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Symmetry-resolved entanglement entropy in critical free-fermion chains

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