All Majorana Models with Translation Symmetry are Supersymmetric

Understand

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.