[Paper Review] The Space of Integrable Systems from Generalised $T\bar{T}$-Deformations

Benjamin Doyon, Joseph Durnin+1 more|arXiv (Cornell University)|May 7, 2021

Quantum many-body systems53 references4 citations

TL;DR

This paper introduces a generalized $T\bar{T}$-deformation framework that extends beyond standard stress-energy tensor deformations by incorporating the complete set of extensive and quasi-local charges in integrable systems. Using statistical mechanics principles, it derives universal flow equations for free energy and free energy fluxes, proving that thermodynamic quantities in deformed models obey thermodynamic Bethe ansatz (TBA) and generalized hydrodynamics (GHD) equations without relying on microscopic integrability techniques. The key result is a model-independent derivation showing that arbitrary S-matrix momentum dependence emerges naturally from this charge-based deformation scheme.

ABSTRACT

We introduce an extension of the generalised $T\bar{T}$-deformation described by Smirnov-Zamolodchikov, to include the complete set of extensive charges. We show that this gives deformations of S-matrices beyond CDD factors, generating arbitrary functional dependence on momenta. We further derive from basic principles of statistical mechanics the flow equations for the free energy and all free energy fluxes. From this follows, without invoking the microscopic Bethe ansatz or other methods from integrability, that the thermodynamics of the deformed models are described by the integral equations of the thermodynamic Bethe-Ansatz, and that the exact average currents take the form expected from generalised hydrodynamics, both in the classical and quantum realms.

Motivation & Objective

• To extend the $T\bar{T}$-deformation framework beyond the stress-energy tensor to include all extensive and quasi-local conserved charges.
• To derive flow equations for free energy and free energy fluxes in deformed integrable systems using only statistical mechanics principles, without invoking Bethe ansatz or integrability techniques.
• To demonstrate that the resulting thermodynamics of deformed models are governed by the thermodynamic Bethe ansatz (TBA) integral equations.
• To show that the exact average currents in both classical and quantum regimes take the form predicted by generalized hydrodynamics (GHD).
• To establish a universal, Hamiltonian-based construction for a broad class of factorized-scattering integrable systems with arbitrary two-body scattering phases.

Proposed method

• Formalism is built on statistical mechanics with an arbitrary number of extensive conserved charges, treating both classical and quantum systems on equal footing.
• Derives flow equations for free energy and free energy fluxes by considering the deformation of densities and currents under a generalised boost generator.
• Uses a complete basis of quasi-local charges labeled by asymptotic particle momenta to ensure completeness in the charge space.
• Applies the thermodynamic limit to derive integral equations for the free energy and fluxes, showing consistency with TBA and GHD.
• Demonstrates that the flow equations for free energy and fluxes match those derived from TBA, proving uniqueness and consistency.
• Introduces a deformation parameter $\lambda_{\theta\eta}$ that controls the change in particle width and momentum dependence, linking to the S-matrix's functional dependence.

Experimental results

Research questions

• RQ1 Can a generalized $T\bar{T}$-deformation be constructed using the full set of extensive and quasi-local charges, rather than just the stress-energy tensor?
• RQ2 Do the resulting flow equations for thermodynamic quantities in deformed models reproduce the integral equations of the thermodynamic Bethe ansatz (TBA)?
• RQ3 Is the exact form of average currents in the deformed systems consistent with predictions from generalized hydrodynamics (GHD)?
• RQ4 Can the S-matrix of the deformed theory exhibit arbitrary functional dependence on momenta through this charge-based deformation?
• RQ5 Is the TBA solution the unique solution to the derived flow equations, and can this be shown without assuming microscopic integrability?

Key findings

• The flow equation for the free energy is derived as $\frac{\delta f}{\delta\lambda_{\theta\eta}} = \rho(\eta)g_{\theta} - \rho(\theta)g_{\eta}$, where $\rho$ is the particle density and $g$ the free energy flux.
• The flow equation for the free energy flux is $\frac{\delta g_i}{\delta\lambda_{\theta\eta}} = \langle j_{i\theta} \rangle g_{\eta} - \langle j_{i\eta} \rangle g_{\theta}$, showing a universal structure across systems.
• The thermodynamic Bethe ansatz (TBA) form for free energy and fluxes satisfies the derived flow equations, proving that TBA provides the unique solution.
• The derivation of flow equations does not require the Bethe ansatz or any microscopic integrability assumptions, relying only on statistical mechanics principles.
• The S-matrix of the deformed theory acquires arbitrary momentum dependence through the deformation, generalizing beyond CDD factors.
• In boost-symmetric cases, the flow equations reduce to $\frac{\delta f}{\delta\lambda_{\theta}} = \int_{\mathbb{R}} d\eta \, \rho(\eta)(g_{\eta+\theta} - g_{\eta-\theta})$, matching known results.