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Marginal quenches and drives in Tomonaga-Luttinger liquids

by Shouvik Datta, Bastien Lapierre, Per Moosavi, Apoorv Tiwari

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Submission summary

Authors (as registered SciPost users): Bastien Lapierre · Per Moosavi · Apoorv Tiwari
Submission information
Preprint Link:  (pdf)
Date submitted: 2022-07-08 14:07
Submitted by: Moosavi, Per
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
Approach: Theoretical


We study Tomonaga-Luttinger liquids thrown out of equilibrium by marginal deformations in the form of interaction modulations. This is modeled by quenching or periodically driving the Luttinger parameter or, equivalently, the compactification radius of the free boson conformal field theory between two different values. We obtain exact analytical results for the evolution of the Loschmidt echo and observables such as the particle and energy densities. Starting from generic initial states, the quench dynamics are shown to exhibit revivals and temporal orthogonalities. For the periodic drive, we show stability or instability of time-evolved physical quantities dependent on the drive parameters. We also compare the corresponding marginally deformed thermal density matrices by non-perturbatively evaluating their R\'{e}nyi divergence as a Euclidean quench. All the dynamics is shown to be crucially dependent on the ratio of the Luttinger parameters, which corresponds to the Zamolodchikov distance in the space of marginal deformations. Our setup is equivalently interpreted as the dynamics of the bosonic string upon instantaneous changes of the target-space radius.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2022-10-16 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2206.11287v1, delivered 2022-10-16, doi: 10.21468/SciPost.Report.5910


Clearly written paper


The paper involves studying a non-equilibrium exactly solvable problem, namely a Luttinger liquid. By now this is rather a matured field, yet the authors do arrive at some interesting new observations. Perhaps the most interesting is the Floquet analysis and the study of the Renyi divergence.

In the Floquet part of the paper, I would suggest the authors do the following
a). compare their results with the famous stability phase diagram of the Matheiu equation, well known from the 1940s.
b). Explicitly compare their exponent and phase diagram with that obtained by Eggert et al ( a paper that has been cited, but not discussed).

After these two changes, the paper may be accepted for publication.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Author:  Per Moosavi  on 2023-01-11  [id 3230]

(in reply to Report 2 on 2022-10-16)

We thank the referee for their useful suggestions and positive recommendation. We now added an explanation as well as a new Figure (Fig. 9 in the updated manuscript) contextualizing the stability analysis carried out in our work with past works (in particular Phys. Rev. Lett. 126, 243401 by Eggert et al.) related to the stability phase diagram of the quantum parametric oscillator. Specifically, the new Fig. 9 contains a stability diagram that parallels Fig. 2 of Phys. Rev. Lett. 126, 243401 by identifying (i) the ratio of Luttinger parameters $K_1/K_2$ as the amplitude in the related continuous drive and (ii) the phase argument of the eigenvalues of the 2x2 su(1,1) matrix obtained from the Floquet unitary as the Mathieu characteristic exponent. We note that it is interesting that such a correspondence can indeed be made due to the shared underlying su(1,1) algebraic structure despite the fact that we use a step-like Floquet protocol as opposed to the quantum parametric oscillator, which is continuously driven.

Please see the list of changes for details.

Anonymous Report 1 on 2022-10-6 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2206.11287v1, delivered 2022-10-06, doi: 10.21468/SciPost.Report.5837


Very detailed paper


References not always complete for the paper to be self-contained


The paper studies the dynamics of Tomonaga-Luttinger liquids (TLLs) after quenching or periodically driving the Luttinger parameter (equivalently, the compactification radius).
Using Bogoliubov transformation and an underlying su(1,1) algebraic structure, various exact results are obtained. These includes results for the Loschmidt echo and the energy density in the aforementioned protocols. A relatively disconnected but interesting section is Section 6, where the same methods are applied to the calculation of Renyi divergences and relative entropy, via the replica limit, between two equilibrium thermal states of two different TLLs.

The paper contains interesting calculations which go clearly beyond the known results in the TLL physics. Moreover, it is clearly written and contains rather detailed calculations.

What is to be slightly improved are references, in particular:

- Recently, other relevant works considering quenches in TTL appeared, e.g., arXiv:2203.06740

- In Section 3, some more references could be included, as the material can be non-standard for physicists which are not mathematical physicists

- When discussing relative entropy, there are no references about the initial works where the replica approach has been used in the context of QFT (more specifically CFT), e.g., Phys. Rev. Lett. 113, 051602 (2014), Phys. Rev. Lett. 117, 041601, JHEP 02 (2017) 039

Once these are included, I recommend the paper for publication in Scipost Physics.

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

Author:  Per Moosavi  on 2023-01-11  [id 3229]

(in reply to Report 1 on 2022-10-06)

We thank the referee for their positive recommendation. We now added several references as suggested by the referee. In particular, we added four references in the introduction, where we discuss additional recent past works on interaction quenches in Tomonaga-Luttinger liquids, and added several references and a clarifying footnote in Sec. 3 which would aid the reader unfamiliar with those tools. We also added the suggested references related to initial works where the replica approach was used to study relative entropy in QFT.

Please see the list of changes for details.

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