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arXiv:1302.4803 · Physical Review Letters

Topological Response Theory of Abelian Symmetry-Protected Topological Phases in Two Dimensions

Meng Cheng, Zheng‐Cheng Gu

Published Apr 9, 2014• 67 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.112.141602

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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.
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​.
According to a recent theoretical conjecture for a classification of symmetry protected topological (SPT) phases which can arise in a collection of lattice boson models called group cohomology models, each model belongs to a distinct SPT phase, and each SPT phase can be realized by a model. Here the authors show that for SPT phases with symmetry groups that are finite, Abelian, and unitary, each group cohomology model can be characterized by a unique set of topological invariants and therefore belongs to a distinct SPT phase. In two dimensions it is further shown that the complete set of group cohomology models accounts for all SPT phases, while in three dimensions evidence pointing toward such a proof is found.
Symmetry-protected topological (SPT) phases (e.g., the well known topological insulators) are a class of energetically gapped condensed matter systems that exhibit interesting topological properties only in the presence of certain global symmetries. While a great understanding of noninteracting fermionic SPT phases has been achieved recently, interacting SPT phases, in particular those that exist only in strongly interacting systems, are much less understood in general. Here, the authors study strongly interacting fermionic SPT phases in two spatial dimensions with finite Abelian unitary symmetries and provide a potentially complete classification of them.
The construction and classification of symmetry-protected topological (SPT) phases in interacting bosonic and fermionic systems have been intensively studied in the past few years. Very recently, a complete classification and construction of space-group SPT phases were also proposed for interacting bosonic systems. In this Rapid Communication, we attempt to generalize this classification and construction scheme systematically into interacting fermion systems. In particular, we construct and classify point-group SPT phases for two-dimensional (2D) interacting fermion systems via lower-dimensional block-state decorations. We discover several intriguing fermionic SPT states that can only be realized in interacting fermion systems (i.e., not in free-fermion or bosonic SPT systems). Moreover, we also verify the recently conjectured crystalline equivalence principle for 2D interacting fermion systems. Finally, the potential experimental realization of these different classes of point-group SPT phases in 2D correlated superconductors is addressed.
Symmetry-protected topological (SPT) phases are bulk-gapped quantum phases with symmetries, which have gapless or degenerate boundary states as long as the symmetries are not broken. The SPT phases in free fermion systems, such as topological insulators, can be classified; however, it is not known what SPT phases exist in general interacting systems. We present a systematic way to construct SPT phases in interacting bosonic systems. Just as group theory allows us to construct 230 crystal structures in three-dimensional space, we use group cohomology theory to systematically construct different interacting bosonic SPT phases in any dimension and with any symmetry, leading to the discovery of bosonic topological insulators and superconductors.
We study the classification of interacting fermionic symmetry-protected topological (SPT) phases with both rotation symmetry and Abelian internal symmetries in one, two, and three dimensions. By working out this classification, on the one hand, we demonstrate the recently proposed correspondence principle between crystalline topological phases and those with internal symmetries through explicit block-state constructions. We find that for the precise correspondence to hold it is necessary to change the central extension structure of the symmetry group by the Z2{\mathbb{Z}}_{2}Z2​ fermion parity. On the other hand, we uncover new classes of intrinsically fermionic SPT phases that are only enabled by interactions, both in 2D and 3D. Several 3D Majorana-type fermionic SPT phases are identified. Moreover, several new instances of Lieb-Schultz-Mattis-type theorems for Majorana-type fermionic SPT phases are obtained and we discuss their interpretations from the perspective of bulk-boundary correspondence.