# Thermalization and Hydrodynamics of Two-Dimensional Quantum Field Theories

### Submission summary

 As Contributors: Luca Delacrétaz Arxiv Link: https://arxiv.org/abs/2105.02229v2 (pdf) Date submitted: 2021-09-08 03:32 Submitted by: Delacrétaz, Luca Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory High-Energy Physics - Theory Approach: Theoretical

### Abstract

We consider 2d QFTs as relevant deformations of CFTs in the thermodynamic limit. Using causality and KPZ universality, we place a lower bound on the timescale characterizing the onset of hydrodynamics. The bound is determined parametrically in terms of the temperature and the scale associated with the relevant deformation. This bound is typically much stronger than $\frac{1}{T}$, the expected quantum equilibration time. Subluminality of sound further allows us to define a thermodynamic $C$-function, and constrain the sign of the $\mathcal T\bar{\mathcal T}$ term in EFTs.

###### Current status:
Has been resubmitted

### List of changes

- Added Footnote 8 explaining the validity of the KPZ correlation function.

- Corrected a statement about the bulk viscosity, Eq. (38).

### Submission & Refereeing History

#### Published as SciPost Phys. 12, 119 (2022)

Resubmission 2105.02229v3 on 4 February 2022

Resubmission 2105.02229v2 on 8 September 2021
Submission 2105.02229v1 on 17 May 2021

## Reports on this Submission

### Anonymous Report 1 on 2021-12-29 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2105.02229v2, delivered 2021-12-29, doi: 10.21468/SciPost.Report.4114

### Report

The paper examines hydrodynamic behaviour in two dimensional QFTs, viewed as CFTs deformed by a relevant operator. The single hydrodynamic mode in the system is the propagating sound mode, arising from the broken conformal symmetry. The aim is to translate a bound derive for the onset of equilibration times in diffusive systems to this case, where the long-time behavior of the sound mode is modified by strong IR effects. The authors translate the known results into a bound for the equilbriation time.

### Requested changes

Overall I think the paper is reasonably written and the results ought to be interesting to the community. I think it deserves publication.

That said, I have a couple of suggestions for the authors to consider:
1. By definition, hydrodynamics is a description that is only valid at low energies compared to temperature, so a qualitative statement that says $\tau_{eq} \grsim T^{-1}$ does not have much content. Early time physics is dominated by transient effects and this may be worth noting.
2. In section 3 the authors argue that in higher dimensions late time physics is governed by diffusion and that the hydrodynamic modes are irrelevant. This is not strictly speaking accurate owing to again to IR effects, known in hydrodynamics as long-time tails, cf., https://arxiv.org/abs/1205.5040
3. It would be clearer if the discussion in Section 4.1 began by noting that that the UV physics is governed by the CFT the authors start with with further specialization subsequently to the free case.
4. Hydrodynamics in holographic theories in 2 dimensions has been discussed in https://arxiv.org/abs/1008.4350, which would worth commenting on in Section 5.4

• validity: good
• significance: good
• originality: good
• clarity: high
• formatting: excellent
• grammar: excellent

### Author:  Luca Delacrétaz  on 2022-02-02  [id 2142]

(in reply to Report 1 on 2021-12-29)

We thank the second referee for the useful feedback. Following their suggestions, we have made clarifications in Secs. 3 and 4.1. We also thank the referee for bringing arxiv:1008.4350 to our attention, which should indeed have been cited in Sec. 5.4; this is now remedied. Finally, we address the two remaining comments:

1. The motivation for studying or bounding the equilibration time is precisely to understand when the "early time physics [...] dominated by transient effects" gives way to hydrodynamics. Hydrodynamics, like any EFT, is indeed valid only for energies up to some cutoff, but it is not obvious what this cutoff is in a given system. Finding universal relations or bounds for this cutoff has therefore been the subject of active research (see e.g. https://arxiv.org/abs/2107.07802 for a recent review).

2. Hydrodynamic interactions in d>2, leading to the well-known hydrodynamic long-time tails, are irrelevant in the RG sense (but interesting!). We have clarified in Sec. 3 that we meant "irrelevant" in the technical, and not colloquial, sense.