Majorana Scars as Group Singlets

TL;DR

This paper explores Majorana fermion lattice systems, identifying special invariant states called scars, which have lower entanglement and distinct symmetry properties, and discusses their spectrum, degeneracies, and potential ergodic behavior.

Contribution

It introduces a group-theoretic framework for Majorana scars, generalizes known scar states to higher M, and analyzes their entanglement and spectral properties.

Findings

Two families of Majorana scars: η-states and ζ-states.

Entanglement entropy of scars grows logarithmically with subsystem size.

Scar states can be made equidistant in energy and influence the density of states.

Abstract

In some quantum many-body systems, the Hilbert space breaks up into a large ergodic sector and a much smaller scar subspace. It has been suggested [arXiv:2007.00845] that the two sectors may be distinguished by their transformation properties under a large group whose rank grows with the system size (it is not a symmetry of the Hamiltonian). The quantum many-body scars are invariant under this group, while all other states are not. Here we apply this idea to lattice systems containing $M$ Majorana fermions per site. The Hilbert space for $N$ sites may be decomposed under the action of the O$(N)\times$O$(M)$ group, and the scars are the SO$(N)$ singlets. For any even $M$ there are two families of scars. One of them, which we call the $\eta$ states, is symmetric under the group O$(N)$. The other, the $\zeta$ states, has the SO$(N)$ invariance. For $M=4$, where our construction reduces to spin-$1/2$ fermions on a lattice with local interactions, the former family are the $N+1$ $\eta$-pairing states, while the latter are the $N+1$ states of maximum spin. We generalize this construction to $M>4$. For $M=6$ we exhibit explicit formulae for the scar states and use them to calculate the bipartite entanglement entropy analytically. For large $N$, it grows logarithmically with the sub-system size. We present a general argument that any group-invariant scars should have the entanglement entropy that is parametrically smaller than that of typical states. The energies of the scars we find are not equidistant in general but can be made so by choosing Hamiltonian parameters. For $M>6$ we find that with local Hamiltonians the scars typically have certain degeneracies. The scar spectrum can be made ergodic by adding a non-local interaction term. We derive the dimension of each scar family and show the scars could have a large contribution to the density of states for small $N$.