Symmetry-Resolved Entanglement Entropy in Critical Free-Fermion Chains

Abstract

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.