$T\overline T$ deformations of non-Lorentz invariant field theories

TL;DR

The paper generalizes the solvable $T\overline T$ ($\det T$) deformation to non-Lorentz invariant 1+1D theories, showing that the finite-size spectrum and $S$-matrix are still tractable. The deformation induces a Burgers-type flow for energy levels and a CDD-dressed $S$-matrix, with the flow adapting to Lifshitz-type dynamics via the dispersion relation. It also extends the framework to arbitrary conserved-current deformations, highlights open-boundary and torus subtleties, and discusses phase structure including a Hagedorn transition for negative deformation parameter. These results suggest a broad, model-independent mechanism for deforming translationally invariant 1+1D systems while preserving solvability and yielding rich spectral and scattering behavior.

Abstract

We point out that the arguments of Zamolodchikov and others on the $T\overline T$ and similar deformations of two-dimensional field theories may be extended to the more general non-Lorentz invariant case, for example non-relativistic and Lifshitz-type theories. We derive results for the finite-size spectrum and $S$-matrix of the deformed theories.

Figures (2)

• Figure 1: Real parts of solutions of (\ref{['0.20']}) for $z=2$ and $R=C=1$. The physical solution has a square root singularity at $t\approx 0.15$. However for $t<0$ there are complex conjugate solutions with lower energy. The solutions for $C=-1$ may be visualized by reflecting the figure in both axes.
• Figure 2: Free energy $F$ per unit length v. inverse temperature $\beta$ for a deformed gapless system with $z=2$. The lower curve corresponds to fixed $t<0$ and exhibits the typical square root singularity of the Hagedorn transition. The upper curve is for $t>0$: the free energy and internal energy are analytic at $\beta=0$, so a continuation to negative temperature makes sense.