## SciPost Submission Page

# Statistical Floquet prethermalization of the Bose-Hubbard model

### by Emanuele G. Dalla Torre

### Submission summary

As Contributors: | Emanuele Dalla Torre |

Arxiv Link: | https://arxiv.org/abs/2005.07207v4 (pdf) |

Date submitted: | 2020-06-18 |

Submitted by: | Dalla Torre, Emanuele |

Submitted to: | SciPost Physics Core |

Discipline: | Physics |

Subject area: | Quantum Physics |

Approaches: | Theoretical, Computational |

### Abstract

The manipulation of many-body systems often involves time-dependent forces that cause unwanted heating. One strategy to suppress heating is to use time-periodic (Floquet) forces at large frequencies. In particular, for quantum spin systems with bounded spectra, it was shown rigorously that the heating rate is exponentially small in the driving frequency. Recently, the exponential suppression of heating has also been observed in an experiment with ultracold atoms, realizing a periodically driven Bose-Hubbard model. This model has an unbounded spectrum and, hence, is beyond the reach of previous theoretical approaches. Here, we develop a semiclassical description of Floquet prethermal states and link the suppressed heating rate to the low probability of finding many particles on a single site. We derive an analytic expression for the exponential suppression of heating valid at strong interactions and large temperatures, which matches the exact numerical solution of the model. Our approach demonstrates the relevance of statistical arguments to Floquet perthermalization of interacting many-body quantum systems.

###### Current status:

### Submission & Refereeing History

## Reports on this Submission

### Anonymous Report 1 on 2020-7-12 Invited Report

### Strengths

The paper gives a clear explanation of the exponential suppression of heating rate with driving frequency for systems whose Hamiltonian is not bounded by an energy scale J. Well ... given the presence of number conservation, the local terms of the Hamiltonian are actually bounded, but 1) not in the thermodynamic limit and 2) from Eq.(1) one would not be able to see the 1/T dependence of the heating rate.

So this paper brings very interesting insight into the problem and significantly pushes forward the field.

### Weaknesses

Fig.1 is not well described.

The explanation of why a high-temperature analysis is relevant is missing.

A more thorough overview of the field of periodically driven systems could be given (see Report).

### Report

I have read with great interest this nice work by Dalla Torre. The paper is fairly well written and the results are interesting and relevant to most current theoretical and experimental research.

The paper gives a clear explanation of the exponential suppression of heating rate with driving frequency for systems whose Hamiltonian is not bounded by an energy scale J. Well ... given the presence of number conservation, the local terms of the Hamiltonian are actually bounded, but 1) not in the thermodynamic limit and 2) from Eq.(1) one would not be able to see the 1/T dependence of the heating rate.

This paper follows fairly naturally from [21], however it is brings enough novelty and insight to deserve a high profile publication.

There are some clarifications I would like to have though:

1) Is an high temperature T analysis relevant to the experiment [23] ?

Is it because the driving first brings the state far from low temperature and then, due to what described in this work, the suppression of heating occurs?

2) Fig.1 is a bit messy:

The caption is not clear. What are the two different panels about? Is panel (b) just an enlarged version of panel (a)? The why there seems to be used 2 different values of J/U? An why in the legend you write Eq.(13)? I think it would be (15) now. Maybe the figure was prepared when Eq.(15) was actually in position (13).

Also it is not clear where the continuous lines come from. Are they derived from (16)?

Furthermore the caption contains 2 typos:

- Caption of Fig.1 -> J/U = 0_05. instead of 0.05 I think ...

- Also in the same caption Eq. (15 no closing parenthesis

3) The suppression observed in Fig.1 seems to be larger than exponential. Any understanding of that?

4) By reading the literature on periodically driven systems, and heating, I cannot avoid incurring in works of A. Eckardt whom however is not cited at all in this work. It is quite surprisiping to me as, for example, he is a co-author also of this experimental work PRL 119, 200402 (2017), and the author of an important review in the field.

5) To give a more comprehensive picture of the field, I also find relevant to mention the works PHYSICAL REVIEW E 97, 022202 (2018), PHYSICAL REVIEW B 101, 064302 (2020) and related ones. This would help a reader.

6) until before Eq.(5) you have hbar and then in 9 you don't. Also the units for Phi should be energy over hbar. Since delta function has units of 1/energy, maybe all you need is to divide the right-hand side by hbar. Overall the use of hbar is a little inconsistent.

7) This is a comment to improve the clarity of the derivation. The author considers corrections to \delta(\Omega - \Delta E) due to tunneling J. Can we have an idea of why only using the corrections in this function compared to the probabilities P in Eq.(8) is a reasonable approach? I think this could help the reader.

8) I am not sure about the exponent in Eq.(14) ... is it \hbar\Omega/U maybe?

9) There may be a factor 1/2 in Eq.(15) coming from the +1 in z^{\Omega/U +1}, but maybe I am wrong.

some typos

- Page 1, second column -> "This effect was explainED in Ref. [21]"

- Put together [23] [24].

- "more than one particleS" remove s (I think)

### Requested changes

See Report