Symmetry resolved entanglement in integrable field theories via form factor bootstrap

Abstract

We consider the form factor bootstrap approach of integrable field theories to derive matrix elements of composite branch-point twist fields associated with symmetry resolved entanglement entropies. The bootstrap equations are determined in an intuitive way and their solution is presented for the massive Ising field theory and for the genuinely interacting sinh-Gordon model, both possessing a ℤ2 symmetry. The solutions are carefully cross-checked by performing various limits and by the application of the ∆-theorem. The issue of symmetry resolution for discrete symmetries is also discussed. We show that entanglement equipartition is generically expected and we identify the first subleading term (in the UV cutoff) breaking it. We also present the complete computation of the symmetry resolved von Neumann entropy for an interval in the ground state of the paramagnetic phase of the Ising model. In particular, we compute the universal functions entering in the charged and symmetry resolved entanglement.

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.…”

“…Notice that each new representation of spθ q from ( 32), corresponds to a new breather with an even index entering the spectrum of the theory. Thus, when writing down the minimal part of F ss pθ ; ξ, nq we have two alternative representations: if we employ the S-matrix representation (32) together with (5) and apply the standard machinery (28) to obtain the minimal form factor, the result possesses no breather bound state poles. This feature, is generally what is meant by "minimal solution".…”

“…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.…”

Finite-size corrections in critical symmetry-resolved entanglement

“…over a large range of values of the local charge, the symmetry-resolved entropies are independent of the charge. The study of symmetry-resolved entanglement entropies has been applied to a variety of situations [6][7][8][9][10][11][12][13][14][15][16][17]. For quantum computing, these entropies also serve as building blocks for the definition of operationally accessible entanglement [18,19].…”