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arXiv:1304.4569 · Physical Review B

Effect of interactions on two-dimensional fermionic symmetry-protected topological phases with Z2 symmetry

Zheng‐Cheng Gu, Michael Levin

Published May 19, 2014• 143 citations
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Abstract

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 ${Z}_{2}$ Ising symmetry. Although these systems are classified by $\mathbb{Z}$ in the noninteracting limit, our results suggest that the classification collapses to ${\mathbb{Z}}_{8}$ 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.

DOI
10.1103/physrevb.89.201113

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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 study braiding statistics between quasiparticles and vortices in two-dimensional charge-2m2m2m (in units of eee) superconductors that are coupled to a Z2m{\mathbb{Z}}_{2m}Z2m​ dynamical gauge field, where mmm is any positive integer. We show that there exist 16m16m16m types of braiding statistics when mmm is odd, but only 4m4m4m types when mmm is even. Based on the braiding statistics, we obtain a classification of topological phases of charge-2m2m2m superconductors---or formally speaking, a classification of symmetry-protected topological phases, as well as invertible topological phases, of two-dimensional gapped fermions with Z2mf{\mathbb{Z}}_{2m}^{f}Z2mf​ symmetry. Interestingly, we find that there is no nontrivial fermionic symmetry-protected topological phase with Z4f{\mathbb{Z}}_{4}^{f}Z4f​ symmetry.
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.
We construct a two-dimensional (2D) quantum spin model that realizes an Ising paramagnet with gapless edge modes protected by Ising symmetry. This model provides an example of a ``symmetry-protected topological phase.'' We describe a simple physical construction that distinguishes this system from a conventional paramagnet: We couple the system to a Z2{\mathbb{Z}}_{2}Z2​ gauge field and then show that the $\ensuremath{\pi}$-flux excitations have different braiding statistics from that of a usual paramagnet. In addition, we show that these braiding statistics directly imply the existence of protected edge modes. Finally, we analyze a particular microscopic model for the edge and derive a field theoretic description of the low energy excitations. We believe that the braiding statistics approach outlined in this paper can be generalized to a large class of symmetry-protected topological phases.
In free fermion systems with given symmetry and dimension, the possible topological phases are labeled by elements of only three types of Abelian groups, 0, Z2, or Z. For example, noninteracting one-dimensional fermionic superconducting phases with S(z) spin rotation and time-reversal symmetries are classified by Z. We show that with weak interactions, this classification reduces to Z4. Using group cohomology, one can additionally show that there are only four distinct phases for such one-dimensional superconductors even with strong interactions. Comparing their projective representations, we find that all these four symmetry-protected topological phases can be realized with free fermions. Further, we show that one-dimensional fermionic superconducting phases with Z(n) discrete S(z) spin rotation and time-reversal symmetries are classified by Z4 when n is even and Z2 when n is odd; again, all these strongly interacting topological phases can be realized by noninteracting fermions. Our approach can be applied to systems with other symmetries to see which one-dimensional topological phases can be realized with free fermions.
We give a brief overview on interacting symmetry-protected topological (SPT) phases. In particular, we will focus on various cases where the classification of non-interacting fermionic systems breaks down in the presence of interactions. The examples include topological (and trivial) superconductors in one, two and three spatial dimensions, protected by various symmetries such as time-reversal, spin parity, and reflection symmetries. Common to these examples is the ‘collapse’ of the non-interacting classification, where an integer classification (‘ ’) at non-interacting level reduces to or in the presence of interactions. We make use of these examples to illustrate various approaches to interacting SPT phases.
We give a brief overview on interacting symmetry-protected topological (SPT) phases. In particular, we will focus on various cases where the classification of non-interacting fermionic systems breaks down in the presence of interactions. The examples include topological (and trivial) superconductors in one, two and three spatial dimensions, protected by various symmetries such as time-reversal, spin parity, and reflection symmetries. Common to these examples is the ‘collapse’ of the non-interacting classification, where an integer classification (‘’) at non-interacting level reduces to or in the presence of interactions. We make use of these examples to illustrate various approaches to interacting SPT phases.