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

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.

Figure 1

The repeat unit cells of a lattice $\mathrm{\Lambda }$ are represented pictorially by squares. The lattice $\mathrm{\Lambda }$ is chosen for simplicity to be one dimensional. (a) The repeat unit cell is decorated with two circles, one empty, the other filled. If periodic boundary conditions are imposed, translations by one repeat unit cell are symmetries. The permutation of the empty and filled circle within all repeat unit cells is not a symmetry. (b) If the filling pattern is smoothly tuned (through an on-site potential whose magnitude is color coded, say) so that both circles in an repeat unit cell have the same filling, then the permutation of the left and right circles within all repeat unit cells is a symmetry. One may then choose a smaller repeat unit cell, a square centered about one circle only. (c) Image of (a) under the permutation of the empty and filled circle within all repeat unit cells.

Figure 2

Example of a path that visits all the sites of a two-dimensional lattice that decorates the surface of a torus.