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arXiv:1505.06598 · Physical Review B

Correlation effects on topological crystalline insulators

Tsuneya Yoshida, Akira Furusaki

Published Aug 7, 2015• 28 citations
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Abstract

We study interaction effects on the topological crystalline insulators protected by time-reversal $(T)$ and reflection symmetry $(R)$ 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 $\mathbb{Z}$ to ${\mathbb{Z}}_{4}$ by interactions where the ${Z}_{2}$ 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 $\mathbb{Z}$ to ${\mathbb{Z}}_{8}$ by interactions.

DOI
10.1103/physrevb.92.085114

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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 Z2\ifmmode×\else\texttimes\fi[U(1)\ensuremath⋊T]{Z}_{2}\ifmmode\times\else\texttimes\fi{}[\text{U(1)}\ensuremath{\rtimes}T]Z2​\ifmmode×\else\texttimes\fi[U(1)\ensuremath⋊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×\else\texttimes\fi[U(1)\ensuremath⋊T]R\ifmmode\times\else\texttimes\fi{}[\text{U(1)}\ensuremath{\rtimes}T]R\ifmmode×\else\texttimes\fi[U(1)\ensuremath⋊T] symmetry is reduced from Z\mathbb{Z}Z to Z8{\mathbb{Z}}_{8}Z8​ by interactions.
We study the effect of electron interactions in topological crystalline insulators (TCIs) protected by mirror symmetry, which are realized in the SnTe material class and host multivalley Dirac fermion surface states. We find that interactions reduce the integer classification of noninteracting TCIs in three dimensions, indexed by the mirror Chern number, to a finite group Z8{Z}_{8}Z8​. In particular, we explicitly construct a microscopic interaction Hamiltonian to gap eight flavors of Dirac fermions on the TCI surface, while preserving the mirror symmetry. Our construction builds on interacting edge states of $U(1)\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry-protected topological phases of fermions in two dimensions, which we classify. Our work reveals a deep connection between three-dimensional topological phases protected by spatial symmetries and two-dimensional topological phases protected by internal symmetries.
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We present the correspondence between symmetry-protected topological (SPT) phases and their anomalous boundary states, based on examples in various spacetime dimensions. Through the study of the effect of interactions on these SPT phases, we discovered a new formalism of quantum anomalies, associated with discrete spacetime (such as time-reversal and spatial reflection) symmetries in particular, to classify distinct interacting topological phases. An example is the Z2 classification of the (2+1)d topological insulator protected by charge U(1) and time-reversal (or CP) symmetries, which can be deduced by the form of the global U(1) gauge anomalies on its edge theories defined on closed unorientable manifolds. In this case, the nontrivial phase (in free systems) is robust against electron interactions. Another example is the (3+1)d topological superconductor protected by only time-reversal or reflection symmetry. For this system, we identified the bulk phase by studying the global gravitational anomalies of the surface theories formulated on unorientable spacetime manifolds, and also discussed its connection to the collapse of the non-interacting classification by an integer Z to Z16, in the presence of interactions. We also revisit the problem of gauging a discrete internal symmetry in theories of chiral (Weyl) fermions in 3+1 dimensions – which have not been fully understood so far – from the perspective of fermionic SPT phases in 4+1 dimensions. Comparing with the previous results, we give a complete answer for the anomalies constraints on the discrete symmetry, as our approach is based on purely geometrical considerations, namely, our assumption is more fundamental and general. Furthermore, our result also provides an understanding of gapped states of fermions with anomalous discrete symmetries, and we present a model, based on weak coupling, for constructing these anomalous gapped states.
Symmetry protected topological (SPT) phases are a minimal generalization of the concept of topological insulators to interacting systems. In this paper, we describe the classification and properties of such phases for three-dimensional (3D) electronic systems with a number of different symmetries. For symmetries representative of all classes in the famous 10-fold way of free-fermion topological insulators/superconductors, we determine the stability to interactions. By combining with results on bosonic SPT phases, we obtain a classification of electronic 3D SPT phases for these symmetries. In cases with a normal U(1)U(1)U(1) subgroup we show that this classification is complete. We describe the nontrivial surface and bulk properties of these states. In particular, we discuss interesting correlated surface states that are not captured in a free-fermion description. We show that in many, but not all, cases, the surface can be gapped while preserving symmetry if it develops intrinsic topological order.
We study the effect of interactions on two-dimensional fermionic symmetry-protected topological (SPT) phases using the recently proposed braiding statistics approach. We focus on a simple class of examples: superconductors with a Z2{Z}_{2}Z2​ Ising symmetry. Although these systems are classified by Z\mathbb{Z}Z in the noninteracting limit, our results suggest that the classification collapses to Z8{\mathbb{Z}}_{8}Z8​ in the presence of interactions---consistent with previous work that analyzed the stability of the edge. Specifically, we show that there are at least eight different types of Ising superconductors that cannot be adiabatically connected to one another, even in the presence of strong interactions. In addition, we prove that each of the seven nontrivial superconductors have protected edge modes.
We introduce a novel class of interaction-enabled topological crystalline insulators in two- and three-dimensional electronic systems, which we call ``topological crystalline magnet.'' It is protected by the product of the time-reversal symmetry T\mathcal{T}T and a mirror symmetry or a rotation symmetry R\mathcal{R}R. A topological crystalline magnet exhibits two intriguing features: (i) it cannot be adiabatically connected to any Slater insulator and (ii) the edge state is robust against coupling electrons to the edge. These features are protected by the anomalous symmetry transformation property ${(\mathcal{R}\mathcal{T})}^{2}=\ensuremath{-}1$ of the edge state. An anisotropic response to the external magnetic field can be an experimental signature.