International Journal of Geometric Methods in Modern Physics

Rényi entanglement entropy in fermionic chains

Filiberto Ares, José G. Esteve, Fernando Falceto

Published Mar 3, 2015• 4 citations

Abstract

In this paper we discuss the asymptotic behavior of the entanglement entropy for a fragment of a fermionic chain in an excited state. We first study the case in which the subsystem consists of a single interval. We review the analytical tools that make the computation feasible, derived from the asymptotics of the determinant of Toeplitz matrices, we also compare with the numerical results and give a physical interpretation of them. Next we move to the case in which the fragment is made of several disjoint intervals. We compute numerically the entropy and, based on these results, we establish a conjecture on the determinant of a submatrix of a Toeplitz matrix that leads to a general expression for the entropy in the multi-interval case. We check that our results agree with those derived from conformal field theory when the latter are available.

DOI: 10.1142/s0219887815600026

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International Journal of Geometric Methods in Modern Physics

Rényi entanglement entropy in fermionic chains

Filiberto Ares, José G. Esteve, Fernando Falceto

Published Mar 3, 2015• 4 citations

Publisher

Abstract

DOI: 10.1142/s0219887815600026

Related papers

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Abstract The symmetry-resolved Rényi entanglement entropy is the Rényi entanglement entropy of each symmetry sector of a density matrix

\rho

ρ . This experimentally relevant quantity is known to have rich theoretical connections to conformal field theory (CFT). For a family of critical free-fermion chains, we present a rigorous lattice-based derivation of its scaling properties using the theory of Toeplitz determinants. We consider a class of critical quantum chains with a microscopic U(1) symmetry; each chain has a low energy description given byNmassless Dirac fermions. For the density matrix,

\rho _A

ρA , of subsystems ofLneighbouring sites we calculate the leading terms in the largeLasymptotic expansion of the symmetry-resolved Rényi entanglement entropies. This follows from a largeLexpansion of the charged moments of

\rho _A

ρA ; we derive

\mathrm {tr}(\mathrm {e}^{\mathrm {i}\alpha Q_A} \rho _A^n)~=~a \mathrm {e}^{\mathrm {i}\alpha \langle Q_A\rangle } (\sigma L)^{-x}(1+O(L^{-\mu }))

tr(eiαQAρAn)=aeiα⟨QA⟩(σL)-x(1+O(L-μ)) , wherea, xand

\mu

μ are universal and

\sigma

σ depends only on theNFermi momenta. We show that the exponentxcorresponds to the expectation from CFT analysis. The error term

O(L^{-\mu })

O(L-μ) is consistent with but weaker than the field theory prediction

O(L^{-2\mu })

O(L-2μ) . However, using further results and conjectures for the relevant Toeplitz determinant, we find excellent agreement with the expansion over CFT operators.