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Statistical Floquet prethermalization of the Bose-Hubbard model

by Emanuele G. Dalla Torre, David Dentelski

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Submission summary

Authors (as registered SciPost users): Emanuele Dalla Torre
Submission information
Preprint Link: scipost_202105_00039v1  (pdf)
Date accepted: 2021-07-20
Date submitted: 2021-05-27 12:21
Submitted by: Dalla Torre, Emanuele
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Atomic, Molecular and Optical Physics - Theory
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
  • Mathematical Physics
  • Quantum Physics
Approaches: Theoretical, Computational


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 driving frequencies. 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 study this model with two semiclassical approaches valid, respectively, at large and weak interaction strengths. In both limits, we compute the heating rates by studying the statistical probability to encounter a many-body resonance, and obtain a quantitative agreement with the exact diagonalization of the quantum model. Our approach demonstrates the relevance of statistical arguments to Floquet perthermalization of interacting many-body quantum systems.

Author comments upon resubmission


REFEREE 1 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.

AUTHORS: We sincerely thank the Referee for her/his assessment of our work and for her/his questions and comments. The Referee report helped us prepare an improved version of the manuscript.

REFEREE 1: 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?

AUTHORS: The Referee correctly points out that the regime of validity of our study does not correspond to the experimental situations. Indeed, our approach is based on a semiclassical approximation, where the particles are considered as classical objects hopping on a one-dimensional lattice. This approach is valid in the regime of strong interactions U≫J. While this regime is theoretically contemplated in Ref. [23] (now [24]), the actual experiment is performed in the opposite regime of U≪J. In the new version of the article, we added an entire section that deals with this regime by using a perturbative analysis of the interaction term U. Our main finding is that the temperature has opposite effects in the two regimes: the heating rate increases with temperature for U≫J and it decreases for U≪J. Hence, in the former case, the heating rate is bounded by the infinite temperature semiclassical result, while in the latter it is bounded by the zero-temperature quantum case.

REFEREE 1: 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

AUTHORS: We thank the Referee for this comment and we sincerely apologize for the many typos in Figure 1 and its caption. We prepared a new version of the Figure, where hopefully all the typos are taken care of.

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

AUTHORS: The suppression of heating is faster than exponential for two main reasons: (1) In the semiclassical approximation, the heating rate is proportional to the probability to find a site with large occupation, Δp=U/Ω. According to our statistical approach, this probability is proportional to the Boltzmann distribution exp⁡(-(E-μN)/T). At large temperatures, the ratio E/T→0, and the heating suppression is entirely due to the ratio μN/T, which tends to a constant. In the limit of infinite temperatures, this leads to the exponential suppression of Eq. (15). At intermediate temperatures, the occupation probability is further suppressed by exp⁡(-E/T)=exp⁡(p^2/2T). This term leads to decays faster than exponential. (2) In the exact diagonalization of the quantum model, the heating rate at large frequencies is suppressed by finite-size effects. If one considers a system of N particle, the maximal value of the particle difference is N, leading to a very fast decay for frequencies larger than NU/ℏ. This point is studied in detail in Appendix D, where we show that the result of the exact diagonalization reproduce the behavior of a large system for Ω<NU/ℏ only. The text now reads: "At intermediate temperatures, the heating rate is additionally suppressed by the fact $Un^2 /(2kBT)$ in Eq.~(8), leading to a faster-than-exponential decay of $\Phi(\Omega)$, see Fig.~1. Hence, Eq.~(15) can be considered as an upper bound of the heating rate at all temperatures".

REFEREE 1: 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.

AUTHORS: We thank the Referee for pointing out these relevant references, which are now mentioned.

REFEREE 1: 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.

AUTHORS: The works mentioned by the referee study situations where the heating rate is suppressed at all temperatures, because of dynamical localization. Following the Referee’s suggestion, we now explain the difference between these works and the present study: "Our semiclassical approach disregards effects associated with quantum coherence. In the case of a single kicked rotor, quantum coherence strongly suppresses heating through the dynamical localization in energy space [37, 38]. Accordingly, it was recently shown that dynamical localization can lead to ergodicity breaking in many-body kicked models, such as coupled rotors [39] and the Bose-Hubbard model [40]. However, as conjectured in Ref. [41], dynamical localization is probably restricted to kicked models and, hence, is not relevant to the present study, where we considered a sinusoidal time dependence. ".

REFEREE 1: 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.

AUTHORS: We have restored the missing hbars, where needed.

REFEREE 1: 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.

AUTHORS: We added a few sentences to explain this point. Is it clear now? "Here, the delta function $\delta(\hbar\Omega-\Delta E)$ imposes the relevant resonance condition. To regularize this function, one needs to take into account the effects of small, but finite, $J/U$: the hopping term in Eq.~(3) transforms the single particle states into ``conduction bands'' of width $\Lambda = 4dJ$. To model this effect, we substitute the delta function in Eq.~(9) by a square function of width $8dJ$, namely …"

REFEREE 1: 8) I am not sure about the exponent in Eq.(14) ... is it \hbar\Omega/U maybe?

AUTHORS: We thank the Referee for this comment, which was now fixed in the text and in the figures legends.

REFEREE 1: 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)

AUTHORS: The Referee is correct, and we have now fixed these problems.   REFEREE 2: This work focuses on interacting lattice bosons, described by the Bose-Hubbard mode, driven by a periodic hopping modulation. This is a problem relevant to recent cold atoms experiments and to the field of Floquet quantum many body systems. The author discusses the linear-response heating rate of the system for high temperatures and large interactions and its dependence from drive frequency. Analytical (statistical) arguments are compared with exact numerical calculations. The results show that, also for this model with unbounded local Hilbert space, the linear response heating rate of the system is exponentially suppressed in drive frequency, with a temperature dependent prefactor (at least in the high-T regime considered here). I think this is an interesting and topical work that deserves publication in SciPost Physics Core. I nevertheless find that the current manuscript could benefit from a better clarification of the points below:

AUTHORS: We thank the Referee for her/his positive assessment of our work and for her/his constructive comments.

REFEREE 2: 1)The basic idea of statistical floquet prethermalization (Eq. 2, Eq. 8) is that the prethermal state is described by a Gibbs state of the Floquet Hamiltonian. In practice, in this work, how is the temperature T evaluated? It seems to me the author uses it as an external parameter, or identifies it as the initial temperature (before the drive is switched on). What is the rationale behind this? It would be useful to further elaborate on this point.

AUTHORS: In the present work, we indeed assume that the system is initially prepared in an equilibrium state at finite temperature. This situation corresponds to the experimental situation, where the optical lattice is, first, adiabatically turned on, allowing the system to reach an equilibrium state with respect to the time-independent part of the Bose-Hubbard model. At a later stage, the periodic pump is turned on and the heating rate is measured. In this situation, we can estimate the heating rate using the temperature of the equilibrium state. In the case of interest, at large driving frequencies, the heating rate is exponentially suppressed and, hence, the temperature varies very slowly and the heating rate is, in turn, roughly constant. After very long times, the heating rate will eventually lead to an increase of the system’s energy and temperature. However, our paper demonstrates that the heating rate is exponentially suppressed at all temperatures and, hence, the system is expected to remain in quasi-equilibrium at all times. We now stress this point: "At large driving frequencies, the heating rate is small and the time-averaged energy of the system is (quasi) conserved. If the system is ergodic, the state of the system can be approximated by the Boltzmann distribution function, […] the temperature $T$ is determined by the initial energy of the system, measured with respect to the average Hamiltonian $H_{\rm av}$".

REFEREE 2: 2) The paper focuses on the linear response heating rate and uses perturbation theory in the drive amplitude to obtain both Eq 9 and Eq16. What happens beyond this regime? Can this statistical approach be generalized? Does the main result of this work still holds? I would expect that for strong drive amplitude the system would be still able to absorb energy even when Omega/U=n>>1 through non-perturbative effects (see also point #5). It would be good if the author could comment in the manuscript on this point and also mention clearly the regime of validity of the present analysis.

AUTHORS: The referee correctly points out that our analytical and numerical calculations are limited to the lowest order in the drive amplitude. While we agree that considering higher orders is very interesting, this calculations goes beyond the scope of the present manuscript. We now highlight this point in the following sentence "For both cases, we compute the heating rate to lowest order in the strength of the periodic drive ($\sim\delta J^2$) and compare it with the exact numerical diagonalization of the model."

REFEREE 2: 3)The comparison with ED shown in the bottom panel of Figure 1 reveals some oscillations of the heating rate which are absent in the statistical approach. Is it clear where do these come from? I suspect they might come from a mechanism of "resonant thermalisation", see point below.

AUTHORS: The statistical approach predicts an oscillating behavior, according to the normalization used for the delta function (in this case, a square “window”). We have now improved the visualization of the statistical result to highlight this point. (We admit that the previous visualization with dots and dotted lines was misleading).

REFEREE 2: 4) In the same figure, I find particularly interesting the deviations at small drive frequencies, discussed in the paper towards its end and attributed to quantum resonances- beyond the current semiclassical approximation- and leading to higher heating rates. This effect has been discussed for example in Phys. Rev. Lett. 120, 197601 (2018) in the context of Fermi-Hubbard model at large interaction, where it has been shown that sweeping the drive frequency across the condition Omega=U (or multiples) could lead to a rapid "resonant" thermalisation and increased heating. The numerical results of this paper seems to indicate a similar mechanism also in the Bose Hubbard, which I think is very interesting.

AUTHORS: We agree with the referee that our approach is analogous to the “resonant thermalization” first discussed in Phys. Rev. Lett. 120, 197601 (2018). We were not aware of this work and we now mention it in the paper. The key difference is that in the case of fermions, the number difference can at most be two. We added the following sentence: "If the maximal occupation of each site is limited to $n_i\le2$, such as in the case of spin-1/2 fermions, the resonant condition can be satisfied only for $n_\Omega=1$ [28]. In contrast, for bosons $n_i$ can be arbitrarily large and energy can be resonantly absorbed at all energies".

REFEREE 2: 5)Related point: given this "resonant" condition, I would then suspect that increasing enough the drive amplitude deltaJ could result in an increased heating rates even at large (resonant) frequencies Omega/U=n>>1...Could the author comment on this point?

AUTHORS: As mentioned above, in this manuscript we focus on the lowest order term in the driving amplitude. However, because we deal with a bosonic system where the occupation difference between neighboring sites can be arbitrarily large, high frequency resonances can be hit even at the lowest order in the drive amplitude.

REFEREE 2: 6)The caption of Figure 1 could be improved, referring to the separate panels and clarifying what has been obtained with the analytical theory and what with exact numerics.

AUTHORS: We thank the Referee for this comment and we sincerely apologize for the many typos in Figure 1 and its caption. We prepared a new version of the Figure, where hopefully all the typos are taken care of.

List of changes

Fig. 1 was revised for clarity.

Added an entire new section dealing with the experimental-relavant regime of small interaction U<<J.
New figures in this section: Fig. 2 - diagrams for t he second and fourth order in the perturbation theory; Fig. 3 - The dependent of the heating rate of U and $\Omega$; Fig. 4 - Temperature dependence of the heating rate.

The Abstract, introduction and discussion were revised accordingly.

Published as SciPost Phys. 11, 040 (2021)

Reports on this Submission

Anonymous Report 2 on 2021-7-19 (Invited Report)


I have read the new manuscript and the reply of the authors to the Referees queries. I am convinced that the authors have made an important effort to address and clarify all the points raised and that the manuscript in the present form is much more clear. For these reasons I am happy to recommend this article for publication in SciPost.

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

Anonymous Report 1 on 2021-5-30 (Invited Report)


I had a positive impression of this work in my first review, as expressed in the report. There were some issues in one of the figures, in some of the explanations and in a few missing citations.
The authors have taken care of all the comments (for example regarding the larger than exponential suppression) and actually they have exceeded my expectations by adding a study for J>>U which lead to Eq.(18).
I can now recommend the paper for publication.

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

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