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Noiseinduced transport in the AubryAndréHarper model
by Devendra Singh Bhakuni, Talía L. M. Lezama and Yevgeny Bar Lev
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Authors (as registered SciPost users):  Yevgeny Bar Lev · Devendra Singh Bhakuni · Talía L. M. Lezama 
Submission information  

Preprint Link:  scipost_202307_00026v1 (pdf) 
Date submitted:  20230720 16:07 
Submitted by:  Bhakuni, Devendra Singh 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We study quantum transport in a quasiperiodic AubryAndréHarper (AAH) model induced by the coupling of the system to a Markovian heat bath. We find that coupling the heat bath locally does not affect transport in the delocalized and critical phases, while it induces logarithmic transport in the localized phase. Increasing the number of coupled sites at the central region introduces a transient diffusive regime, which crosses over to logarithmic transport in the localized phase and in the delocalized regime to ballistic transport. On the other hand, when the heat bath is coupled to equally spaced sites of the system, we observe a crossover from ballistic and logarithmic transport to diffusion in the delocalized and localized regimes, respectively. We propose a classical master equation, which captures our numerical observations for both coupling configurations on a qualitative level and for some parameters, even on a quantitative level. Using the classical picture, we show that the crossover to diffusion occurs at a time that increases exponentially with the spacing between the coupled sites, and the resulting diffusion constant decreases exponentially with the spacing.
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This paper seems to have solid and original results. They are not groundbreaking, but will be of interest to some other researchers working in this general area (such as myself).
I suggest a few changes to help with the clarity and completeness of the presentation:
The classical approximations that are developed in section IIIC appear to be only for the localized phase, which is fine. But this restriction should be explicitly stated where this is mentioned in the abstract, discussion and in section IIIC, since other parts of the paper also treat the critical and ballistic phases.
The study of the entanglement entropy uses an initial random product state. This initial state should be described more completely. Since another part of the paper uses an initial mixed state, here it should be said explicitly that an initial pure state is used. The ensemble from which this random pure product state is drawn should be explicitly stated. I suspect they mean initial local eigenstates of the occupation, randomly occupied or empty with equal probabilities for these two possibilities, but this must be explicit. Or are the initial local pure states uniformly sampled from the local Bloch sphere?
The caption of Fig. 2 should say what bath and couplings are being used (I did not find those details in the main text either).
Fig. 6(b), especially the lower trace, shows a possible odd/even effect vs lambda/2. Or is this all within the error bars? No error bars are shown in any of the figures. They should be roughly estimated and indicated, or where they are not visible this should be stated. This is crucial and required for any numerical work where the errors can be estimated (as they can here).
Strengths
1)Numerical results are compared to phenomenological classical master equation, showing excellent agreement
2)Qualitative interpretation of the results clearly explained and discussed
Weaknesses
1)Incremental with respect to previous literature, lack of major advance/breakthrough
2)Introduction and motivations can be improved
3)Results on entanglement entropy only for one type of systembath coupling
Report
This work studies transport in the AubryAndre'Harper (AAH) model in presence of a Markovian dephasing bath. Different types of coupling geometries are considered. The focus is on spreading of density excitations (encoded in the root mean squared displacement) and partially on entanglement entropy.
The numerical results, obtained through a unitary unravelling of the Lindblad master equation, are well presented and interpreted through a classical master equation. The Authors find that when the coupling to the bath affects a finite portion of the chain transport at short times becomes diffusive, independent on disorder, while at longer time scales (outside of the zone of the influence of the bath) the usual transport behavior of the AAH is restored.
On the other hand, when the coupling to the bath is distributed among equally spaced sites diffusion emerges at long times.
I find this work interesting for those working in the field of disorder and dynamics. However I do not see the case for publication in Scipost Physics, based on its acceptance criteria. In particular I do not see a major result/breakthrough here. The work appears to be incremental with respect to previous literature (in particular Ref 58 from a subset of the authors, where similar analysis was done for the one dimensional Anderson model in presence of a local bath). I think the manuscript could be published in SciPostCore, provided the Authors address the points below.
Requested changes
1)Introduction and general motivations could be improved: why precisely the Authors are considering this type of coupling to the bath? What is the connection with MBL?Shouldn't thermal inclusions be somehow randomly distributed ?
2)Related question: in the conclusions the Authors mention their results provide an upper bound on transport in MBL. Why? How this comes about? Can the Authors elaborate on this point?
3)While the unravelling of the density matrix in Eq.5 is quite clear, it's less obvious how one should interpret Eq. 6, particularly given the Authors consider directly an infinite temperature density matrix. Is this correlation function averaged over the noise? Should this be equivalent to computing the same object through Linbdlad master equation because of some quantum regression theorem? It seems that some clarification is needed here.
4)Results on entanglement entropy are presented only in Fig 2, for the local (singlesite) bath coupling? Why? As it is, this part seems quite decoupled from the rest of the article.
5)I am a bit surprised that a timestep dt=0.1 (in units of the hopping I guess?) is enough to have converged data and even more that averaging over N=10 trajectories is enough to converge to the Lindblad result. Could the Authors provide some supporting evidence?
Author: Devendra Singh Bhakuni on 20231106 [id 4096]
(in reply to Report 1 on 20230827)
We thank the Referee for their thorough review of the manuscript and recommendation for publication in SciPost Physics Core. It is important to emphasize that Ref.[58] was focused on a different question: How does a local coupling to a heat bath affect transport in a localized model? In contrast, our current work presents a comprehensive analysis of various types of couplings for the bath, motivated by both theoretical and experimental questions. Moreover, it provides a broader theoretical picture, which we verify using an experimentally relevant model which exhibits different transport regimes. We show that using the configuration of the couplings, transport can be tuned to logarithmic, subdiffusive, or diffusive. We provide a qualitatively good estimate of the diffusion coefficient in the diffusive regime.
Besides the broad implication on transport in open quantum systems, our study also captures the physics of ergodic bubbles in the manybody localized phase. It sheds light on the nature of transport in this phase. It is worth mentioning that similar setups to study the effect of ergodic bubbles have been previously proposed theoretically (Phys. Rev. Lett. 119, 150602) and experimentally (Nature Physics 19, 481–485 (2023)) but with no consideration of quantum transport.
Given the above, our study goes beyond Ref. [58] and is directly relevant to the fields of quantum transport in the presence of dephasing, Anderson localization, and manybody localization. Moreover, it has direct theoretical and experimental applications. As such, we believe it matches the criteria of SciPost Physics.
The referee writes:
Introduction and general motivations could be improved: why precisely the Authors are considering this type of coupling to the bath? What is the connection with MBL? Shouldn't thermal inclusions be somehow randomly distributed?
Our response: The main goal of our work was to consider the effect of the spatial configuration of the coupling to the heat bath on quantum transport, which is a question of great interest in the field of open quantum systems. As such, we considered types of couplings relevant to several experimental and theoretical settings. Local coupling can be relevant to systems of point contacts (Phys. Rev. Lett. 92, 156801, 2004, Phys. Rev. B 85, 155327 (2012)) and the thermalization of nonergodic chains by a single ergodic bubble (see, ((Phys. Rev. Lett. 119, 150602), for theoretical study and Nature Physics 19, 481–485 (2023) for experimental)). Periodic coupling is relevant, for example, when a system is coupled to an underlying periodic lattice/crystal (Phys. Rev. Lett. 125, 180605). In the context of manybody localization, the ergodic bubbles can be modeled by coupling to an external bath. The positions of the ergodic bubbles are indeed random for disordered models (see the Appendix of our work for results on randomly distributed couplings). However, for the AubryAndre model that we consider in this work, the location of the ergodic bubbles is deterministic, highlighting the importance of studying other configurations.
We have considerably reworked the introduction, providing a more general motivation for our work.
The referee writes:
Related question: in the conclusions the Authors mention their results provide an upper bound on transport in MBL. Why? How this comes about? Can the Authors elaborate on this point?
Our response: In the manybody localized phase, the ergodic bubbles serve as a \textit{finite} thermal bath and induce quantum transport. We model those rare regions/ergodic bubbles by an \textit{infinite} dephasing bath and consider the simplest case of a noninteracting localized system. Unlike the ergodic bubbles, which cool off during the dynamics (and therefore can stop being ergodic), the system does not affect the heat bath. As such, our model provides a (nonrigorous) upper bound on ergodic bubblesinduced transport in a manybody localized system. We have clarified this point in the text.
The referee writes:
While the unravelling of the density matrix in Eq.$5$ is quite clear, it's less obvious how one should interpret Eq.$6$, particularly given the Authors consider directly an infinite temperature density matrix. Is this correlation function averaged over the noise? Should this be equivalent to computing the same object through Linbdlad master equation because of some quantum regression theorem? It seems that some clarification is needed here.
Our response: The referee is right that for a system coupled to a bath, the evolution of the correlation function is obtained using the quantum regression theorem. This allows us to compute the correlation function directly in the stationary state of the Lindblad equation, which for Hermitian jump operators is the infinite temperature. For the problem at hand, it is much more efficient to use the unraveling of the dynamics, which is unitary and allows computation of the correlation function in a standard fashion, as explained after Eq.~(6). We have added the appropriate clarification in the text.
The referee writes:
Results on entanglement entropy are presented only in Fig $2$, for the local (singlesite) bath coupling? Why? As it is, this part seems quite decoupled from the rest of the article.
Our response: We agree that this part looks decoupled from the rest of the article. Since we are primarily focused on the quantum transport, we have moved the analysis of the entanglement entropy from the main text to the Appendix.
The referee writes:
I am a bit surprised that a timestep $dt=0.1$ (in units of the hopping I guess?) is enough to have converged data and even more that averaging over $N=10$ trajectories is enough to converge to the Lindblad result. Could the Authors provide some supporting evidence?
Our response: The unitary unraveling used in the work is exact for any $dt$ for Hermitian Lindblad jump operators. Increasing the number of trajectories suppresses the statistical noise, and for our purpose, averaging over $10$ trajectories along with $10$ different choices of the onsite potential was enough to suppress the statistical noise. A relationship between the unraveling and the Lindblad master for any choice of $dt$ is provided in Phys. Rev. Lett. 118,140403 (2017). For completeness, we plot (see attached file) the root mean squared displacement dynamics for the case where the bath is coupled to all the sites. It can be seen that the qualitative features of the dynamics are independent of the choice of $dt$.
Author: Devendra Singh Bhakuni on 20231106 [id 4097]
(in reply to Report 2 on 20230906)We thank the Referee for the positive comments and a constructive report. We have incorporated all of the suggestions.
The referee writes:
Our response: As shown in Appendix A, the classical model applies to any system driven by white noise. However, for delocalized models, by definition, the states $\phi_{\alpha}(k)$ are not ordered by the "centerofmass" of the states, such that the spatial structure of the rates is lost. Moreover, the rates $\Gamma_{\alpha\beta}$ are all roughly equal. Therefore, for such models, transport is expected to transition to diffusion on timescale $\gamma^{1}$. We have added a clarifying statement in Section IIIC.
The referee writes:
Our response: We have added a more detailed description of the initial state.
The referee writes:
Our response: We consider $\gamma=1$. We have now added this in the revised manuscript.
The referee writes:
Our response: We are unsure about the even/odd effect. The error bars in our simulations are less than the line width. We have now mentioned this.