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arXiv:2310.14186 · Journal of High Energy Physics

Multi-charged moments and symmetry-resolved Rényi entropy of free compact boson for multiple disjoint intervals

Himanshu Gaur, Urjit A. Yajnik

Published Jan 9, 2024• 8 citations
arXivHTMLPDF

Abstract

A bstract We study multi-charged moments and symmetry-resolved Rényi entropy of free compact boson for multiple disjoint intervals. The Rényi entropy evaluation involves computing the partition function of the theory on Riemann surfaces with genus g > 1. This makes Rényi entropy sensitive to the local conformal algebra of the theory. The free compact boson possesses a global U(1) symmetry with respect to which we resolve Rényi entropy. The multi-charged moments are obtained by studying the correlation function of flux-generating vertex operators on the associated Riemann surface. Symmetry-resolved Rényi entropy is then obtained from the Fourier transforms of the charged moments. Rényi entropy is shown to have the familiar equipartition into local charge sectors upto the leading order. The multi-charged moments are also essential in studying the symmetry resolution of mutual information. The multi-charged moments of the self-dual compact boson and massless Dirac fermion are also shown to match for the cases when the associated reduced density moments are known to be the same. Finally, we numerically check our results against the tight-binding model.

DOI
10.1007/jhep01(2024)042

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A bstract We study multi-charged moments and symmetry-resolved Rényi entropy of free compact boson for multiple disjoint intervals. The Rényi entropy evaluation involves computing the partition function of the theory on Riemann surfaces with genus g > 1. This makes Rényi entropy sensitive to the local conformal algebra of the theory. The free compact boson possesses a global U(1) symmetry with respect to which we resolve Rényi entropy. The multi-charged moments are obtained by studying the correlation function of flux-generating vertex operators on the associated Riemann surface. Symmetry-resolved Rényi entropy is then obtained from the Fourier transforms of the charged moments. Rényi entropy is shown to have the familiar equipartition into local charge sectors upto the leading order. The multi-charged moments are also essential in studying the symmetry resolution of mutual information. The multi-charged moments of the self-dual compact boson and massless Dirac fermion are also shown to match for the cases when the associated reduced density moments are known to be the same. Finally, we numerically check our results against the tight-binding model.
Abstract In this paper, we study the symmetry decomposition of Rényi negativity into charge imbalance sectors for the 1+1 dimensional free compact boson field with a global U(1) symmetry in the ground state for the case of two disjoint intervals. We obtain multicharged and charged Rényi negativity moments by computing the four-point correlator of flux-generating vertex operators on the Riemann surface. We then obtain charge imbalance resolved Rényi negativity by taking the Fourier transform of the charged moments. Finally, we match our results against the tight-binding model as a numerical check.
Abstract We study the multi-charged moments for two disjoint intervals in the ground state of two 1 + 1 dimensional CFTs with central charge c = 1 and global U(1) symmetry: the massless Dirac field theory and the compact boson (Luttinger liquid). For this purpose, we compute the partition function on the higher genus Riemann surface arising from the replica method in the presence of background magnetic fluxes between the sheets of the surface. We consider the general situation in which the fluxes generate different twisted boundary conditions at each branch point. The obtained multi-charged moments allow us to derive the symmetry resolution of the Rényi entanglement entropies and the mutual information for non complementary bipartitions. We check our findings against exact numerical results for the tight-binding model, which is a lattice realisation of the massless Dirac theory.
We investigate the second Rényi entropy of two intervals in the massless Thirring model describing a self-interacting Dirac fermion in two dimensions. Boson-fermion duality relating this model to a free compact boson theory enables us to simplify the calculation of the second Rényi entropy, reducing it to the evaluation of the partition functions of the bosonic theory on a torus. We derive exact results on the second Rényi entropy, and examine the dependence on the sizes of the intervals and the coupling constant of the model both analytically and numerically. We also explore the mutual Rényi information, a measure quantifying the correlation between the two intervals, and find that it generally increases as the coupling constant of the Thirring model becomes larger.
Abstract In this paper, we consider the computation of charged moments of the reduced density matrix of two disjoint intervals in the 1+1 dimensional free compactified boson conformal field theory (CFT) by studying the four-point function of the fluxed twist fields. We obtained the exact scaling function of this four-point function and discussed its decompactification limit. This scaling function was used to obtain the charged moments of the partial transpose which we refer as charged Rényi negativity. These charged moments and the charged moments of the partial transpose are essential for the problem of symmetry decomposition of the corresponding entanglement measures. We test our analytic formula against exact numerical computation in the complex harmonic chain, finding perfect agreements.
We investigate the second R\'enyi entropy of two intervals in the massless Thirring model describing a self-interacting Dirac fermion in two dimensions. Boson-fermion duality relating this model to a free compact boson theory enables us to simplify the calculation of the second R\'enyi entropy, reducing it to the evaluation of the partition functions of the bosonic theory on a torus. We derive exact results on the second R\'enyi entropy, and examine the dependence on the sizes of the intervals and the coupling constant of the model both analytically and numerically. We also explore the mutual R\'enyi information, a measure quantifying the correlation between the two intervals, and find that it generally increases as the coupling constant of the Thirring model becomes larger.
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.