# Non-equilibrium Sachdev-Ye-Kitaev model with quadratic perturbation

### Submission summary

 As Contributors: Lunkin Aleksey Arxiv Link: https://arxiv.org/abs/2106.11245v3 (pdf) Date accepted: 2021-12-14 Date submitted: 2021-11-11 09:15 Submitted by: Aleksey, Lunkin Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Approach: Theoretical

### Abstract

We consider a non-equilibrium generalization of the mixed SYK$_4$+SYK$_2$ model and calculate the energy dissipation rate $W(\omega)$ that results due to periodic modulation of random quadratic matrix elements with a frequency $\omega$. We find that $W(\omega)$ possesses a peak at $\omega$ close to the polaron energy splitting $\omega_R$ found recently (PRL 125, 196602), demonstrating the physical significance of this energy scale. Next, we study the effect of energy pumping with a finite amplitude at the resonance frequency $\omega_R$ and calculate, in presence of this pumping, non-equilibrium dissipation rate due to low-frequency parametric modulation. We found an unusual phenomenon similar to "dry friction" in presence of pumping.

Published as SciPost Phys. 12, 031 (2022)

Resubmission latter.

We would like to thank the referees for their reports and comments.

To the report #1:
Minor comments: a&b – we have done all the corrections proposed
Main comments: We have retained S_2 terms as it changes the quadratic action of the soft modes as this contribution is dominant for low frequencies. It can be seen from Eq.10 where the Green function of the soft modes is presented. The second term in the brackets (with function $\psi$) comes from the S_2 and is dominant in the assumption $\Gamma \gg T$. We do not need to take into account higher terms as they are small as $N\gg1$. The term $S_2$ is also important as it mixes low and high frequencies in the non-linear regime (see eq. (28) ).
To the report #2:
In our model, interaction (described by the SYK model) is much stronger than quadratic perturbation. As a result, the starting point of perturbative analysis is an SYK model, which does not have a standard diagrammatic description with an ordinary fermionic Green function. Moreover, the fluctuation propagator in the SYK model has a “zero mode” due to asymptotic symmetry. This mode could be taken into account using the path integral approach for the variable $\phi$ (as action is non-local we could not write a Hamiltonian). As a result, for the SYK model, the diagrammatic description could be used in terms of fluctuation propagator of $\phi$. To clarify the origin of the key relation shown in Eq.(18), we added the text between equations (18)-(22) which shows how susceptibility can be obtained working in the functional integral representation for the $\phi$ field.
We also added the citation of the mentioned paper by Skvortsov et al. It is referred to in the Introduction after the words: “In other words, we propose here a generalization of the approach well-known [6–9]”

Aleksey Lunkin and Mikhail Feigel’man

### List of changes

1. The citation of the paper by Skvortsov et al. was added in the introduction.
2. The equation (5) was enumerated
3. Intervals in the equation (8) were increased
5. The last line in the equation (35) was corrected.

### Submission & Refereeing History

#### Published as SciPost Phys. 12, 031 (2022)

Resubmission 2106.11245v3 on 11 November 2021
Submission 2106.11245v2 on 21 September 2021

## Reports on this Submission

### Report

The authors have included my suggestions in somewhat different way that I have expected. However, I agree that the resubmitted version has a better readability as compared to the previous one. Comments of the 1st Referee have also lead to the improvement of an overall presentation. Thus, I suggest that the manuscript can be now published in its present form.

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### Report

The authors addressed my questions. I recommend publication.

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