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arXiv:2407.06427 · Journal of Statistical Mechanics Theory and Experiment

Entanglement asymmetry in the critical XXZ spin chain

Marco Lastres, Sara Murciano, Filiberto Ares, Pasquale Calabrese

Published Jan 1, 2025• 21 citations
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Abstract

Abstract We study the explicit breaking of a SU(2) symmetry to a U(1) subgroup employing the entanglement asymmetry, a recently introduced observable that measures how much symmetries are broken in a part of extended quantum systems. We consider as specific model the critical XXZ spin chain, which breaks the SU(2) symmetry of spin rotations except at the isotropic point, and is described by the massless compact boson in the continuum limit. We examine the U(1) subgroup of SU(2) that is broken outside the isotropic point by applying conformal perturbation theory, which we complement with numerical simulations on the lattice. We also analyse the entanglement asymmetry of the full SU(2) group. By relying on very generic scaling arguments, we derive an asymptotic expression for it.

DOI
10.1088/1742-5468/ada497

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Abstract We study the explicit breaking of a SU(2) symmetry to a U(1) subgroup employing the entanglement asymmetry, a recently introduced observable that measures how much symmetries are broken in a part of extended quantum systems. We consider as specific model the critical XXZ spin chain, which breaks the SU(2) symmetry of spin rotations except at the isotropic point, and is described by the massless compact boson in the continuum limit. We examine the U(1) subgroup of SU(2) that is broken outside the isotropic point by applying conformal perturbation theory, which we complement with numerical simulations on the lattice. We also analyse the entanglement asymmetry of the full SU(2) group. By relying on very generic scaling arguments, we derive an asymptotic expression for it.
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Entanglement asymmetry in the gapped XYZ spin-12\frac1221​ chain
The entanglement asymmetry measures how strongly a symmetry is broken inside a subsystem. Analytic results at equilibrium have so far covered free theories and, perturbatively, the critical XXZ chain. We compute the Rényi entanglement asymmetries of a large interval in the gapped, U(1)U(1)U(1)-breaking phase of the interacting XYZ chain. The calculation combines three ingredients. A charged-moment identity, which we prove for fermionic Gaussian and for injective matrix-product ground states, ties the asymmetry to the static susceptibility of the broken charge. A non-conservation sum rule then evaluates the susceptibility from sine-Gordon form factors, its two-kink and one-breather channels providing a lower bound on the universal amplitude. The Baxter--Johnson--Krinsky--McCoy solution supplies the kink mass for different couplings. Infinite-system density-matrix renormalization group simulations built on these masses reproduce the master formula.
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Abstract Entanglement asymmetry is a novel quantity that, using entanglement methods, measures how much a symmetry is broken in a part of an extended quantum system. So far, it has only been used to characterise the breaking of continuous Abelian symmetries. In this paper, we extend the concept to cyclic Z N groups. As an application, we consider the XY spin chain, in which the ground state spontaneously breaks the Z 2 spin parity symmetry in the ferromagnetic phase. We thoroughly investigate the non-equilibrium dynamics of this symmetry after a global quantum quench, generalising known results for the standard order parameter.
A bstract We examine the behavior of the entanglement asymmetry in the ground state of a (1+1)-dimensional conformal field theory with a boundary condition that explicitly breaks a bulk symmetry. Our focus is on the asymmetry of a subsystem A originating from the symmetry-breaking boundary and extending into a semi-infinite bulk. By employing the twist field formalism, we derive a universal expression for the asymmetry, showing that the asymptotic behavior for large subsystems is approached algebraically, with an exponent which is twice the conformal dimension of a boundary condition-changing operator. As a secondary result, we also establish a similar asymptotic behavior for the string order parameter. Our exact analytical findings are validated through numerical simulations in the critical Ising and 3-state Potts models.