Entanglement of Free Fermions on Hamming Graphs

TL;DR

This paper analyzes entanglement entropy in free fermions on Hamming graphs, deriving analytical formulas for specific subsystems and connecting the problem to integrable models like BC-Gaudin magnets.

Contribution

It provides new analytical expressions for entanglement entropy in Hamming graph fermion systems and links the problem to integrable Hamiltonians.

Findings

Analytical formulas for entanglement entropy of Hamming subgraph subsets

Closed-form entanglement entropy for neighborhood-based subsystems

Construction of a block-tridiagonal operator related to BC-Gaudin magnet

Abstract

Free fermions on Hamming graphs $H(d,q)$ are considered and the entanglement entropy for two types of subsystems is computed. For subsets of vertices that form Hamming subgraphs, an analytical expression is obtained. For subsets corresponding to a neighborhood, i.e. to a set of sites at a fixed distance from a reference vertex, a decomposition in irreducible submodules of the Terwilliger algebra of $H(d,q)$ also yields a closed formula for the entanglement entropy. Finally, for subsystems made out of multiple neighborhoods, it is shown how to construct a block-tridiagonal operator which commutes with the entanglement Hamiltonian. It is identified as a BC-Gaudin magnet Hamiltonian in a magnetic field and is diagonalized by the modified algebraic Bethe ansatz.