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.
Highlights
Is in a pure state, the bipartite entanglement of a subsystem A may be quantified by the Rényi entanglement entropies [1,2,3,4]
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
Summary
A primary focus of this work is on the composite branch-point twist field and their FFs in free models, before discussing these objects, it is natural to review the standard branch-point twist fields and the corresponding FFs first. [46], we introduce the bootstrap equations for branch-point twist field and comment on their solution. Since it takes little effort and helps keep connection with interacting IQFTs, we keep the following discussion more general, which describes the case of interacting theories with diagonal but non-trivial scattering as well. The asymptotic states are spanned by multi-particle excitations whose dispersion relation can be parametrised as (E, p) = (mβi cosh θ, mβi sinh θ), where βi indicates the particle species and θ is the rapidity of the particle In the absence of bound states, the two-particle form factors (which are the most relevant FFs) for the branch-point twist field, satisfying the kinematic poles axioms, have been determined as [46].
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