Form factors of the U(1) composite branch-point twist field in the soliton-antisoliton sector of the sine-Gordon model

In the replica-trick formulation, the symmetry-resolved entanglement of the 1+1-dimensional sine-Gordon (sG) model is controlled by two-point functions of a composite branch-point twist field (CTF): the fusion of the standard branch-point twist field with the U(1) symmetry (vertex) field of sG. Horvath, Calabrese and Castro-Alvaredo (SciPost Phys. 12, 088 (2022)) solved the associated form factor equations only in the U(1)-neutral breather sector of the attractive regime and explicitly left open the charged sector: the CTF form factors on soliton/antisoliton states, whose Watson equations carry nontrivial $e^{i\kappa\alpha/n}$ monodromy phases and whose solutions are expected to involve non-integer powers of the rapidity variables fixed by the U(1) charge and the coupling, defeating standard ansatz methods. Equally open is the vacuum expectation value $\langle\mathcal{T}_n^\alpha\rangle$ that normalizes these form factors and is required for a full non-perturbative evaluation of the charged moments. No work computes sG CTF form factors in the soliton sector or the CTF VEV. This is the missing ingredient for symmetry-resolved entanglement of sG in the repulsive regime, where the soliton-antisoliton doublet is the entire particle spectrum.

Background

The sine-Gordon (sG) model is a 1+1-dimensional integrable quantum field theory with action $\mathcal{A}=\int d^2x\,[\tfrac{1}{2}(\partial\varphi)^2+\mu\cos(g\varphi)]$; it is the simplest massive integrable theory whose two-particle scattering is non-diagonal, i.e. soliton-antisoliton scattering includes backscattering. Its particle spectrum consists of a soliton $s$ and antisoliton $\bar{s}$ of opposite $U(1)$ (topological) charge, plus, in the attractive regime ($\xi<1$ with $\xi=g^2/(8\pi-g^2)$), a tower of neutral bound states called breathers $b_k$ of masses $m_k=2m\sin(\pi\xi k/2)$; at $\xi=1$ the theory reduces to a free massive Dirac fermion, and for $\xi>1$ (the repulsive regime, equivalent to the massive Thirring model) the soliton doublet is the whole spectrum.

Entanglement entropies in 1+1-dimensional QFT are computed by the replica trick: $\mathrm{Tr}\,\rho_A^n$ of a subsystem is a two-point function of branch point twist fields (BPTFs) $\mathcal{T}_n$, $\tilde{\mathcal{T}}_n$, which are symmetry fields of the $n$-copy (replica) theory implementing cyclic permutation of the copies (Cardy, Castro-Alvaredo and Doyon, J. Stat. Phys. 130, 129 (2008)). Symmetry-resolved entanglement decomposes the entropy into contributions of the sectors of an internal symmetry (Goldstein and Sela, Phys. Rev. Lett. 120, 200602 (2018)); the needed charged moments $Z_n(\alpha)=\mathrm{Tr}(\rho_A^n e^{i\alpha\hat{Q}_A})$ are two-point functions of a composite twist field (CTF) $\mathcal{T}_n^\alpha$, the leading field in the operator product expansion of the BPTF with the $U(1)$ symmetry (vertex) field $e^{i\alpha g\varphi/(2\pi)}$ of sG. The form factor bootstrap programme gives these correlators via a spectral expansion in matrix elements $F^{\alpha}_{\underline{a}}(\underline{\theta};\xi,n)=\langle 0|\mathcal{T}_n^\alpha|\underline{a}\rangle$ (form factors, FFs), which are fixed by Watson's equations (exchange with the exact S-matrix and crossing), kinematic residue equations, and bound-state residue equations.

For the standard BPTF in sG the FF equations were solved in the breather sector and, at two-particle level, in the soliton-antisoliton sector by Castro-Alvaredo and Horvath (SciPost Phys. 10, 132 (2021)) (Part I). For the CTF, Horvath, Calabrese and Castro-Alvaredo (SciPost Phys. 12, 088 (2022)) (Part II) first wrote the composite-field FF equations for a non-diagonal theory: crossing acquires the per-particle phase $e^{i\kappa\alpha/n}$, where $\kappa=+1,-1,0$ is the $U(1)$ charge of $s$, $\bar{s}$, and breathers respectively, so the breather sector is governed by the same equations as the standard BPTF while the charged sector is genuinely different. They solved only the breather sector (by analytic continuation from the sinh-Gordon model, where they used angular quantisation, a scheme in which the theory is quantised on radial half-lines so that the Hilbert space becomes a bosonic Fock space), computed one- and two-particle breather FFs, and used them to show equipartition of the leading symmetry-resolved entropies (the equality of the leading symmetry-resolved entropies across all charge sectors). Two ingredients were left undetermined, and are the subject of this problem. First, the CTF form factors on soliton/antisoliton states: at two-particle level the same-copy Watson equations reduce to $F^\alpha_{s\bar{s}}(\theta)=e^{i\alpha}F^\alpha_{\bar{s}s}(2\pi i n-\theta)$ and $F^\alpha_{\bar{s}s}(\theta)=e^{-i\alpha}F^\alpha_{s\bar{s}}(2\pi i n-\theta)$ with the kinematic residue $-i\,\mathrm{Res}_{\theta=i\pi}F^\alpha_{s\bar{s}}=\langle\mathcal{T}_n^\alpha\rangle$; the authors note that, unlike in the breather sector, the solutions are expected to depend on non-integer powers of the variables $e^{\theta/n}$ set by the $U(1)$ charge and the coupling $\xi$ (as is already true for the FFs of the plain exponential field computed by Lukyanov, Mod. Phys. Lett. A 12, 2543 (1997), and Lukyanov and Zamolodchikov, Nucl. Phys. B 607, 437 (2001)), which enlarges the space of admissible solutions so strongly that simple product ansatz methods become inadequate. Second, the vacuum expectation value $\langle\mathcal{T}_n^\alpha\rangle$ is unknown for sG; it enters as the overall normalization of every CTF form factor and controls the non-perturbative evaluation of the charged moments, although certain derived quantities (e.g. differences between symmetry-resolved and conventional entropies, which are governed by normalized FFs) are independent of it.

Problem Statement

Determine the form factors of the $U(1)$ composite branch-point twist field $\mathcal{T}_n^\alpha$ of the sine-Gordon model on charged (soliton/antisoliton) asymptotic states, together with the vacuum expectation value $\langle\mathcal{T}_n^\alpha\rangle$ that normalises them.

Concretely, solve the composite-twist-field form factor equations of Horvath, Calabrese and Castro-Alvaredo (SciPost Phys. 12, 088 (2022)) in the soliton/antisoliton sector, beginning with the two-particle functions $F^\alpha_{s\bar{s}}(\theta;\xi,n)$ and $F^\alpha_{\bar{s}s}(\theta;\xi,n)$ (particles in the same replica) and the multi-particle soliton/antisoliton tower generated from them, for general coupling $\xi$ and replica number $n$. The requested solutions must satisfy:

1. Watson's equations in the two-channel (backscattering) soliton sector, including the crossing relation $F_{\underline{a}}^\alpha(\theta_1+2\pi i,\theta_2,\ldots)=e^{i\kappa_1\alpha/n}F^\alpha_{a_2\ldots a_k\hat{a}_1}(\theta_2,\ldots,\theta_k,\theta_1;\xi,n)$ with $\kappa=\pm1$ for $s,\bar{s}$ (in the multi-index convention $a_i=(\gamma_i,\mu_i)$ of the source, $\hat{a}_i=(\gamma_i,\mu_i+1)$ denotes the same particle species on the next replica, while the antiparticle is denoted $\bar{a}_i=(\bar{\gamma}_i,\mu_i)$), which at two-particle level gives $F^\alpha_{s\bar{s}}(\theta)=e^{i\alpha}F^\alpha_{\bar{s}s}(2\pi i n-\theta)$ and $F^\alpha_{\bar{s}s}(\theta)=e^{-i\alpha}F^\alpha_{s\bar{s}}(2\pi i n-\theta)$, together with the exchange relations involving the full non-diagonal $S$-matrix;
2. the kinematic residue equations $-i\,\mathrm{Res}_{\theta=i\pi}F^\alpha_{s\bar{s}}(\theta;\xi,n)=\langle\mathcal{T}_n^\alpha\rangle$, so that the overall normalisation (the VEV) is fixed as part of the same bootstrap problem;
3. the bound-state residue equations relating soliton-sector form factors to the already known breather-sector composite-twist-field form factors in the attractive regime;
4. the $U(1)$-neutrality selection rules (e.g. $F^\alpha_{ss}=F^\alpha_{sb_k}=F^\alpha_s=0$) and cluster-decomposition asymptotics at large rapidity (the form factor must factorise into products of lower-particle form factors when one rapidity is sent to infinity);
5. the $n\to1$ limit reproducing the known form factors of the sG $U(1)$ exponential field, and the free-fermion point $\xi=1$ matching the $U(1)$ twist-field form factors of the free massive Dirac theory.

A complete answer covers all couplings, in particular the repulsive regime $\xi>1$ where the soliton-antisoliton doublet is the only sector, and thereby enables the non-perturbative evaluation of the charged moments $Z_n(\alpha)$ and of the symmetry-resolved entropies of the sine-Gordon model in all regimes (in the attractive regime these contributions are subleading; in the repulsive regime they are the entire contribution).

References

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