Symmetry-resolved entanglement entropy in critical free-fermion chains

TL;DR

This paper rigorously derives the scaling properties of symmetry-resolved entanglement entropy in critical free-fermion chains, connecting lattice calculations with conformal field theory predictions.

Contribution

It provides a lattice-based derivation of the asymptotic behavior of symmetry-resolved entanglement entropy in critical free-fermion chains, confirming CFT predictions.

Findings

Universal scaling exponents match CFT predictions

Derived explicit asymptotic expansion for charged moments

Confirmed the universality of the leading terms in entropy scaling

Abstract

The symmetry-resolved R\'enyi entanglement entropy is the R\'enyi entanglement entropy of each symmetry sector of a density matrix $\rho$. 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 by $N$ massless Dirac fermions. For the density matrix, $\rho_A$, of subsystems of $L$ neighbouring sites we calculate the leading terms in the large $L$ asymptotic expansion of the symmetry-resolved R\'enyi entanglement entropies. This follows from a large $L$ expansion of the charged moments of $\rho_A$; we derive $tr(e^{i \alpha Q_A} \rho_A^n) = a e^{i \alpha \langle Q_A\rangle} (\sigma L)^{-x}(1+O(L^{-\mu}))$, where $a, x$ and $\mu$ are universal and $\sigma$ depends only on the $N$ Fermi momenta. We show that the exponent $x$ corresponds to the expectation from CFT analysis. The error term $O(L^{-\mu})$ is consistent with but weaker than the field theory prediction $O(L^{-2\mu})$. However, using further results and conjectures for the relevant Toeplitz determinant, we find excellent agreement with the expansion over CFT operators.