Symmetry-Resolved Entanglement Entropy in Critical Free-Fermion Chains

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 by N massless Dirac fermions. For the density matrix, $$\rho _A$$ 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 $$\rho _A$$ ρ A ; 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 ( e i α Q A ρ A n ) = a e i α ⟨ Q A ⟩ ( σ L ) - x ( 1 + O ( L - μ ) ) , where a, x and $$\mu $$ μ are universal and $$\sigma $$ σ depends only on the N Fermi momenta. We show that the exponent x corresponds to the expectation from CFT analysis. The error term $$O(L^{-\mu })$$ O ( L - μ ) is consistent with but weaker than the field theory prediction $$O(L^{-2\mu })$$ O ( L - 2 μ ) . However, using further results and conjectures for the relevant Toeplitz determinant, we find excellent agreement with the expansion over CFT operators.

Multi-charged moments of two intervals in conformal field theory

“…Also at more practical level, it can help to speed-up the numerical algorithms to simulate quantum many-body systems [40]. All that has been the breeding ground for a plethora of works that analyse the resolution of entanglement from different perspectives: spin chains [38,[43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59], integrable quantum field theories [60][61][62][63][64], CFTs [39,40,56,[65][66][67][68][69][70][71][72][73][74][75][76], holography [77][78][79][80], out-ofequilibrium [81][82][83][84][85][86] and disordered...…”

Exact hydrodynamic description of symmetry-resolved Rényi entropies after a quantum quench

“…1 + 1 conformal field theories [28][29][30][38][39][40][41][42][43][44][45][46][47], free [48,49] and interacting integrable quantum field theories [50][51][52] and also holographic settings [53][54][55]. These studies are generically carried out in lattice models [28,[38][39][40][56][57][58][59][60][61][62][63][64][65][66][67], but other systems exhibiting more exotic types of dynamics have been also considered [69][70][71][72][73][74][75]. Moreover, the interest for symmetry-resolved entanglement measures on the theoretical side is accompanied and consolidated by their experimental feasibility, see e.g.…”

Symmetry-resolved entanglement in fermionic systems with dissipation

“…The question of how the different symmetry sectors contribute to the entanglement entropies has attracted a lot of attention, theoretically and experimentally [14,[20][21][22]. On the theoretical side, several works focus on the symmetry-resolved entropies, especially in field theory [18,19, and spin chains, both at equilibrium [46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65], and in out-of-equilibrium situations [67][68][69][70][71][72][73][74]. For mixed states, it was proven that, whenever there is a conserved extensive charge, the negativity admits a resolution in terms of the charge imbalance between the two subsystems (see section 4.2 for a rigorous definition) [75].…”