$\mathrm{T}\overline{\mathrm{T}}$-deformed 1d Bose gas

TL;DR

This paper extends the $ ext{T}ar{ ext{T}}$ deformation to 1D integrable particle models like Lieb-Liniger, analyzing how it affects spectrum, thermodynamics, and particle size, revealing UV breakdown and Hagedorn-like behavior.

Contribution

It constructs $ ext{T}ar{ ext{T}}$ deformation for 1D integrable particle systems and studies its effects on spectrum and thermodynamics, broadening the scope beyond 2D QFTs.

Findings

Deformation causes spectrum to become complex at small volumes for $ ext{sign}( ext{lambda})<0$.

For $ ext{sign}( ext{lambda})>0$, an upper temperature bound appears, similar to Hagedorn behavior.

Deformation effectively alters particle size, increasing volume for $ ext{lambda}>0$ and creating finite-sized particles for $ ext{lambda}<0$.

Abstract

$\mathrm{T}\overline{\mathrm{T}}$ deformation was originally proposed as an irrelevant solvable deformation for 2d relativistic quantum field theories (QFTs). The same family of deformations can also be defined for integrable quantum spin chains which was first studied in the context of integrability in AdS/CFT. In this paper, we construct such deformations for yet another type of models, which describe a collection of particles moving in 1d and interacting in an integrable manner. The prototype of such models is the Lieb-Liniger model. This shows that such deformations can be defined for a very wide range of systems. We study the finite volume spectrum and thermodynamics of the $\mathrm{T}\overline{\mathrm{T}}$-deformed Lieb-Liniger model. We find that for one sign of the deformation parameter $(\lambda<0)$, the deformed spectrum becomes complex when the volume of the system is smaller than certain critical value, signifying the break down of UV physics. For the other sign $(\lambda>0)$, there exists an upper bound for the temperature, similar to the Hagedorn behavior of the $\mathrm{T}\overline{\mathrm{T}}$ deformed QFTs. Both behaviors can be attributed to the fact that $\mathrm{T}\overline{\mathrm{T}}$ deformation changes the size the particles. We show that for $\lambda>0$, the deformation increases the spaces between particles which effectively increases the volume of the system. For $\lambda<0$, $\mathrm{T}\overline{\mathrm{T}}$ deformation fattens point particles to finite size hard rods. This is similar to the observation that the action of $\mathrm{T}\overline{\mathrm{T}}$-deformed free boson is the Nambu-Goto action, which describes bosonic strings -- also an extended object with finite size.