We study the relative entanglement entropies of one interval between excited states of a 1+1 dimensional conformal field theory (CFT). To compute the relative entropy S(ρ 1∥ρ 0) between two given reduced density matrices ρ 1 and ρ 0 of a quantum field theory, we employ the replica trick which relies on the path integral representation of Tr(ρ 1 ρ 0 − 1 ) and define a set of Rényi relative entropies S n (ρ 1∥ρ 0). We compute these quantities for integer values of the parameter n and derive via the replica limit the relative entropy between excited states generated by primary fields of a free massless bosonic field. In particular, we provide the relative entanglement entropy of the state described by the primary operator i∂ϕ, both with respect to the ground state and to the state generated by chiral vertex operators. These predictions are tested against exact numerical calculations in the XX spin-chain finding perfect agreement.
Abstract We develop a systematic approach to compute the subsystem trace distances and relative entropies for subsystem reduced density matrices associated to excited states in different symmetry sectors of a 1+1 dimensional conformal field theory having an internal U(1) symmetry. We provide analytic expressions for the charged moments corresponding to the resolution of both relative entropies and distances for general integer n. For the relative entropies, these formulas are manageable and the analytic continuation to n = 1 can be worked out in most of the cases. Conversely, for the distances the corresponding charged moments become soon untreatable as n increases. A remarkable result is that relative entropies and distances are the same for all symmetry sectors, i.e. they satisfy entanglement equipartition, like the entropies. Moreover, we exploit the OPE expansion of composite twist fields, to provide very general results when the subsystem is a single interval much smaller than the total system. We focus on the massless compact boson and our results are tested against exact numerical calculations in the XX spin chain.
In this paper, we consider the Rényi entanglement asymmetry of excited states in the 1+1 dimensional free compact boson conformal field theory (CFT) at equilibrium. We obtain a universal CFT expression written by correlation functions for the charged moments via the replica trick. We provide detailed analytic computations of the second Rényi entanglement asymmetry in the free compact boson CFT for excited states
Ψ=Vβ+V−β and
Φ=Vβ+J with
Vβ and
J=i∂ϕ being the vertex operator and current operator respectively. We make numerical tests of the universal CFT computations using the XX spin chain model. Taking the non-Hermite fake RDMs into consideration, we propose an effective way to test them numerically, which can be applied to other excited states. The CFT predictions are in perfect agreement with the exact numerical calculations.
In this paper, we consider the Rényi entanglement asymmetry of excited states in the ( 1 + 1 ) dimensional free compact boson conformal field theory (CFT) at equilibrium. We obtain a universal CFT expression written by correlation functions for the charged moments via the replica trick. We provide detailed analytic computations of the second Rényi entanglement asymmetry in the free compact boson CFT for excited states Ψ = V β + V − β and Φ = V β + J with V β and J = i ∂ ϕ being the vertex operator and current operator, respectively. We make numerical tests of the universal CFT computations using the XX spin chain model. Taking the non-Hermite fake reduced density matrices into consideration, we propose an effective way to test them numerically, which can be applied to other excited states. The CFT predictions are in perfect agreement with the exact numerical calculations. Published by the American Physical Society 2024
We study the symmetry resolution of the entanglement entropy of an interval in two-dimensional conformal field theories (CFTs), by relating the bipartition to the geometry of an annulus with conformal boundary conditions. In the presence of extended symmetries such as Kac-Moody type current algebrae, symmetry resolution is possible only if the boundary conditions on the annulus preserve part of the symmetry group, i.e. if the factorization map associated with the spatial bipartition is compatible with the symmetry in question. The partition function of the boundary CFT (BCFT) is then decomposed in terms of the characters of the irreducible representations of the symmetry group preserved by the boundary conditions. We demonstrate that this decomposition already provides the symmetry resolution of the entanglement spectrum of the corresponding bipartition. Considering the various terms of the partition function associated with the same representation, or charge sector, the symmetry-resolved Rényi entropies can be derived to all orders in the UV cutoff expansion without the need to compute the charged moments. We apply this idea to the theory of a free massless boson with U(1) U ( 1 ) , \mathbb{R} ℝ and \mathbb{Z}_2 ℤ 2 symmetry.
Abstract The symmetry-resolved Rényi entanglement entropy is the Rényi entanglement entropy of each symmetry sector of a density matrix
ρ 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,
ρA 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
ρA ρA ; we derive
tr(eiαQAρAn) = aeiα⟨QA⟩(σL)−x(1+O(L−μ)) tr(eiαQAρAn)=aeiα⟨QA⟩(σL)-x(1+O(L-μ)) , wherea, xand
μ μ are universal and
σ σ depends only on theNFermi momenta. We show that the exponentxcorresponds to the expectation from CFT analysis. The error term
O(L−μ) O(L-μ) is consistent with but weaker than the field theory prediction
O(L−2μ) O(L-2μ) . However, using further results and conjectures for the relevant Toeplitz determinant, we find excellent agreement with the expansion over CFT operators.