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

Abstract

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.

Branch point twist field form factors in the sine-Gordon model I: Breather fusion and entanglement dynamics

“…As anticipated in the introduction, our immediate goal now is to extend these results to symmetry resolved twist fields [28,29].…”

“…These equations were first given in [27] for diagonal theories and then in [30] for non-diagonal ones. They have been generalized to symmetry resolved branch point twist fields in [28,29]. We will not review all these equations and their properties here but only those relations that are repeateadly used in the current paper, in particular the equations for one-and two-particle form factors.…”

“…The works we have referred to so far are concerned with "standard" local fields of the sine-Gordon theory, such as the sine-Gordon field ϕ, its powers and, especially, exponential fields of the form e iaϕ which are of particular interest as they are related to the trace of the stress-energy tensor. In the present work our main aim is to generalize these results to branch point twist fields, starting with the branch point twist field and associated form factor program introduced in [27], and then continuing (in part II) with the symmetry resolved branch point twist field recently introduced in [28,29]. Twist field form factors of the sine-Gordon model were first studied in [30] but only in the so-called repulsive regime where no breathers are present.…”

Symmetry-resolved entanglement entropy in Wess-Zumino-Witten models

“…The possibility of measuring in an experiment the internal symmetry structure of the entanglement [20][21][22][23] went together with new theoretical frameworks developed to address the same problem [24,25]. These progresses paved the way to study different symmetryresolved contributions in various theoretical contexts such as CFTs [24][25][26][27][28][29][30][31], free [32,34] and interacting integrable quantum field theories [33,35], holographic settings [36,37], spin chains [21,22,[38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54], disordered systems [55][56][57][58] and for non-trivial topological phases [59][60][61].…”

Symmetry decomposition of relative entropies in conformal field theory

“…This operator can be seen as the composition of the branch point twist field T n and the U(1) twist field V µ and we denote it by T n,µ [26,50]. The form factors and vacuum expectation values (VEVs) of the composite twist field in integrable field theories have been obtained in [51,52] recently. If subsystem A is an interval [u, v], then one can identify…”