$\mathrm{T}\overline{\mathrm{T}}$-deformed 1d Bose gas
Yunfeng Jiang
Abstract
\mathrm{T}\bar{\mathrm{T}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="normal"> <mml:mi>T</mml:mi> </mml:mstyle> <mml:mover> <mml:mstyle mathvariant="normal"> <mml:mi>T</mml:mi> </mml:mstyle> <mml:mo accent="true">‾</mml:mo> </mml:mover> </mml:mrow> </mml:math> 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}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="normal"> <mml:mi>T</mml:mi> </mml:mstyle> <mml:mover> <mml:mstyle mathvariant="normal"> <mml:mi>T</mml:mi> </mml:mstyle> <mml:mo accent="true">¯</mml:mo> </mml:mover> </mml:mrow> </mml:math> -deformed Lieb-Liniger model. We find that for one sign of the deformation parameter (\lambda<0) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>λ</mml:mi> <mml:mo><</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> , 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) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> , there exists an upper bound for the temperature, similar to the Hagedorn behavior of the \mathrm{T}\overline{\mathrm{T}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="normal"> <mml:mi>T</mml:mi> </mml:mstyle> <mml:mover> <mml:mstyle mathvariant="normal"> <mml:mi>T</mml:mi> </mml:mstyle> <mml:mo accent="true">¯</mml:mo> </mml:mover> </mml:mrow> </mml:math> deformed QFTs. Both behaviors can be attributed to the fact that \mathrm{T}\overline{\mathrm{T}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="normal"> <mml:mi>T</mml:mi> </mml:mstyle> <mml:mover> <mml:mstyle mathvariant="normal"> <mml:mi>T</mml:mi> </mml:mstyle> <mml:mo accent="true">¯</mml:mo> </mml:mover> </mml:mrow> </mml:math> deformation changes the size the particles. We show that for \lambda>0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , the deformation increases the spaces between particles which effectively increases the volume of the system. For \lambda<0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo><</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , \mathrm{T}\overline{\mathrm{T}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="normal"> <mml:mi>T</mml:mi> </mml:mstyle> <mml:mover> <mml:mstyle mathvariant="normal"> <mml:mi>T</mml:mi> </mml:mstyle> <mml:mo accent="true">¯</mml:mo> </mml:mover> </mml:mrow> </mml:math> deformation fattens point particles to finite size hard rods. This is similar to the observation that the action of \mathrm{T}\overline{\mathrm{T}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="normal"> <mml:mi>T</mml:mi> </mml:mstyle> <mml:mover> <mml:mstyle mathvariant="normal"> <mml:mi>T</mml:mi> </mml:mstyle> <mml:mo accent="true">¯</mml:mo> </mml:mover> </mml:mrow> </mml:math> -deformed free boson is the Nambu-Goto action, which describes bosonic strings — also an extended object with finite size.