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Register a new user$mathrm{T}overline{mathrm{T}}$-deformed 1d Bose gas
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Yunfeng Jiang
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$mathrm{T}overline{mathrm{T}}$ 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.
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We provide a simple geometric meaning for deformations of so-called $T{overline T}$ type in relativistic and non-relativistic systems. Deformations by the cross products of energy and momentum currents in integrable quantum field theories are known to modify the thermodynamic Bethe ansatz equations by a CDD factor. In turn, CDD factors may be interpreted as additional, fixed shifts incurred in scattering processes: a finite width added to the fundamental particles (or, if negative, to the free space between them). We suggest that this physical effect is a universal way of understanding $T{overline T}$ deformations, both in classical and quantum systems. We first show this in non-relativistic systems, with particle conservation and translation invariance, using the deformation formed out of the densities and currents of particles and momentum. This holds at the level of the equations of motion, and for any interaction potential, integrable or not. We then argue, and show by similar techniques in free relativistic particle systems, that $Toverline T$ deformations of relativistic systems produce the equivalent phenomenon, accounting for length contractions. We also show that, in both the relativistic and non-relativistic cases, the width of particles is equivalent to a state-dependent change of metric, where the distance function discounts the particles widths, or counts the additional free space. This generalises and explains the known field-dependent coordinate change describing $Toverline T$ deformations. The results connect such deformations with generalised hydrodynamics, where the relations between scattering shifts, widths of particles and state-dependent changes of metric have been established.
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