Universal thermalization dynamics in (1+1)d QFTs

We are very grateful to the two referees for carefully reading our manuscript. Our responses to their comments are below.

REFEREE 2

Referee Comment: 1) In the introduction, the authors seem to identify the concepts of thermalization and hydrodynamics. As they note much later, in the discussion section, a CFT can still thermalize (e.g. in the sense of correlators approaching the thermal value), even though it does not admit hydrodynamics. It would be better to be more clear about this already in the introduction.

Response: 1) Indeed, even certain free and integrable models can be said to thermalize under such a criterion. Because this criterion is not very differentiating, we find it less useful in the study of the thermal dynamics of QFTs. Nevertheless, the question of whether a 2d CFT should be viewed as thermalizing is an important point which we devoted an entire subsection to in the Discussion. Following the referee's comment, we have further added a footnote clarifying the distinction, and why the notion of thermalization we use is appealing.


Referee Comment: 2) The authors invoke frequently the notion of breakdown of perturbation theory. Also here I think that they should clarify in what sense they mean this: to my understanding, it is simply that in the hydrodynamic regime the effective coupling becomes large, but the resummation to all-orders of the CPT can still describe the physics without the need of non-perturbative effects.

Response: 2) The referee is correct, for large-$c$ QFTs: in that case we have a proposed resummation that should save CPT, and which is necessary even to only get the leading order correlators at late times. At finite $c$, we expect the breakdown to be more severe, see [1]. We have added a clarification in the introduction on this point.


Referee Comment: 3) When one considers higher orders in CPT, generically one has to consider that the RG flow will introduce mixing of different operators; is there an argument for why it is sufficient to consider the contribution of a single operator as illustrated in Fig. 1? Relatedly, in the large-c limit illustrated in Fig. 2, can one ignore diagrams with O O fusing into O’, and O O’ fusing into T?

Response: 3) Such channels will give subleading contributions to the late time correlator near the sound front. Every additional operator insertion comes at the small CPT parameter $\lambda$. This cost is balanced by the enhancement of by a factor of $t$ when a stress tensor (or other twist-zero operator) is produced. The dominant channels therefore come from when stress tensors are produced by as a few operator $\mathcal O$'s as possible. Channels with intermediate operators in other multiplets are therefore suppressed.


Referee Comment: 4) On page 18, the authors advocate for a definition of the equilibration time purely within hydrodynamics, which would be more universal than the one defined from perturbation theory. However with this definition, it would still depend in principle on the full set of transport coefficients, which in turn contain the information of the UV theory, so it is not clear to me to what extent it would be more universal.

Response: 4) This bottom-up'' definition of the thermalization time is more universal in that it can be applied to any theory where hydrodynamics emerges, regardless of how it emerges. The {\em value} of the thermalization time $\tau_{\rm eq}$ obtained from this definition is indeed not universal, it is different in every system and depends sensitively on the UV: it is therefore a useful characteristic of a QFT. It distinguishes free or integrable systems ($\tau_{\rm eq} = \infty$) from weakly interacting $\tau_{\rm eq}T \gg 1$ and strongly interacting $\tau_{\rm eq}T\sim 1$ systems.


Referee Comment: 5) On page 25 they write “we do not expect the TTbar deformation to generate viscous effects”. In fact ref [24] found a non-vanishing momentum diffusion in TTbar-deformed CFT at leading order in CPT, see eq (15)-(16).

Response: 5). Thank you for this very interesting observation. We believe these results are actually consistent with our statement (which, like all of our Sec. 4, assumes large $c$). To see this, we first need to fix normalizations in order that the results in [24] have a sensible large$-c$ limit. This requires their $\sigma\sim 1/c$ (see e.g. eq (14) in [24]). With this normalization, the viscosity is $\sim c^0$. Although this is finite, it is suppressed relative to the other terms in the constitutive relation which are $O(c)$, and is also suppressed relative to the $O(c)$ viscosity generated by the deformation with a primary scalar that we study. We have added a footnote on p26 to explain this.


Referee Comment: 6) In appendix B, they state that the UV divergence in the Fourier transform of the scalar two-point function for $\Delta\geq 1$ can be reabsorbed in a local counterterm. How does it work, since the counterterm action they write would generate a contact term in the correlator, while the divergence comes from the light-cone ?

Response: 6) Our wording was indeed somewhat imprecise. It is true that when $\Delta \geq 1$, the Fourier transform of $G^R_{OO}(t,x)$ is UV divergent, a divergence which can be absorbed by adding a counterterm $\int d^2 x \, J^2(x)$. However, in this situation the divergence also affects any observable in CPT, including the two-point function of the stress tensor under consideration, or of any other operator: $\langle \mathcal O'(t,x)\mathcal O'\rangle$. The $O(\lambda^2)$ CPT correction to such an observable involves two $\mathcal O$ insertions, integrated over the thermal cylinder. The UV divergence arises from the two $\mathcal O$'s fusing into the identity. This produces a UV divergent factor, multiplying the original CFT correlator. It can therefore be absorbed by wavefunction renormalization of the operator $\mathcal O'$.


Referee Comment: 7) In eq. (4.20), the scalar correlator on the right-hand side should determine all the transport coefficients, but those are encoded in two functions of $\omega,k$: can the author explain in more detail how one single equation fixes both functions?

Response: 7) Equation (4.20) should be interpreted as valid when both sides are expanded at small $\omega$ and $k^2$. Although there are two functions on the left hand side $\Omega(\omega,k^2)$ and $\kappa(\omega,k^2)$, these do not have generic expansions in this limit. This is seen in equation (4.13): for example, in $\Omega$ there are no terms with more than one factor of $\omega$. By counting, we can see that the combined number of transport coefficients $\Omega_n$ and $\kappa_{n,m}$ appearing at each order in the small $\omega,k^2$ limit of the left hand side of equation (4.20) agrees with the number of Taylor coefficients of the expansion of a generic function of $\omega$ and $k^2$ (i.e~the right hand side of equation (4.20)). And so this equation is sufficient to determine all transport coefficients.

The counting works as follows. We scale $\omega\sim k\sim\epsilon$. If $n$ is even, then at order $n$ in the expansion of a generic function we have terms of the form $k^{n},\omega^2 k^{n-2}$, etc. In total there are $(n+2)/2$ such terms. If $n$ is odd, then at order $n$ in the expansion the terms are of the form $\omega k^{n-1},\omega^3 k^{n-3}$, etc. In total there are $(n+1)/2$ such terms.

The (first) index on each transport coefficient $\Omega_n$, $\kappa_{n,m}$ tells the order that this term first appears on the left hand side of equation (4.20) in this expansion. For example, at order $4$ there is one parameter coming from $\Omega$ and two coming from $\kappa$, giving a total of 3 new parameters at this order. This agrees with the number of terms in a generic expansion above $=(4+2)/2$. The same can be checked for other orders.

Thank you for raising this important point -- we have added a footnote on p25 to clarify this.



REFEREE 1

Referee Comment: Introduction: Be more accurate concerning current understanding of how long to thermalise, etc, including the fact that $1/\lambda^2$ is expected and that weak coupling is not necessary, and refer to papers mentioned (and / or others if found to be relevant).

Response: We agree with the referee that the slow thermalization in our setup is not a surprise: the intuitive argument for this is given around our equation (1.1). The main result of the first half of our paper is not so much the thermalization time (1.2) itself but rather the universal mechanism by which the thermalization occurs. We agree with the referee that the argument for slow thermalization around (1.1) also applies to weakly perturbed integrable models and we are very grateful to the referee for drawing our attention to the papers they mentioned. We have added a comment regarding this as well as the reference Phys. Rev. Lett. 127, 130601 (2021). To try and prevent confusion we have not added the second reference suggested by the author. From our understanding, this paper studies the crossover from one equilibrium-like state to the final true equilibrium, due to the weak breaking of a conservation law of the first. It does not study the initial emergence of the first equilibrium-like description (instead this is assumed via the assumption of fast equilibration'' referred to in that paper). In our language, it is the initial emergence that we would refer to as thermalization. We agree that even when the coupling is strong there is theoretical control over the first state, but there is not any controlled description of how it emerges -- instead this is just assumed.


Referee Comment: Introduction: Make more clear that the QFT is assumed to be described by a nontrivial CFT at both UV and IR. This is important, because at low temperature, if there is a gap (i.e. the CFT is the trivial one-dimensional CFT spanned by the operator identity), as is the case in many one-dimensional QFT’s studied in the literature, the results are very different.

Response: Our analysis of thermalization at high and low temperatures are separate. In particular: in the high temperature limit the dynamics are controlled by the UV CFT, and we do not need to know anything about the IR physics. Similarly, in the low temperature limit the dynamics are controlled by the IR CFT and we do not need to know anything about the UV of the theory. So we are assuming only that either the IR or UV physics is controlled by a non-trivial CFT, rather than both. We have edited the second paragraph of the discussion to emphasize that the CFT must be non-trivial.


Referee Comment: Page 7: “We will mostly focus on the generic case given in Eq. (2.7).” Mostly? Not always? Could you be more specific?

Response: We comment on $\Delta \to 1$ below (4.24).


Referee Comment: Page 8: GE or GE? The former is a typo?

Response: The typo is now fixed, thank you.


Referee Comment: Page 8: why not define Oeq mathematically? It is just put in text and explained in words, which it very inefficient.

Response: We have added the mathematical definition of $\mathcal O_{\rm eq}$ in Sec.~2.


Referee Comment: Page 8: eq 2.11 seems to be important. However the derivation presented on pages 7-8 leading to it is very hard to follow. It is said in words, with some equations in App A. However, it is not clear where 2.11 actually come from. At least the full derivation of 2.11 should be given in App A and referred to in the text.

Response: We thank the referee for the suggestion, we have added a detailed step by step derivation of Eq.~(2.11) below Eqs.~(A.5) and (A.6) in Appendix A.


Referee Comment: Section 3.2: I can more or less follow the arguments made in section 3.2, which are very interesting, but it would be useful to put more key equations to support the arguments. Currently, I could not, with the limited time I had, follow all derivations because I would have had to write the equations.

Response: We have edited the 2nd and 3rd paragraphs of Section 3.2, inserting equation (3.16) (which was previously equation (B.10) of the Appendix) and a reference to equation (B.5) of the Appendix to make the mathematical basis for our statements clearer.


Referee Comment: Pages 14-15, it is not clear what “schematic” means in equation 3.17 and 3.18. Does it mean that coefficients are not all 1? Please be more precise there. Also, it is unclear where these equations come from.

Response: The OPE of every operator with itself contains the Virasoro multiplet of the identity, with coefficient 1. The Virasoro descendents that appear in the OPE are entirely fixed by conformal symmetry -- their precise form (involving derivatives $\partial_-$) as well as the relative numerical factors have been dropped for simplicity. It is in this sense that these equations are schematic. We have added clarifications above and below Eq.~(3.17).


Referee Comment: Page 15: the terminology double-twist and higher-twist is not so universal. Please explain.

Response: In the context of large-$c$ QFTs, we simply mean double-trace and higher-trace. However, the notion of higher-twist operators is meaningful even away from large-$c$ (see, e.g., Convexity and Liberation at Large Spin'' by Komargodski and Zhiboedov). We expect the OPE channels we have identified to also play a role at finite $c$, which is why we have adopted this language.


Referee Comment: Pages 14-15: in fact I find the paragraph “large-c scaling” completely unclear and hard to follow. Perhaps it is better placed in section 4, and with more explanations?

Response: We agree that the fact that this discussion appeals to results obtained in the later hydrodynamic section (Sec.~4), may make this part confusing in a first read. We have explored moving this paragraph to Sec.~4 as the referee suggests, but could not find a good home for it. We have expanded the paragraph after equation (3.19) to contain more explanation of how the OPE structure given in (3.18) and (3.19) is used to conclude what are the dominant channels at large $c$.

Edited the second paragraph of the introduction to emphasize that the CFT must be non-trivial.

Added a sentence, and ref [1], in the second paragraph of the introduction to connect with recent work on weakly perturbed integrable systems.

Added a clarification in the introduction concerning the nature of the breakdown of CPT.

Fixed a typo in the formula for the speed of sound before equation (2.5)

Edited the second and third paragraphs of Section 3.2, added equation (3.16) and a reference to equation (B.5) to make the mathematical basis for our arguments clear without having to read the whole Appendix B.

Stated above (3.18) that we are focusing on the identity contribution to the scalar OPE, and specified below (3.18) that we do so because this is the dominant CPT correction.

Paragraph after (3.19) expanded to explain better how to deduce the dominant thermalization channels at large $c$.

Added a footnote on p.19 clarifying the notion of thermalization used in this paper, and why it is an appealing definition.

Added a footnote on p25 to clarify why a single generating functional is sufficient to fix both hydrodynamic functions $\Omega(\omega,k^2)$ and $\kappa(\omega,k^2)$.

Added a footnote on p26 to explain how our statement that we do not expect the TTbar deformation to generate a viscosity is consistent with the results of ref [24].

Added a detailed derivation of Eq.~(2.11) in Appendix A.