Thermodynamic form factor bootstrap for integrable quantum field theories with bound states and non-diagonal scattering

The thermodynamic form factor (TDFF) bootstrap program determines exact matrix elements of local operators between states of finite energy density in integrable quantum field theories, and thereby the dynamical correlation functions of thermal and post-quench (generalized Gibbs) states. As established by Cortés Cubero and Panfil, and completed by Panfil and Konik (JHEP 10 (2023) 072), the program — axioms including the annihilation pole with its residues, a crossing relation, and a reparameterization invariance of the thermodynamic state — is valid only for theories with diagonal scattering and without bound states, and its axioms leave undetermined multiplicative CDD-type soft-mode factors. The open problem is the extension beyond this established domain: formulate and establish the axiom system and the crossing/reparameterization relations for integrable QFTs with bound states (e.g. sine-Gordon breathers) and non-diagonal scattering (nested-Bethe-ansatz models), with validation against finite-volume or microscopic form factors — the extension to bound-state form factors that the source paper explicitly defers to future work.

Background

In a (1+1)-dimensional integrable quantum field theory, scattering is elastic and factorized: the exact two-particle S-matrix $S(\theta_1-\theta_2)$, written in terms of rapidities $\theta$ that parametrize energy and momentum, encodes the full interaction. Smirnov's form factor bootstrap determines vacuum matrix elements $F(\theta_1,\dots,\theta_n)=\langle 0|\mathcal{O}|\theta_1,\dots,\theta_n\rangle$ of local operators from axioms: normalization, periodicity and analyticity in the rapidities, factorized scattering, and annihilation-pole residues when a particle rapidity coincides with an antiparticle rapidity shifted by $i\pi$. These matrix elements are the building blocks of spectral expansions of correlation functions.

Away from the vacuum, integrable systems at finite energy density — thermal states, or the generalized Gibbs ensembles (GGEs, the maximal-entropy states fixed by all extensive conserved charges) reached after a quantum quench — are described by thermodynamic Bethe ansatz (TBA) root densities: a filled rapidity density $\rho_p(\theta)$ with hole density $\rho_h(\theta)$. The natural excitations are particle-hole pairs, a particle moved from an occupied rapidity to a hole, with all rapidities shifted by the backflow (dressing) function $F$ that accounts for interactions with the sea. Thermodynamic form factors (TDFFs) $f_\rho(\alpha_1,\dots,\alpha_n)$ are matrix elements of local operators between such a thermodynamic state $\rho$ and the state carrying additional particle-hole excitations at rapidities $\alpha_i$; assembled into LeClair–Mussardo-type spectral series (finite-temperature expansions of correlation functions over multi-particle states), they give dynamical correlation functions at finite temperature and in non-equilibrium steady states.

Cortés Cubero and Panfil generalized Smirnov's axioms to this setting, founding the thermodynamic bootstrap program: TDFFs obey normalization, periodicity, scattering, and annihilation-pole axioms, admit minimal solutions, and reproduce the Euler-scale correlators of generalized hydrodynamics (GHD, the exact hydrodynamic theory of integrable systems). Panfil and Konik then verified the annihilation-pole axiom against a finite-volume regularization in the low-energy regime of small particle-hole displacement, established a crossing relation connecting TDFFs with different particle-hole content — nontrivial because the finite-energy-density background breaks Lorentz invariance — and showed that crossing follows from a reparameterization invariance expressing that the detailed representation of the thermodynamic state is unphysical. Together these results complete the program for theories with diagonal scattering, no bound states, and (in the relativistic treatment) a single self-conjugate particle species, such as sinh-Gordon and, through a non-relativistic limit, the Lieb-Liniger gas.

Two structural limitations remain. First, the axiom system has not been formulated for generic integrable QFTs: theories with bound states, such as the sine-Gordon model whose S-matrix has breather (bound-state) poles related to the soliton sector by fusion, and theories with non-diagonal scattering, where the S-matrix is matrix-valued and mixes particle species so that nested Bethe ansatz is required (e.g. the Yang-Gaudin model, the two-component delta-interacting gas, or integrable spin chains). The annihilation-pole residues, crossing, and reparameterization arguments of the existing program all rely on the diagonal, bound-state-free structure. Second, the axioms do not determine TDFFs uniquely: any multiplicative symmetric factor $W(\{\alpha_i\})$ obeying mild limiting conditions can be inserted — a CDD-type ambiguity, analogous to Castillejo–Dalitz–Dyson pole ambiguities in S-matrix theory; the entropic factor $\exp(\sum_i \delta S(\alpha_i))$ expected from resumming soft modes (exponentially low-energy excitations, with $\delta S$ the differential entropy) is compatible with, but not fixed by, the axioms.

Problem Statement

Extend the thermodynamic form factor bootstrap program, established only for integrable quantum field theories with diagonal scattering and without bound states, to the generic case of theories with bound states and non-diagonal scattering at finite energy density. A complete answer should, as jointly supporting components of this single objective: (1) formulate the axiom system for TDFFs on finite-energy-density backgrounds — normalization, periodicity/analyticity, scattering, and the annihilation pole with its residue structure — incorporating multi-species and breather bound-state (fusion) structure and, for non-diagonal theories, the matrix-valued scattering treated by nested Bethe ansatz; (2) establish the crossing relation and reparameterization invariance in this generalized setting, or determine precisely how they are modified; and (3) demonstrate the resulting bootstrap in at least one nontrivial theory with bound states or non-diagonal scattering — for instance the sine-Gordon model with breathers, or a nested-Bethe-ansatz model such as Yang-Gaudin — validating the TDFFs against finite-volume or microscopic form-factor computations. The common objective is a complete TDFF bootstrap for generic integrable QFTs — the extension beyond the source paper's completion claim, whose bound-state form factors the authors explicitly defer to future work. Partial progress on individual components (for example, axioms for a breather sector alone) is meaningful but does not by itself complete the extension.

References

1. Miłosz Panfil and Robert M. Konik, "Extending the Thermodynamic Form Factor Bootstrap Program: Multiple particle-hole excitations, crossing symmetry, and reparameterization invariance", JHEP 10 (2023) 072, DOI 10.1007/JHEP10(2023)072, arXiv:2306.10748, https://arxiv.org/abs/2306.10748
2. Axel Cortés Cubero and Miłosz Panfil, "Thermodynamic bootstrap program for integrable QFT's: form factors and correlation functions at finite energy density", JHEP 01 (2019) 104, DOI 10.1007/JHEP01(2019)104, arXiv:1809.02044, https://arxiv.org/abs/1809.02044
3. Axel Cortés Cubero and Miłosz Panfil, "Generalized hydrodynamics regime from the thermodynamic bootstrap program", SciPost Phys. 8 (2020) 004, DOI 10.21468/SciPostPhys.8.1.004, arXiv:1909.08393, https://arxiv.org/abs/1909.08393
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12. Miłosz Panfil and Felipe Taha Sant'Ana, "The relevant excitations for the one-body function in the Lieb–Liniger model", J. Stat. Mech. (2021) 073103, DOI 10.1088/1742-5468/ac0f70, arXiv:2104.10491, https://arxiv.org/abs/2104.10491