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.
Coupled-wire constructions have been widely applied to quantum Hall systems and symmetry-protected topological (SPT) phases. In this Letter, we use the coupled one-dimensional nonchiral Luttinger liquids with domain-wall structured mass terms as quantum wires to construct crystalline higher-order topological superconductors (HOTSCs) in two-dimensional (2D) interacting fermionic systems by two representative examples: a
D4-symmetric class-D HOTSC and a
C4-symmetric class-BDI HOTSC, with Majorana corner modes on the edge. Furthermore, based on the coupled-wire constructions, the quantum phase transitions between different phases of 2D HOTSCs by tuning the interwire coupling are investigated in a straightforward way.
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
T2=±i.