All Majorana Models with Translation Symmetry are Supersymmetric

TL;DR

This paper proves that all Majorana models with translation symmetry inherently exhibit supersymmetry and guaranteed degeneracy, revealing fundamental constraints and potential experimental signatures in such systems.

Contribution

It demonstrates that translation-symmetric Majorana systems always possess supersymmetry and degeneracy, providing explicit constructions and extending fundamental theorems.

Findings

All states are at least doubly degenerate in 1D and 2D Majorana arrays with translation symmetry.

Such systems inherently have an underlying $ abla=2$ supersymmetry.

Degeneracy manifests as a zero-bias peak in tunneling conductance.

Abstract

We establish results similar to Kramers and Lieb-Schultz-Mattis theorems but involving only translation symmetry and for Majorana modes. In particular, we show that all states are at least doubly degenerate in any one and two dimensional array of Majorana modes with translation symmetry, periodic boundary conditions, and an odd number of modes per unit cell. Moreover, we show that all such systems have an underlying $\mathcal{N}=2$ supersymmetry and explicitly construct the generator of the supersymmetry. Furthermore, we establish that there cannot be a unique gapped ground state in such one dimensional systems with anti-periodic boundary conditions. These general results are fundamentally a consequence of the fact that translations for Majorana modes are represented projectively, which in turn stems from the anomalous nature of a single Majorana mode. An experimental signature of the degeneracy arising from supersymmetry is a zero-bias peak in tunneling conductance.