The reduced density matrix of many-body systems possessing an additive conserved quantity can be decomposed in orthogonal sectors which can be independently analyzed. Recently, these have been proven to equally contribute to entanglement entropy for one-dimensional conformal and integrable systems. In this paper, we extend this equipartition theorem to the disordered critical systems by studying the random singlet phase. We analytically compute the disorder averaged symmetry resolved Renyi entropies and show the leading orders are independent of the symmetry sector. Our findings are cross-checked with simulations within the numerical strong disorder renormalization group. We also identify the first subleading term breaking equipartition which is of the form s2/ln l where s is the magnetization of a subsystem of length l.

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Entanglement equipartition in critical random spin chains

The time evolution of the entanglement entropy is a key concept to understand the structure of a nonequilibrium quantum state. In a large class of models, such evolution can be understood in terms of a semiclassical picture of moving quasiparticles spreading the entanglement throughout the system. However, it is not yet known how the entanglement splits between the sectors of an internal local symmetry of a quantum many-body system. Here, guided by the examples of conformal field theories and free-fermion chains, we show that the quasiparticle picture can be adapted to this goal, leading to a general conjecture for the charged entropies whose Fourier transform gives the desired symmetry-resolved entanglement ${S}_{n}(q)$. We point out two physically relevant effects that should be easily observed in atomic experiments: a delay time for the onset of ${S}_{n}(q)$ which grows linearly with $|\mathrm{\ensuremath{\Delta}}q|$ (the difference between the charge $q$ and its mean value) and an effective equipartition when $|\mathrm{\ensuremath{\Delta}}q|$ is much smaller than the subsystem size.

Quasiparticle dynamics of symmetry-resolved entanglement after a quench: Examples of conformal field theories and free fermions

We present a thorough analysis of the entanglement entropies related to different symmetry sectors of free quantum field theories (QFT) with an internal U(1) symmetry. We provide explicit analytic computations for the charged moments of Dirac and complex scalar fields in two spacetime dimensions, both in the massive and massless cases, using two different approaches. The first one is based on the replica trick, the computation of the partition function on Riemann surfaces with the insertion of a flux $\alpha$, and the introduction of properly modified twist fields, whose two-point function directly gives the scaling limit of the charged moments. With the second method, the diagonalisation in replica space maps the problem to the computation of a partition function on a cut plane, that can be written exactly in terms of the solutions of non-linear differential equations of the Painleve V type. Within this approach, we also derive an asymptotic expansion for the short and long distance behaviour of the charged moments. Finally, the Fourier transform provides the desired symmetry resolved entropies: at the leading order, they satisfy entanglement equipartition and we identify the subleading terms that break it. Our analytical findings are tested against exact numerical calculations in lattice models.

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Entanglement and symmetry resolution in two dimensional free quantum field theories

We consider symmetry-resolved entanglement entropy in AdS3/CFT2 coupled to U(1) Chern-Simons theory. We identify the holographic dual of the charged moments in the two-dimensional conformal field theory as a charged Wilson line in the bulk of AdS3, namely the Ryu-Takayanagi geodesic minimally coupled to the U(1) Chern-Simons gauge field. We identify the holonomy around the Wilson line as the Aharonov-Bohm phases which, in the two-dimensional field theory, are generated by charged U(1) vertex operators inserted at the endpoints of the entangling interval. Furthermore, we devise a new method to calculate the symmetry resolved entanglement entropy by relating the generating function for the charged moments to the amount of charge in the entangling subregion. We calculate the subregion charge from the U(1) Chern-Simons gauge field sourced by the bulk Wilson line. We use our method to derive the symmetry-resolved entanglement entropy for Poincare patch and global AdS3, as well as for the conical defect geometries. In all three cases, the symmetry resolved entanglement entropy is determined by the length of the Ryu-Takayanagi geodesic and the Chern-Simons level k, and fulfills equipartition of entanglement. The asymptotic symmetry algebra of the bulk theory is of $$ \hat{\mathfrak{u}}{(1)}_k $$
 Kac-Moody type. Employing the $$ \hat{\mathfrak{u}}{(1)}_k $$
 Kac-Moody symmetry, we confirm our holographic results by a calculation in the dual conformal field theory.

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Symmetry-resolved entanglement in AdS3/CFT2 coupled to U(1) Chern-Simons theory

We report on the calculation of the symmetry resolved entanglement entropies in two-dimensional many-body systems of free bosons and fermions by \emph{dimensional reduction}. When the subsystem is translational invariant in a transverse direction, this strategy allows us to reduce the initial two-dimensional problem into decoupled one-dimensional ones in a mixed space-momentum representation. While the idea straightforwardly applies to any dimension $d$, here we focus on the case $d=2$ and derive explicit expressions for two lattice models possessing a $U(1)$ symmetry, i.e., free non-relativistic massless fermions and free complex (massive and massless) bosons. Although our focus is on symmetry resolved entropies, some results for the total entanglement are also new. Our derivation gives a transparent understanding of the well known different behaviours between massless bosons and fermions in $d\geq2$: massless fermions presents logarithmic violation of the area which instead strictly hold for bosons, even massless. This is true both for the total and the symmetry resolved entropies. Interestingly, we find that the equipartition of entanglement into different symmetry sectors holds also in two dimensions at leading order in subsystem size; we identify for both systems the first term breaking it. All our findings are quantitatively tested against exact numerical calculations in lattice models for both bosons and fermions.

Symmetry resolved entanglement in two-dimensional systems via dimensional reduction

We report a throughout analysis of the entanglement entropies related to different symmetry sectors in the low-lying primary excited states of a conformal field theory (CFT) with an internal U(1) symmetry. Our findings extend recent results for the ground state. We derive a general expression for the charged moments, i.e. the generalised cumulant generating function, which can be written in terms of correlation functions of the operator that define the state through the CFT operator-state correspondence. We provide explicit analytic computations for the compact boson CFT (aka Luttinger liquid) for the vertex and derivative excitations. The Fourier transform of the charged moments gives the desired symmetry resolved entropies. At the leading order, they satisfy entanglement equipartition, as in the ground state, but we find, within CFT, subleading terms that break it. Our analytical findings are checked against free fermions calculations on a lattice, finding excellent agreement. As a byproduct, we have exact results for the full counting statistics of the U(1) charge in the considered excited states.

Symmetry resolved entanglement entropy of excited states in a CFT

We generalise the form factor bootstrap approach to integrable field theories with U(1) symmetry to derive matrix elements of composite branch-point twist fields associated with symmetry resolved entanglement entropies. The bootstrap equations are solved for the free massive Dirac and complex boson theories, which are the simplest theories with U(1) symmetry. We present the exact and complete solution for the bootstrap, including vacuum expectation values and form factors involving any type and arbitrarily number of particles. The non-trivial solutions are carefully cross-checked by performing various limits and by the application of the ∆-theorem. An alternative and compact determination of the novel form factors is also presented. Based on the form factors of the U(1) composite branch-point twist fields, we re-derive earlier results showing entanglement equipartition for an interval in the ground state of the two models.

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U(1) symmetry resolved entanglement in free 1+1 dimensional field theories via form factor bootstrap

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 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.

Symmetry resolved relative entropies and distances in conformal field theory

https://link.springer.com/content/pdf/10.1007/JHEP12(2022)127.pdf

Luca Capizzi

Papers

Symmetry resolved entanglement entropy of excited states in a CFT

U(1) symmetry resolved entanglement in free 1+1 dimensional field theories via form factor bootstrap

Symmetry resolved relative entropies and distances in conformal field theory

Symmetry resolved entanglement entropy of excited states in a CFT

Symmetry resolved entanglement of excited states in quantum field theory. Part I. Free theories, twist fields and qubits