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arXiv:2212.09805 · SciPost Physics Proceedings

Computation of entanglement entropy in inhomogeneous free fermions chains by algebraic Bethe ansatz

Pierre-Antoine Bernard, Gauvain Carcone, Nicolas Crampé, Luc Vinet

Published Nov 23, 2023• 4 citations
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Abstract

The computation of the entanglement entropy for inhomogeneous free fermions chains based on q-Racah polynomials is considered. The eigenvalues of the truncated correlation matrix are obtained from the diagonalization of the associated Heun operator via the algebraic Bethe Ansatz. In the special case of chains based on dual q-Hahn polynomials, the eigenvectors and eigenvalues are expressed in terms of symmetric polynomials evaluated on the Bethe roots.

DOI
10.21468/scipostphysproc.14.018

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The computation of the entanglement entropy for inhomogeneous free fermions chains based on q-Racah polynomials is considered. The eigenvalues of the truncated correlation matrix are obtained from the diagonalization of the associated Heun operator via the algebraic Bethe Ansatz. In the special case of chains based on dual q-Hahn polynomials, the eigenvectors and eigenvalues are expressed in terms of symmetric polynomials evaluated on the Bethe roots.
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We introduce an inhomogeneous model of free fermions on a (D−1)-dimensional lattice with D(D−1)/2 continuous parameters that control the hopping strength between adjacent sites. We solve this model exactly, and find that the eigenfunctions are given by multidimensional generalizations of Krawtchouk polynomials. We construct a Heun operator that commutes with the chopped correlation matrix, and compute the entanglement entropy numerically for D=2,3,4, for a wide range of parameters. For D=2, we observe oscillations in the sub-leading contribution to the entanglement entropy, for which we conjecture an exact expression. For D>2, we find logarithmic violations of the area law for the entanglement entropy with nontrivial dependence on the parameters.
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Free Fermions on vertices of distance-regular graphs are considered. Bipartitions are defined by taking as one part all vertices at a given distance from a reference vertex. The ground state is constructed by filling all states below a certain energy. Borrowing concepts from time and band limiting problems, algebraic Heun operators and Terwilliger algebras, it is shown how to obtain, quite generally, a block tridiagonal matrix that commutes with the entanglement Hamiltonian. The case of the Hadamard graphs is studied in detail within that framework and the existence of the commuting matrix is shown to allow for an analytic diagonalization of the restricted two-point correlation matrix and hence for an explicit determination of the entanglement entropy.