Entanglement of Free Fermions on Hamming Graphs

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.

Time and band limiting operator and Bethe ansatz

“…In many cases, including the original sinc kernel and its discrete counterpart, the commuting operator can be identified as an algebraic Heun operator [29,30]. It has been also used for the computation of the entanglement entropy of free Fermions on different chains [20,21,9] and on graphs associated to various association schemes [19,10,11]. These results provide an algebraic explanation of the existence of a tridiagonal matrix commuting with Q ± which has been found by direct computations [25].…”

“…Their spectrum therefore contains the necessary information to compute von Neumann entanglement entropies. Indeed, the methods used in [19,10,11] associated to different graphs may be used in the context of the 2n-gon.…”

Entanglement of Free Fermions on Johnson Graphs

“…They were used for models of free fermions hopping on chains [9,10] or on the vertices of distance-regular graphs. In the latter case, the Hadamard [8] and the Hamming graphs [4,18,19] were specifically studied and in some instances analytical expressions for the entanglement entropy and thermodynamic limits were obtained. Bethe ansatz techniques were also shown to be useful to study such problems [3].…”

“…Translated in algebraic terms, this statement implies that T can be embedded in two copies of the Terwilliger algebra of the hypercube. Since the decomposition in irreducible modules is known for the latter [4,14], it will also yield the decomposition of T . This perspective has the advantage of being related to the coupling of two su(2) representations and establishes a relation between the Terwilliger algebra of the Johnson scheme and the Hahn algebra h [16].…”

The Terwilliger algebra of symplectic dual polar graphs, the subspace lattices and $U_q(sl_2)$

“…In recent years, much efforts have been dedicated to the decomposition of the standard module of distance-regular graphs in irreducible submodules of their Terwilliger algebra. The Hamming [2,9,10,16] and Johnson [3,7,15,18,22] cases have been worked out in great details. Distance-regular graphs associated to certain q-polynomials of the Askey-scheme have also received some attention.…”