All Majorana Models with Translation Symmetry are Supersymmetric

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 antiperiodic 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.

Figure 1

Left: Translationally invariant Majorana modes, with periodic boundary conditions, have at least twofold degeneracy in the energy spectrum. The underlying supersymmetry requires that each energy level contains pairs of fermionic and bosonic superpartners. Right: The same system, with antiperiodic boundary conditions (depicted by a slash through a bond), cannot have a unique gapped ground state in the thermodynamic limit.

Figure 2

Our work applies to effective Hamiltonians (green) describing the Majorana modes (blue) emerging at the end points of topological superconductors (vertical chains). Each oval is a spinless fermion consisting of two Majorana fermions (black dots).