Abstract
Supersymmetry, originally proposed in particle physics, refers to a dual relation that connects fermionic and bosonic degrees of freedom in a system. Recently, there has been considerable interest in applying the idea of supersymmetry to topological phases, motivated by the attempt to gain insights from the fermion side into the boson side and vice versa. We present a systematic study of this construction when applied to band topology in noninteracting systems. First, on top of the conventional ten-fold way, we find that topological insulators and superconductors are divided into three classes depending on whether the supercharge can be local and symmetric, must break a symmetry to preserve locality, or needs to break locality. Second, we resolve the apparent paradox between the nontriviality of free fermions and the triviality of free bosons by noting that the topological information is encoded in the identification map. We also discuss how to understand a recently revealed supersymmetric entanglement duality in this context. These findings are illustrated by prototypical examples. Our work sheds new light on band topology from the perspective of supersymmetry.
Highlights
Supersymmetric (SUSY) models play an important role in physics
We should impose the locality such that hf,b(k) is a smooth map from T d ≡ (2π R/Z)d to a matrix space constrained by the gap condition and symmetries
We have examined the role of topology in the SUSY construction of quadratic fermion and boson Hamiltonians
Summary
Perhaps the most well-known is their use in relativistic quantum field theories, where they resolve a number of theoretical problems [1]. It is a powerful tool in the analysis of disordered systems [6] or to solve an array of problems in quantum and statistical physics [7]. The idea of SUSY has been applied to extract the topological indices of free-boson systems, such as photonic and magnonic crystals, from their free-fermion counterparts [16–19]. This strategy works well for individual bands, which may still be nontrivial despite the fact that the ground state of free bosons is always trivial [20]. The entanglement problem gets resolved as the SUSY map, called the identification map, can lead to very strong squeezing, if it is locality preserving
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