Entanglement of Free Fermions on Johnson Graphs

Abstract

Free fermions on Johnson graphs J(n, k) are considered and the entanglement entropy of sets of neighborhoods is computed. For a subsystem composed of a single neighborhood, an analytical expression is provided by the decomposition in irreducible submodules of the Terwilliger algebra of J(n, k) embedded in two copies of su(2). For a subsytem composed of multiple neighborhoods, the construction of a block-tridiagonal operator which commutes with the entanglement Hamiltonian is presented, its usefulness in computing the entropy is stressed and the area law pre-factor is discussed.

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.…”

Bethe ansatz diagonalization of the Heun-Racah operator

“…They also naturally lead to the commuting tridiagonal operator [27] arising in the computation of the entanglement entropy for 1D free Fermion models [23,24] with couplings given by the recurrence coefficients of orthogonal polynomials of the Askey scheme. This has been generalized to free fermions on vertices of distance regular graphs [21,10,11] or to multidimensional networks associated to the multivariate Krawtchouk polynomials [8].…”

The Terwilliger algebra of symplectic dual polar graphs, the subspace lattices and U(sl2)

“…P -polynomial association schemes arise in the description of the neighborhoods of vertices in distance-regular graphs. They play an important role in combinatorics, coding theory [1,4] and have found applications in the study of quantum systems [2,3,5,6]. In the case of schemes that are also Q-polynomial, Leonard's theorem [14] implies that they are related to the hypergeometric orthogonal polynomials of the Askey-scheme [12].…”