Lieb-Schultz-Mattis type theorems for Majorana models with discrete symmetries

TL;DR

This paper establishes Lieb-Schultz-Mattis type theorems for fermionic lattice models with Majorana fermions, showing constraints on ground state degeneracy, symmetry, and gap conditions based on internal symmetries and lattice structure.

Contribution

It extends LSM theorems to Majorana models with discrete symmetries, revealing new constraints on ground states in fermionic systems with specific symmetry representations.

Findings

Ground state cannot be nondegenerate, symmetric, and gapped if the internal symmetry is realized projectively.

Ground state cannot be nondegenerate, gapped, and translation symmetric with an odd number of Majoranas per unit cell.

Results apply to translationally invariant, local fermionic lattice Hamiltonians with broken fermion-number conservation.

Abstract

We prove two Lieb-Schultz-Mattis type theorems that apply to any translationally invariant and local fermionic $d$-dimensional lattice Hamiltonian for which fermion-number conservation is broken down to the conservation of fermion parity. We show that when the internal symmetry group $G^{\,}_{f}$ is realized locally (in a repeat unit cell of the lattice) by a nontrivial projective representation, then the ground state cannot be simultaneously nondegenerate, symmetric (with respect to lattice translations and $G^{\,}_{f}$), and gapped. We also show that when the repeat unit cell hosts an odd number of Majorana degrees of freedom and the cardinality of the lattice is even, then the ground state cannot be simultaneously nondegenerate, gapped, and translation symmetric.