SciPost Submission Page
The Space of Integrable Systems from Generalised $T\bar{T}$Deformations
by Benjamin Doyon, Joseph Durnin, Takato Yoshimura
Submission summary
As Contributors:  Takato Yoshimura 
Arxiv Link:  https://arxiv.org/abs/2105.03326v3 (pdf) 
Date submitted:  20210917 23:46 
Submitted by:  Yoshimura, Takato 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We introduce an extension of the generalised $T\bar{T}$deformation described by SmirnovZamolodchikov, to include the complete set of extensive charges. We show that this gives deformations of Smatrices beyond CDD factors, generating arbitrary functional dependence on momenta. We further derive from basic principles of statistical mechanics the flow equations for the free energy and all free energy fluxes. From this follows, without invoking the microscopic Bethe ansatz or other methods from integrability, that the thermodynamics of the deformed models are described by the integral equations of the thermodynamic BetheAnsatz, and that the exact average currents take the form expected from generalised hydrodynamics, both in the classical and quantum realms.
Current status:
Author comments upon resubmission
List of changes
1. We rearranged the way we cite these references so that it's more accurate. We also cited the paper mentioned by the referee as well as another article on nonrelativistic $T\Bar{T}$deformations.
2. We added the definition of $T\Sigma^\mathrm{Int}$.
3. We corrected the typo.
4. We meant ``supplement material" by SM. We rephrased it as the appendix.
5. The reference to eq (5) is corrected.
6. We added the reference to a particular section in the appendices.
7. The definition of the indicator function $\chi$ is added.
8. The definition of $\rho(\theta)$ is added.
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 2 on 20211122 (Invited Report)
Strengths
 proposes an interesting generalisation of TTbar deformations
 shows the implications on the infinitevolume Smatrix
 derives flow equations for the deformed theory
Weaknesses
 imprecise on the properties of these generalised transformations
 does not compare with previously determined flow equations
 no examples or physical discussion provided
Report
Dear Editor,
this article proposes a generalisation of the currentcurrent deformations discussed by Smirnov and Zamolodchikov (that already generalise the celebrated TTbar deformation). The authors derive the effects of these deformations on the Smatrix of the theory (in infinite volume) and write down flow equations for the charges.
The topic is interesting and some of the authors' result seem correct. However, this work needs revision in several points, which I discuss below. It is my recommendation that the paper should not be accepted for publication until such a major revision has been made.
The referee
Requested changes
1. In the introduction, the authors state that "A physical insight into TTbar was gained [...]" by relating them to changes of particle width. To put it mildly, this is a very partial statement. Physical insights in TTbar include their description as quasilocal deformations, their relation to twodimensional gravity to string theories on the worldsheet and in target space, and to holography. The authors should mention all that.
2. In section two the authors talk about a "more judiciously chosen" set of charges. However, it becomes clear later that these charges do not necessarily satisfy physical unitarity and crossing (not to mention real / Hermitian analyticity). The authors should make it clear "what are this charges good for", see also my points 3. and 6. below.
3. Related to point 2., it would be good if the authors discussed in some detail the deformation of one simple theory (such a SinhGordon), for some example of deformations that they propose that were not previously in the literature. In particular, the authors should present and discuss the finitevolume spectrum for such deformations (for instance , the kappa, eta and lambda deformations that they introduce), also as a way to put their newlydeveloped formalism to the test.
4. The names eta and lambda deformations, and to a lesser extent kappa, are commonly used in the literature of integrable deformations of sigma models (they are types of quantum deformations). The authors should probably pick new names.
5. In section 4 and 5 the authors discuss the flow equations and thermodynamic Bethe ansatz for their deformations. Throughout the discussion it is unclear to me whether the theory is in finite volume or at finite temperature. If I recall correctly, this made quite a difference in the authors ref. [16]. The authors should clarify this, and explain in detail whether or not their results match the one of [16] in the case where they are both applicable. This is far from immediately clear.
6. Related to point 2., I find the discussion of section 6 imprecise. The requirement of crossing symmetry and of physical unitarity (for real momenta) seem sufficient to rule out these newlyconstructed deformations. These requirement are considerably weaker than imposing that the Smatrix is analytic in the whole physical strip. More generally, for two dimensional integrable QFTs there is a well define list of properties that may be demanded of the Smatrix, related to a welldefined list of physical principles: Poincare' invariance, locality, causality, unitarity, parity, timereversal, particletoantiparticle symmetry, existence of boundstates. The authors should clarify which of these properties are broken by their new deformations with respect to the "usual TTbar" ones, and if possible provide example of known theories of such a type.
Anonymous Report 1 on 20211117 (Invited Report)
Report
This work extends the notion of generalised $T\bar{T}$ deformations, including the complete set of extensive charges. They show that the deformation leads to a general deformation of the Smatrix. The authors derive flow equations to the free energy and its fluxes. Moreover, they show that the substitution of the deformed Smatrix in the TBA equation leads to the same results.
The article meets the publication criteria of SciPost Physics, and I do recommend the publication on SciPost Physics after some clarification (see Requested changes).
Requested changes
1 In the first paragraph of Section 5 the authors write the following confusing sentence: "Here we show that the generalised $T\bar{T}$deformation provides a novel derivation of TBA", but in the Conclusion they write "We showed [...] that the thermodynamics of the deformed theories coincides with that obtained by TBA". The former sentence should be rephrased since the derivation in Appendix F starts with stating the deformed TBA equations and results in the flow equation. I also suppose that in the last paragraph of Section 5 the authors intended to refer to Appendix F instead of C.