Symmetry-Resolved Entanglement Entropy in Critical Free-Fermion Chains

Abstract

AbstractThe 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$$\mu $$μare universal and$$\sigma $$σdepends 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.