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arXiv:2206.09393

Non-abelian bosonization of fermion symmetry-protected topological states

Yen-Ta Huang, Dung‐Hai Lee

Published Jun 19, 2022• 0 citations
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Abstract

Applying the results of Ref.[1], we carry out the non-abelian bosonization for a class of free fermion symmetry-protected topological states (SPTs). The resulting boson theories are non-linear sigma models with topological $θ$ term, thus unifying the field theory description of the (free) fermionic and bosonic SPTs. Tuning $θ$ from $0$ to $2π$ triggers the topological trivial to non-trivial phase transitions. In addition, applying the same idea to the symmetry-protected critical spin liquids, we obtain the NL$\s$ models for spin SPTs living in one spatial dimension higher.

DOI
10.48550/arxiv.2206.09393

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Applying the results of Ref.[1], we carry out the non-abelian bosonization for a class of free fermion symmetry-protected topological states (SPTs). The resulting boson theories are non-linear sigma models with topological θθθ term, thus unifying the field theory description of the (free) fermionic and bosonic SPTs. Tuning θθθ from 000 to 2π2π2π triggers the topological trivial to non-trivial phase transitions. In addition, applying the same idea to the symmetry-protected critical spin liquids, we obtain the NL\s\s\s models for spin SPTs living in one spatial dimension higher.
Applying the method of [Nucl. Phys. B 972, 115565 (2021)], which bosonizes massless relativistic free fermions, we derive the (non-Abelian) bosonized theory for free fermion topological insulators and superconductors that have, in addition to the U(1) charge, time reversal and charge conjugation symmetries and flavor symmetries. For the case we consider, the flavor symmetries render the topological classification Z\mathbb{Z}Z. The results are nonlinear $\ensuremath{\sigma}$ models with the topological $\ensuremath{\theta}$ term. In addition, we present the theory of a class of bosonic symmetry-protected topological states, whose boundaries are critical spin liquids.
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.
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.
We study the classification for a large class of interacting fermionic and bosonic symmetry-protected topological (SPT) states, focusing on the cases where interaction reduces the classification of free-fermion SPT states. We define a SPT state as whether it is separated from the trivial state through a bulk phase transition, which is a general definition applicable to SPT states with or without spatial symmetries. We show that in all dimensions short-range interactions can reduce the classification of free-fermion SPT states, and we demonstrate these results by making a connection between the fermionic and bosonic SPT states. We first demonstrate that our formalism gives the correct classification for several known SPT states, with or without interaction, and then we generalize our method to SPT states that involve the spatial inversion symmetry.
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., ${\mathbb{Z}}_{{N}_{1}}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{N}_{2}}\ifmmode\times\else\texttimes\fi{}...$). 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., ${a}^{1}\ensuremath{\wedge}{a}^{2}\ensuremath{\wedge}d{a}^{2}$ and ${a}^{1}\ensuremath{\wedge}{a}^{2}\ensuremath{\wedge}{a}^{3}\ensuremath{\wedge}{a}^{4}$) in TQFT actions, where 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.
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​.