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Quenches in initially coupled Tomonaga-Luttinger Liquids: a conformal field theory approach

by Paola Ruggiero, Pasquale Calabrese, Laura Foini, Thierry Giamarchi

Submission summary

As Contributors: Paola Ruggiero
Preprint link: scipost_202103_00028v1
Date submitted: 2021-03-28 12:36
Submitted by: Ruggiero, Paola
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Quantum Physics
Approach: Theoretical


We study the quantum quench in two coupled Tomonaga-Luttinger Liquids (TLLs), from the off-critical to the critical regime, relying on the conformal field theory approach and the known solutions for single TLLs. We exploit the factorization of the initial state in terms of a massive and massless modes which emerges in the low energy limit, and we encode the non-equilibrium dynamics in a proper rescaling of the time. In this way, we compute several correlation functions, which at leading order factorize into multipoint functions evaluated at different times for the two modes. Depending on the observable, the contri- bution from the massive or from the massless mode can be the dominant one, giving rise to exponential or power-law decay in time, respectively. Our results find a direct application in all the quench problems where, in the scaling limit, there are two independent massless fields: these include the Hubbard model, the Gaudin-Yang gas, and tunnel-coupled tubes in cold atoms experiments.

Current status:
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Submission scipost_202103_00028v1 on 28 March 2021

Reports on this Submission

Anonymous Report 1 on 2021-5-1 Invited Report


1- Well structured presentation with detailed overview of available methods, results in main text and technical calculations in appendices.
2- Good account of relevant literature.
3- Main finding: reduction of massive-to-massless quench in two coupled Luttinger liquids to two decoupled quenches (one massive + one massless) allowing use of earlier results.


1- Motivation, objectives, new results not explained in sufficient clarity/detail in the introduction: which results or methods are new/distinct compared to earlier Ref. [116]?
2- Effects of off-diagonal corrections not investigated in detail.
3- Absence of illustration figures and plots.


The manuscript presents an analysis of quench dynamics in systems described by two coupled Luttinger liquids from the non-critical to the critical phase, which is a subject relevant to a number of significant experimental and theoretical applications. Based on a reasonable ansatz for the form of the initial state, the qualitative behavior of correlations of different types of observables is studied in detail. In particular, the large time asymptotics of vertex operators and derivative fields is derived and classified as following exponential or power-law scaling with respect to the time and distances between the points. The method used in these works relies on the approximate factorization of the initial state that reduces the problem to a combination of a massive and a massless quench that have already been analyzed in the relevant literature. The present work is an extension of that of Ref. [116] on the same problem, including methodological simplifications and additional results.

Even though it is not clearly stated which of the results presented here are new in comparison with the earlier work of Ref. [116], the main findings are the reduction of the quench problem to that of two decoupled sectors, one massive and one massless quench, which allows the derivation of the asymptotics of correlation functions using earlier methods. There are also new results on the asymptotics of derivative field correlations (Sec. 5.2). The presentation is well-structured, presenting the more technical parts of the calculation in appendices, and the introduction includes a good overview of the subject and account of earlier literature. Nevertheless, despite its length, a sufficiently detailed explanation of the motivation, objectives, and summary of results or their significance is missing from the introduction.

Given the above remarks, this submission does not clearly meet the criteria of SciPost Physics, but does meet those of SciPost Physics Core, in which I would recommend publication after the following requested changes.

Requested changes

1- The authors should explain in the introduction in what ways the present work goes beyond Ref. [116]: they should motivate and state the objectives of the present analysis in comparison with those of the previous one in a more detailed way and should also provide a summary of the new results.

2- In Eq. (28) it is argued that at next-to-leading order in a small momentum expansion the initial state receives corrections that are linear in momentum, with one of the coefficients being proportional to the inverse initial mass, as expected by RG considerations, however, these coefficients are not given explicitly. It is important to calculate the parameters $\tau_A, \tau_B$ and $\gamma$ exactly for the quench from (3) to (1), verify the agreement with the RG predictions regarding the functional dependence on the original parameters, and evaluate the relevance of the off-diagonal contributions to practical applications.

3- Even though a discussion of off-diagonal corrections to the initial state is presented in Sec. 5.3, it is not conclusively clarified what their effects are on the asymptotics of the observables studied here, if these corrections can be ignored or if they would result in some significant or noticeable deviation.

4- Unlike in CFT, in the Luttinger liquid description of the quench of Ref. [34] the parameters of the model are momentum dependent functions, which is standard in Luttinger liquid theory. This dependence can be ignored at equilibrium as it corresponds to irrelevant (higher order derivative) perturbations, resulting in the simple free boson CFT model, however this is not necessarily valid away from equilibrium. In fact such irrelevant perturbations may be expected to play an increasingly important role at large times. How would the asymptotic results presented here be modified by such considerations?

5- There is some minor ambiguity over what is meant by leading order asymptotics and in which regimes the reported results are expected to hold. Power-law multiplicative corrections to exponential scaling may not be considered as subleading corrections by all readers. Also, in the case of power-law asymptotics where there is a cancellation of the exponents, the leading asymptotics is controlled by what is considered here as subleading corrections. Lastly, it should be clarified in what regime the results are expected to hold in applications, i.e. how far from the lines separating the different space-time regions (at a finite distance or at distances increasing with time?).

6- Eq. (98) is chopped.

7- There is a rather large number of typos and English errors, most of which can be corrected by an automated language check.

  • validity: top
  • significance: good
  • originality: good
  • clarity: high
  • formatting: good
  • grammar: reasonable

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