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

Abstract

The quantum sine-Gordon model is the simplest massive interacting integrable quantum field theory whose two-particle scattering matrix is generally non-diagonal. As such, it is a model that has been extensively studied, especially in the context of the bootstrap program. In this paper we compute low particle-number form factors of a special local field known as the branch point twist field, whose correlation functions are building blocks for measures of entanglement. We consider the attractive regime where the theory possesses a particle spectrum consisting of a soliton, an antisoliton (of opposite U(1) charges) and several (neutral) breathers. In the breather sector we exploit the fusion procedure to compute form factors of heavier breathers from those of lighter ones. We apply our results to the study of the entanglement dynamics after a small mass quench and for short times. We show that in the presence of two or more breathers the von Neumann and Rényi entropies display undamped oscillations in time, whose frequencies are proportional to the even breather masses and whose amplitudes are proportional to the breather's one-particle form factor.

Branch Point Twist Field Form Factors in the sine-Gordon Model II: Composite Twist Fields and Symmetry Resolved Entanglement

“…In the context of the bootstrap program for IQFTs, the matrix elements of local operators, known as form factors (FFs) of the sG model have been extensively studied by many authors employing many different techniques and in various contexts [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Particularly interesting operators are the branch point twist fields (BPTFs), whose form factors (FFs) in for the sG model were derived in [21,22]. The multi-point correlation functions of these fields, which are in principle calculable via their FFs, are directly related to entanglement entropies.…”

“…As our title suggests, the aim of the present work is to further develop the program initiated in [22], extending it to CTFs with the aim of studying symmetry resolved quantities. In the sG model we will focus on the U(1) symmetry associated with the the topological charge carried by the soliton/antisoliton.…”

“…Accounting for this symmetry, we can write CTF form factor equations, as originally proposed in [76]. Concerning the solution of these equations, we focus as in [22], on the attractive regime of the sG model, where breathers are present in the spectrum. Indeed, in the present work we find solutions to the form factor bootstrap equations only for the breather sector.…”

“…This choice might seem limiting but is justified by the fact that, unlike for the standard BPTF studied in [22] where the first breather's one-particle form factor is zero by symmetry, in the present case the leading contribution of the charge moments is determined by the first breather. The solution of the bootstrap equations can actually be obtained by analytic continuation in the coupling from the sinh-Gordon (shG) model solution for an analogous field, whose validity can be easily checked using the ∆-sum rule [103].…”

Post-Quantum Quench Growth of Renyi Entropies in Low Dimensional Continuum Bosonic Systems

“…where |Ψ i are the states just discussed of the unperturbed bosonic theory. Our focus on using the states of the unperturbed Luttinger liquids to describe entanglement dynamics differs from the form factor bootstrap approach where the emphasis is on the basis of gapped states of the sine-Gordon model [30,31]. The corresponding density matrix of the system is…”