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$T\bar T$-deformation and long range spin chains

Balázs Pozsgay, Yunfeng Jiang, Gábor Takács

TL;DR

The paper reveals a deep correspondence between TTbar-type deformations in 2D IQFTs and a class of integrable long-range deformations in quantum spin chains, showing they share an algebraic origin, preserve integrability, and produce non-local theories with deformed scattering. It develops a lattice framework that mirrors the continuum factorization, proving a lattice factorization formula for antisymmetric current–density combinations and showing the S-matrix acquires a deformation phase analogous to a CDD factor. The work analyzes both local and long-range deformations, derives their action on Bethe states and finite-volume spectra via asymptotic Bethe Ansatz, and demonstrates the continuum-limit relevance of these lattice constructions. Overall, it positions the long-range spin-chain deformation as the natural lattice counterpart of TTbar, unlocking new avenues to study UV behavior, non-locality, and potential connections to lattice gravity and generalized hydrodynamics.

Abstract

We point out that two classes of deformations of integrable models, developed completely independently, have deep connections and share the same algebraic origin. One class includes the $T\bar T$-deformation of 1+1 dimensional integrable quantum field theory and related solvable irrelevant deformations proposed recently. The other class is a specific type of long range integrable deformation of quantum spin chains introduced a decade ago, in the context of $\mathcal{N} = 4$ super-Yang-Mills theory. We show that the detailed structures of the two deformations are formally identical and therefore share many features. Both deformations preserve integrability and lead to non-local deformed theories, resulting in a change of the corresponding factorized S-matrices. We also prove a factorisation formula for the expectation value of the operators which trigger the deformation on the lattice; similar results in quantum field theory play an essential role in the solvability of such deformations. We point out that the long range deformation is a natural counterpart of the $T\bar T$-deformation for integrable spin chains, and argue that this observation leads to interesting new avenues to explore.

$T\bar T$-deformation and long range spin chains

TL;DR

The paper reveals a deep correspondence between TTbar-type deformations in 2D IQFTs and a class of integrable long-range deformations in quantum spin chains, showing they share an algebraic origin, preserve integrability, and produce non-local theories with deformed scattering. It develops a lattice framework that mirrors the continuum factorization, proving a lattice factorization formula for antisymmetric current–density combinations and showing the S-matrix acquires a deformation phase analogous to a CDD factor. The work analyzes both local and long-range deformations, derives their action on Bethe states and finite-volume spectra via asymptotic Bethe Ansatz, and demonstrates the continuum-limit relevance of these lattice constructions. Overall, it positions the long-range spin-chain deformation as the natural lattice counterpart of TTbar, unlocking new avenues to study UV behavior, non-locality, and potential connections to lattice gravity and generalized hydrodynamics.

Abstract

We point out that two classes of deformations of integrable models, developed completely independently, have deep connections and share the same algebraic origin. One class includes the -deformation of 1+1 dimensional integrable quantum field theory and related solvable irrelevant deformations proposed recently. The other class is a specific type of long range integrable deformation of quantum spin chains introduced a decade ago, in the context of super-Yang-Mills theory. We show that the detailed structures of the two deformations are formally identical and therefore share many features. Both deformations preserve integrability and lead to non-local deformed theories, resulting in a change of the corresponding factorized S-matrices. We also prove a factorisation formula for the expectation value of the operators which trigger the deformation on the lattice; similar results in quantum field theory play an essential role in the solvability of such deformations. We point out that the long range deformation is a natural counterpart of the -deformation for integrable spin chains, and argue that this observation leads to interesting new avenues to explore.

Paper Structure

This paper contains 15 sections, 3 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. $T\bar{T}$-deformation of QFT
  3. Definitions
  4. Factorization formula
  5. Deformed spectrum
  6. Deformed $S$-matrix and CDD factor
  7. Local spin chains
  8. Factorization on the lattice
  9. Long range spin chains
  10. Action on Bethe states
  11. Finite volume: asymptotic Bethe Ansatz
  12. Discussion
  13. Continuum limit
  14. Alternative formulations and relation to lattice gravity
  15. Other observables and non-equilibrium dynamics

Key Result

Theorem 1

The function does not depend on $x$.

Theorems & Definitions (5)

  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • proof