Free-Fermion entanglement and orthogonal polynomials

Abstract

We present a simple construction for a tridiagonal matrix $T$ that commutes with the hopping matrix for the entanglement Hamiltonian $\mathcal{H}$ of open finite free-Fermion chains associated with families of discrete orthogonal polynomials. It is based on the notion of algebraic Heun operator attached to bispectral problems, and the parallel between entanglement studies and the theory of time and band limiting. As examples, we consider Fermionic chains related to the Chebychev, Krawtchouk and dual Hahn polynomials. For the former case, which corresponds to a homogeneous chain, the outcome of our construction coincides with a recent result of Eisler and Peschel; the latter cases yield commuting operators for particular inhomogeneous chains. Since $T$ is tridiagonal and non-degenerate, it can be readily diagonalized numerically, which in turn can be used to calculate the spectrum of $\mathcal{H}$, and therefore the entanglement entropy.

17 pages, final version

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Cited by in corpus (32)

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• Free Fermions, vertex Hamiltonians, and lower-dimensional AdS/CFT
• Entanglement in Fermionic Chains and Bispectrality
• Entanglement of inhomogeneous free fermions on hyperplane lattices
• Local and non-local properties of the entanglement Hamiltonian for two disjoint intervals
• Heun operator of Lie type and the modified algebraic Bethe ansatz
• Ising analogues of quantum spin chains with multispin interactions
• Inhomogeneous XX spin chains and quasi-exactly solvable models
• Entanglement of Free Fermions on Johnson Graphs
• Elementary Integral Series for Heun Functions. With an Application to Black-Hole Perturbation Theory
• The Heun-Racah and Heun-Bannai-Ito algebras
• Entanglement entropy of inhomogeneous XX spin chains with algebraic interactions
• Entanglement of Free Fermions on Hadamard Graphs
• On the Bisognano-Wichmann entanglement Hamiltonian of nonrelativistic fermions
• Fermionic logarithmic negativity in the Krawtchouk chain
• Absence of logarithmic enhancement in the entanglement scaling of free fermions on folded cubes
• Exactly solvable inhomogeneous fermion systems
• Bethe ansatz diagonalization of the Heun-Racah operator
• Classical and quantum walks on paths associated with exceptional Krawtchouk polynomials
• The rational Sklyanin algebra and the Wilson and para-Racah polynomials
• Sklyanin-like algebras for ($q$-)linear grids and ($q$-)para-Krawtchouk polynomials
• Computation of entanglement entropy in inhomogeneous free fermions chains by algebraic Bethe ansatz
• Time and band limiting operator and Bethe ansatz
• Entanglement scaling behaviors of free fermions on hyperbolic lattices
• A $q$-version of the relation between the hypercube, the Krawtchouk chain and Dicke states
• Entanglement of Inhomogeneous Free Bosons and Orthogonal Polynomials
• Local fermion density in inhomogeneous free-fermion chains: a discrete WKB approach
• Entanglement of free-fermion systems, signal processing and algebraic combinatorics
• $su(2)$ symmetry of XX spin chains