Form factors of branch-point twist fields in quantum integrable models and entanglement entropy

TL;DR

This work introduces branch-point twist fields in replica-described IQFTs to compute entanglement entropy in 1+1 dimensions. Using a two-particle form-factor expansion in the $n$-copy theory, it reveals a universal infrared correction to $S_A$ that depends only on the particle spectrum, not on the detailed $S$-matrix. The authors explicitly solve the twist-field form-factor equations for Ising and sinh-Gordon models, verify UV dimensions via the $oldsymbol{ riangle}$-sum rule, and derive the IR entropy correction $-rac{1}{8}K_0(2rm)$ with model-dependent constants captured by two-particle data. They generalize to theories with multiple particle types and outline extensions to bound states and non-diagonal theories, highlighting the potential for extracting spectral information from entanglement data in lattice models. The results establish a robust, universal link between entanglement and the mass spectrum in integrable quantum field theories.

Abstract

In this paper we compute the leading correction to the bipartite entanglement entropy at large sub-system size, in integrable quantum field theories with diagonal scattering matrices. We find a remarkably universal result, depending only on the particle spectrum of the theory and not on the details of the scattering matrix. We employ the "replica trick" whereby the entropy is obtained as the derivative with respect to n of the trace of the n-th power of the reduced density matrix of the sub-system, evaluated at n=1. The main novelty of our work is the introduction of a particular type of twist fields in quantum field theory that are naturally related to branch points in an n-sheeted Riemann surface. Their two-point function directly gives the scaling limit of the trace of the n-th power of the reduced density matrix. Taking advantage of integrability, we use the expansion of this two-point function in terms of form factors of the twist fields, in order to evaluate it at large distances in the two-particle approximation. Although this is a well-known technique, the new geometry of the problem implies a modification of the form factor equations satisfied by standard local fields of integrable quantum field theory. We derive the new form factor equations and provide solutions, which we specialize both to the Ising and sinh-Gordon models.

Figures (9)

• Figure 1: [Color online] A representation of the Riemann surface ${\cal M}_{3,a_1,a_2}$.
• Figure 2: [Color online] The effect of ${\cal T}$ on other local fields.
• Figure 3: [Color online] A pictorial representation of the effect of adding $2{ \if@compatibility \mathchar"0119 {} \mathchar"0119 } i$ to rapidity ${ \if@compatibility \mathchar"0112 {} \mathchar"0112 }_1$ in form factors of the twist field ${\cal T}$.
• Figure 4: [Color online] The kinematic poles come from the structure of the wave function far from the local fields, at positive and negative infinity. Adding $i{ \if@compatibility \mathchar"0119 {} \mathchar"0119 }$ to rapidity ${ \if@compatibility \mathchar"0112 {} \mathchar"0112 }_1$ puts the particle in the "out" region. With a particle in that region, there are delta-functions representing particles in the "in" region going through without interacting with the local fields. Those occur from the $e^{ipx}$ form of the wave function at positive and negative infinity. But if the coefficients at both limits are different, $S_- e^{ipx}$ and $S_+e^{ipx}$ with $S_-\neq S_+$, then there are also poles in addition to these delta-functions. Only these poles are seen in the analytic continuation ${ \if@compatibility \mathchar"0112 {} \mathchar"0112 }_1\mapsto{ \if@compatibility \mathchar"0112 {} \mathchar"0112 }_1+i{ \if@compatibility \mathchar"0119 {} \mathchar"0119 }$. Different coefficients come from the semi-locality of the twist field and the non-free scattering matrix, as represented here.
• Figure 5: [Color online] The structure of the function $F_2^{{\cal T}|11}({ \if@compatibility \mathchar"0112 {} \mathchar"0112 })$ in the extended sheet ${\rm Im}({ \if@compatibility \mathchar"0112 {} \mathchar"0112 })\in[0,2{ \if@compatibility \mathchar"0119 {} \mathchar"0119 } n]$, in the case $n=3$. Crosses indicate the positions of the kinematic singularities. Shaded regions represent the physical sheets of the form factors $F_2^{{\cal T}|11}({ \if@compatibility \mathchar"0112 {} \mathchar"0112 })$, $F_2^{{\cal T}|12}({ \if@compatibility \mathchar"0112 {} \mathchar"0112 })$ and $F_2^{{\cal T}|13}({ \if@compatibility \mathchar"0112 {} \mathchar"0112 })$.