Topological crystalline superconductors with linearly and projectively represented $C_{n}$ symmetry

Chen Fang; B. Andrei Bernevig; Matthew J. Gilbert

paper

Topological crystalline superconductors with linearly and projectively represented $C_{n}$ symmetry

arXiv:1701.01944

Abstract

We study superconductors with $n$ -fold rotational invariance both in the presence and in the absence of spin-orbit interactions. More specifically, we classify the non-interacting Hamiltonians by defining a series of $Z$ -numbers for the Bogoliubov-de Gennes (BdG) symmetry classes of the Altland-Zimbauer classification of random matrices in $1$ D, $2$ D, and $3$ D in the presence of discrete rotational invariance. Our analysis emphasizes the important role played by the angular momentum of the Cooper pairs in the system: for pairings of nonzero angular momentum, the rotation symmetry may be represented projectively, and a projective representation of rotation symmetry may have anomalous properties, including the anti-commutation with the time-reversal symmetry. In 1D and 3D, we show how an $n$ -fold axis enhances the topological classification and give additional topological numbers; in 2D, we establish a relation between the Chern number (in class D and CI) and the eigenvalues of rotation symmetry at high-symmetry points. For each nontrivial class in 3D, we write down a minimal effective theory for the surface Majorana states.

arXiv comment21 pages, 5 figures

Topological crystalline superconductors with linearly and projectively represented Cn​ symmetry

Abstract

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Topological crystalline superconductors with linearly and projectively represented $C_{n}$ symmetry