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arXiv:1406.3032

Interaction effects on 3D topological superconductors: surface topological order from vortex condensation, the 16 fold way and fermionic Kramers doublets

Max A. Metlitski, Lukasz Fidkowski, Xie Chen, Ashvin Vishwanath

Published Jun 11, 2014• 50 citations
arXivHTMLPDF

Abstract

Three dimensional topological superconductors with time reversal symmetry (class DIII) are indexed by an integer $ν$, the number of surface Majorana cones, according to the free fermion classification. The superfluid B phase of He$^3$ realizes the $ν=1$ phase. Recently, it has been argued that this classification is reduced in the presence of interactions to Z$_{16}$. This was argued from the symmetry respecting surface topological orders of these states, which provide a non-perturbative definition of the bulk topological phase. Here, we verify this conclusion by focusing on the even index case, $ν=2m$, where a vortex condensation approach can be used to explicitly derive the surface topological orders. We show a direct relation to the well known result on one dimensional topological superconductors (class BDI), where interactions reduce the free fermion classification from Z down to Z$_8$. Finally, we discuss in detail the fermionic analog of Kramers time reversal symmetry, which allows semions of some surface topological orders to transform as $T^2=\pm i$.

DOI
10.48550/arxiv.1406.3032

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Three dimensional topological superconductors with time reversal symmetry (class DIII) are indexed by an integer ννν, the number of surface Majorana cones, according to the free fermion classification. The superfluid B phase of He3^33 realizes the ν=1ν=1ν=1 phase. Recently, it has been argued that this classification is reduced in the presence of interactions to Z16_{16}16​. This was argued from the symmetry respecting surface topological orders of these states, which provide a non-perturbative definition of the bulk topological phase. Here, we verify this conclusion by focusing on the even index case, ν=2mν=2mν=2m, where a vortex condensation approach can be used to explicitly derive the surface topological orders. We show a direct relation to the well known result on one dimensional topological superconductors (class BDI), where interactions reduce the free fermion classification from Z down to Z8_88​. Finally, we discuss in detail the fermionic analog of Kramers time reversal symmetry, which allows semions of some surface topological orders to transform as T2=±iT^2=\pm iT2=±i.
In this paper we show how the classification of topological phases in insulators and superconductors is changed by interactions, in the case of one-dimensional systems. We focus on the time-reversal-invariant Majorana chain (BDI symmetry class). While the band classification yields an integer topological index kkk, it is known that phases characterized by values of kkk in the same equivalence class modulo 8 can be adiabatically transformed one to another by adding suitable interaction terms. Here we show that the eight equivalence classes are distinct and exhaustive, and provide a physical interpretation for the interacting invariant modulo 8. The different phases realize different Altland-Zirnbauer classes of the reduced density matrix for an entanglement bipartition into two half chains. We generalize these results to the classification of all one-dimensional gapped phases of fermionic systems with possible antiunitary symmetries, utilizing the algebraic framework of central extensions. We use matrix product state methods to prove our results.
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Classification of symmetry-protected topological phases such as the topological superconductor (TSC) is well known in the free fermion picture. Recently, it was observed that including many-body interactions that preserves symmetry gives rise to novel surface phases and breaks down the classification scheme. A complete understanding of the microscopic theory for such phases is missing. This paper is an attempt to understand the interaction that gives rise to these phases in the case of the three-dimensional TSC. The authors build a coupled wire model Hamiltonian for the gapless surface and show that the addition of a time-reversal symmetric two-body interaction, constructed in this paper, opens an energy gap in the spectrum. They also show this gapped surface phase is topologically nontrivial and is classified differently compared to the noninteracting case.
The surface electronic structure of a three-dimensional topological superconductor is characterized by a Majorana cone---Majorana-fermion-like electronic excitations whose energies form a gapless continuum, rendering the surface conducting. It was believed that a gap in that electronic structure could only be created if time-reversal symmetry were broken. Theorists now show that, in the presence of strong electronic interaction, ``gapping'' and time-reversal symmetry can coexist if the surface has a special type of electronic excitations that are ``anyons.''
The natural existence of crystalline symmetry in real materials manifests the importance of understanding crystalline symmetry-protected topological phases, especially for interacting systems. In this paper, we systematically construct and classify all the point-group crystalline topological superconductors and insulators in three-dimensional (3D) interacting fermion systems using the novel concept of a topological crystal. The corresponding higher-order topological surface theory can also be systematically studied via higher-order bulk-boundary correspondence. In particular, we discover several intrinsically interacting topological phases that cannot be realized in any free-fermion systems. Moreover, the crystalline equivalence principle for 3D interacting fermionic systems is also verified, with an additional subtle “twist” on the spin of fermions.