# Marginal quenches and drives in Tomonaga-Luttinger liquids

### Submission summary

 Authors (as Contributors): Bastien Lapierre · Per Moosavi · Apoorv Tiwari
Submission information
Date accepted: 2023-02-13
Date submitted: 2023-01-11 18:27
Submitted by: Moosavi, Per
Submitted to: SciPost Physics
Ontological classification
Specialties:
• Condensed Matter Physics - Theory
• High-Energy Physics - Theory
• Mathematical Physics
Approach: Theoretical

### Abstract

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 are 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:
Accepted in target Journal

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

Resubmission with minor adjustments, fixed typos, improved discussions, and added references following the referee reports. We thank both referees for their comments and positive recommendations.

### List of changes

1. In Sec. 1:
a) Added the following four references on interaction quenches in Tomonaga-Luttinger liquids (TLLs):
- Bernier et al., Phys. Rev. Lett. 112 (2014) 065301.
- Dóra and Pollmann, Phys. Rev. Lett. 115 (2015) 096403.
- Ruggiero et al., SciPost Phys. 13 (2022) 111, mentioned by Referee 1.
- Moosavi, arXiv:2208.14467.
b) Added the following reference on conformal interfaces:
- Bachas et al., J. High Energ. Phys. 2002 (2002) 027.

2. In Sec. 3:
Added several references in Sec. 3 that would aid the reader with potentially unfamiliar topics. The added references include:
a) For our overall notation and introduction to CFT in the beginning of Sec. 3, we referenced:
- Di Francesco, Mathieu, and Senechal, Conformal field theory (1997),
- Gawƒôdzki et al., J. Stat. Phys. 172 (2018) 353.
- Moosavi, Ann. Henri Poincar√© (2021).
b) Added the following reference discussing marginal operators and their conformal weight in Sec. 3.1:
- Ginsparg, Nucl. Phys. B 295 (1988) 153.
c) Added the following reference for the Sugawara construction in Sec. 3.2:
- Di Francesco, Mathieu, and Senechal, Conformal field theory (1997).
d) Added the following new references for the su(1,1) algebra in Sec. 3.3:
- Perelomov, Generalized coherent states and their applications (1986).

3. Other changes in Sec. 3:
a) Clarified the definition of the Zamolodchikov
metric in Sec. 3.1.
b) Added Footnote 9 at the end of the first paragraph in Sec. 3.5 to make the discussion more self-contained.
c) Updated the notation for the su(1,1) generators.
d) Added explicit expression for the su(1,1) Cartan-Killing form as the new Eq. (3.21).

4. In Sec 5:
a) Added brief remarks in the beginning about the parallels with quantum parametric oscillators to put the use of the su(1,1) algebraic tools in context, together with the following two new references:
- Perelomov and Popov, Theor. Math. Phys. 1, (1969) 275.
- Gritsev and Polkovnikov, SciPost Phys. 2 (2017) 021.
b) Added a new paragraph and a new figure (Fig. 9) in Sec. 5 describing the connection between our stability analysis and the stability phase diagram obtained for the continuously driven TLL based on the Mathieu equation.
c) Corrected sign typos in formulas in the beginning of Sec. 5.

5. In Sec.6:
a) Added the following references for the replica approach used to study relative entropy in QFT:
- Lashkari, Phys. Rev. Lett. 113, (2014) 051602.
- Lashkari, Phys. Rev. Lett. 117, (2016) 041601.
- Ruggiero and Calabrese, JHEP 02 (2017) 039.
b) Updated the discussion in Sec. 6.1 to improve consistency and clarity.

6. In Sec. 7:
Added a discussion on extensions of our work to driven dissipative TLLs.

7. Optimized the text at various places.