
A large class of symmetry-protected topological phases (SPT) in boson/spin systems have been recently predicted by the group cohomology theory. In this work, we consider bosonic SPT states at least with charge symmetry [U(1) or ZN{}_{N}N​] or spin-Sz{S}^{z}Sz rotation symmetry [U(1) or ZN{}_{N}N​] in two (2D) and three dimensions (3D) and the surface of 3D. If both are U(1), we apply external electromagnetic field/``spin gauge field'' to study the charge/spin response. For the SPT examples we consider {i.e., Uc{}_{c}c​(1)$\ensuremath{\rtimes}$Z2T{}_{2}^{T}2T​, Us{}_{s}s​(1)$\ifmmode\times\else\texttimes\fi{}$Z2T{}_{2}^{T}2T​, Uc{}_{c}c​(1)$\ifmmode\times\else\texttimes\fi{}$[Us{}_{s}s​(1)$\ensuremath{\rtimes}$Z2{}_{2}2​]; subscripts ccc and sss are short for charge and spin; Z2T{}_{2}^{T}2T​ and Z2{}_{2}2​ are time-reversal symmetry and $\ensuremath{\pi}$ rotation about Sy{S}^{y}Sy, respectively}, many variants of Witten effect in the 3D SPT bulk and various versions of anomalous surface quantum Hall effect are defined and systematically investigated. If charge or spin symmetry reduces to ZN{}_{N}N​ by considering charge-NNN or spin-NNN condensate, instead of the linear response approach, we gauge the charge/spin symmetry, leading to a dynamical gauge theory with some remaining global symmetry. The 3D dynamical gauge theory describes a symmetry-enriched topological phase (SET), i.e., a topologically ordered state with global symmetry which admits nontrivial ground-state degeneracy depending on spatial manifold topology. For the SPT examples we consider, the corresponding SET states are described by dynamical topological gauge theory with topological BF\mathit{BF}BF term and axionic $\ensuremath{\Theta}$ term in 3D bulk. In addition, the surface of SET is described by the chiral boson theory with quantum anomaly.
Abelian Chern-Simons theory, characterized by the so-called KKK matrix, has been quite successful in characterizing and classifying Abelian fractional quantum Hall effect as well as symmetry protected topological (SPT) phases, especially for bosonic SPT phases. However, there are still some puzzles in dealing with fermionic SPT (fSPT) phases. In this paper, we utilize the Abelian Chern-Simons theory to study the fSPT phases protected by arbitrary Abelian total symmetry Gf{G}_{f}Gf​. Comparing to the bosonic SPT phases, fSPT phases with Abelian total symmetry Gf{G}_{f}Gf​ have three new features: (1) they may support gapless Majorana fermion edge modes, (2) some nontrivial bosonic SPT phases may be trivialized if Gf{G}_{f}Gf​ is a nontrivial extension of bosonic symmetry Gb{G}_{b}Gb​ by Z2f{\mathbb{Z}}_{2}^{f}Z2f​, and (3) certain intrinsic fSPT phases can only be realized in interacting fermionic system. We obtain edge theories for various fSPT phases, which can also be regarded as conformal field theories with proper symmetry anomaly. In particular, we discover the construction of Luttinger liquid edge theories with central charge $n\ensuremath{-}1$ for type-III bosonic SPT phases protected by (Zn)3{({\mathbb{Z}}_{n})}^{3}(Zn​)3 symmetry and the Luttinger liquid edge theories for intrinsically interacting fSPT protected by unitary Abelian symmetry. The ideas and methods used in these examples could be generalized to derive the edge theories of fSPT phases with arbitrary unitary Abelian total symmetry Gf{G}_{f}Gf​.
Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry GGG, 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 T2=1{T}^{2}=1T2=1 time-reversal Z2T{Z}_{2}^{T}Z2T​ and fermion-number-parity Z2f{Z}_{2}^{f}Z2f​ 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 KKK-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=1c=1c=1 gapless edge excitations, if the symmetry is not broken.
Recently, a new class of quantum phases of matter---symmetry protected topological states, such as topological insulators---has attracted much attention. In presence of interactions, group cohomology provides a classification of these [Chen et al., Phys. Rev. B 87, 155114 (2013)]. These phases have short-ranged entanglement and no topological order in the bulk. However, when long-range entangled topological order is present, it is much less understood how to classify quantum phases of matter in the presence of global symmetries. Here we present a classification of bosonic gapped quantum phases with or without long-range entanglement in the presence or absence of on-site global symmetries. In 2+12+12+1 dimensions, the quantum phases in the presence of a global symmetry group SGSGSG, and with topological order described by a finite gauge group GGGGGG, are classified by the cohomology group ${H}^{3}(SG\ifmmode\times\else\texttimes\fi{}GG,U(1))$. Generally, in d+1d+1d+1 dimensions, such quantum phases are classified by ${H}^{d+1}(SG\ifmmode\times\else\texttimes\fi{}GG,U(1))$. Although we only partially understand to what extent our classification is complete, we present an exactly solvable local bosonic model, in which the topological order is emergent, for each given class in our classification. When the global symmetry is absent, the topological order in our models is described by the general Dijkgraaf-Witten discrete gauge theories. When the topological order is absent, our models become the exactly solvable models for symmetry protected topological phases [Chen et al., Phys. Rev. B 87, 155114 (2013)]. When both the global symmetry and the topological order are present, our models describe symmetry enriched topological phases. Our classification includes, but goes beyond, the previously discussed projective symmetry group classification. Measurable signatures of these symmetry enriched topological phases and generalizations of our classification are discussed.
Symmetry-Protected Topological relationship between SU(3)SU(3)SU(3) and SU(2)×U(1)SU(2)\times{U(1)}SU(2)×U(1) in Two Dimension
Symmetry-protected topological (SPT)\left(SPT\right)(SPT) phases are gapped short-range entangled states with symmetry GGG, which can be systematically described by group cohomology theory. SU(3)SU(3)SU(3) and SU(2)×U(1)SU(2)\times{U(1)}SU(2)×U(1) are considered as the basic groups of Quantum Chromodynamics and Weak-Electromagnetic unification, respectively. In two dimension (2D)(2D)(2D), nonlinear-sigma models with a quantized topological Theta term can be used to describe nontrivial SPT phases. By coupling the system to a probe field and integrating out the group variables, the Theta term becomes the effective action of Chern-Simons theory which can derive the response current density. As a result, the current shows a spin Hall effect, and the quantized number of the spin Hall conductance of SPT phases SU(3)SU(3)SU(3) and SU(2)×U(1)SU(2)\times{U(1)}SU(2)×U(1) are same. In addition, relationships between SU(3)SU(3)SU(3) and SU(2)×U(1)SU(2)\times{U(1)}SU(2)×U(1) which maps SU(3)SU(3)SU(3) to SU(2)SU(2)SU(2) with a rotation U(1)U(1)U(1) will be given.