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Particle fluctuations and the failure of simple effective models for many-body localized phases
by Maximilian Kiefer-Emmanouilidis, Razmik Unanyan, Michael Fleischhauer, Jesko Sirker
This is not the current version.
|As Contributors:||Maximilian Kiefer-Emmanouilidis|
|Arxiv Link:||https://arxiv.org/abs/2108.03142v1 (pdf)|
|Date submitted:||2021-08-12 13:29|
|Submitted by:||Kiefer-Emmanouilidis, Maximilian|
|Submitted to:||SciPost Physics|
We investigate and compare the particle number fluctuations in the putative many-body localized (MBL) phase of a spinless fermion model with potential disorder and nearest-neighbor interactions with those in the non-interacting case (Anderson localization) and in effective models where only interaction terms diagonal in the Anderson basis are kept. We demonstrate that these types of simple effective models cannot account for the particle number fluctuations observed in the MBL phase of the microscopic model. This implies that assisted and pair hopping terms---generated when transforming the microscopic Hamiltonian into the Anderson basis---cannot be neglected. As a consequence, it appears questionable if the microscopic model possesses an exponential number of exactly conserved local charges. If such exactly conserved local charges do not exist, then particles are expected to ultimately delocalize for any finite disorder strength.
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Anonymous Report 2 on 2021-9-28 (Invited Report)
1) Thorough numerical analysis on three related models.
2) New questions are posed about local conserved charges in many-body localisation.
1) Some points need more detailed description.
In this paper one-dimensional quantum models are considered in the presence of disorder and their dynamical properties are studied through numerical exact diagonalization. In particular the entanglement entropy, the number entropy and the time-averaged particle number fluctuations are investigated and compared for the following three models.
(a) The t-V model, with fermionic hopping, nearest-site interaction (V) and a random potential term (D)
(b) An effective model, which is obtained from (a) by keeping interaction terms which are diagonal in the V=0 basis. This model is expected to be accurate in the small-V regime.
(c) The non-interacting (V=0) version of the model (a), which describes Anderson localisation.
According to the numerical results the entanglement entropy for (a) and (b) for small V has the same type of logarithmic increase in time, while (c) has a saturated value. On contrary, the time-dependence of the Hartley entropy and the number entropy is different for the three models. Similar conclusion is obtained for the particle number fluctuations, which increases for (a), but saturates for the effective model (b). From this the authors conclude that during the transformation with the Anderson basis the effect of the off-diagonal terms can not be neglected. These either renormalise the orbitals in the Anderson model with local conserved charges, or these charges became non-local.
This paper represents the continuation of a series of papers of the authors about many-body localization [28-31]. The subject of the present paper is interesting, the obtained results could give new impulses to clarify this phenomena more deeply. The paper is basically well written. I suggest publication of this work after the authors have successfully addressed the following points.
i) The authors should comment on the numerical methods they used. In Fig.11 I expect free-fermionic methods were used.
ii) A comment on the necessary number of disorder realizations would be in order.
iii) For the number entropy and the particle number fluctuations for model (a) double-logarithmic time-dependence fit is used. Such type of scaling works at infinite disorder fixed points (Phys. Rev. B85, 094417 (2012), Phys. Rev. B 93, 205146 (2016)). Can the authors explain such type of scaling in this problem?
iv) Somewhat related point: in Phys. Rev. B102, 100202(R) (2020) the double-logarithmic increase of the number entropy in the thermodynamic limit is debated. The authors should comment on this.
v) Small point: a few typos should be fixed. see an an, ...
Anonymous Report 1 on 2021-9-10 (Invited Report)
1- This work probes in a more thorough way an approximation scheme previously introduced in the literature for conserved quantities in interacting one-dimensional systems in a disordered potential.
2- The work looks at interesting quantities to characterise the dynamical behaviour of the system.
1- Some statements are formulated in a confusing way, or need to be better motivated.
2- Some context and account of previous literature is missing.
This paper deals with the numerical study of particle number fluctuations in finite-size fermionic systems with disorder. The Authors consider three different models:
(i) a canonical model for Many-Body Localization (MBL), with pairwise interacting fermions in a random potential;
(ii) the non interacting, Anderson version of the model;
(iii) an approximate version of (i), which is expected to be provide good results in the weakly-interacting regime.
They compare the dynamical and the time-averaged behaviour of the particle number fluctuations, measured via the von-Neumann and Hartley number entropy, and the number variance. They argue that the approximate model (iii), while reproducing fairly accurately the dynamics of the half-system entanglement entropy of model (i), it does not account for the behaviour of its particle number fluctuations. Based on this observation, they comment on the emergent integrability of the model (i) and on its localization properties. This manuscript belongs to a series of numerical works by the same Authors, Refs. [28-31], in which the occurrence of a truly MBL phase in the thermodynamic limit is considered critically.
I believe that this work is potentially interesting, as it probes certain approximate schemes previously introduced [32,34] by looking at particle number fluctuations, which are interesting quantities to characterize the dynamics of interacting systems in finite-size. Nevertheless, I find that the wording and the presentation of the numerical results are sometimes confusing: I would therefore strongly suggest to the Authors to better motivate certain claims (or eventually to soften them) before publication, in order to “Provide citations to relevant literature in a way that is as representative and complete as possible” and to “summarize the results with objective statements on their reach and limitations”, as required by the SciPost criteria. More detailed comments are given below.
Comments on the presentation of the results:
1) In the abstract and throughout the paper, the Authors insist on the fact that MBL systems are characterised by the presence of an exponential number of local charges. I believe that this statement is confusing (I interpret it as “exponentially-many in the system size L”). Integrability in MBL is usually understood in terms of linearly-many local conserved quantities, which are obtained as quasi-local rotations of the L conserved quantities associated to the quadratic, non-interacting part of the Hamiltonian. Could the Authors specify what they mean by “exponential number” here?
2) In Sec. 2 the Authors define the different Hamiltonians (i-iii) and write that they are used to determine the time evolution. I would clarify here how this time evolution is determined: is it via exact diagonalisation of the Hamiltonian and explicit calculation of the evolution operator? Given that the Authors consider quenches from a fixed pure initial state, could for example Krylov methods be used to have access to larger system sizes (L=18)?
3) In Figs. 4 (d) and 5 (d) the Authors show the scaling with system size of the Hartley number entropy and of the number variance for model (iii), and claim that the curves are overlapping for the two larger system sizes. I suggest to modify (diminish) the y-range of these plots, to make this sister-size independence clearly evident from the plot. The same for Figs. 4 (c) and 5 (c).
4) In Sec. 3.1 the Authors claim a different L dependence of the saturation values of the number entropy in models (i) and (iii): is it correct to say that these conclusions rely on the double-logarithmic fit of the number entropy behaviour in time? Is there any analytical argument justifying this scaling, similar to what one has for the log(t) behaviour of the entanglement entropy?
Comments on the scientific context and interpretation of the results:
5) The approximate model (iii) considered in this work is obtained neglecting the assisted and pair hopping-terms that appear in the Hamiltonian of model (i), when the latter is expressed in the Anderson basis. The numerical data presented in this work indicate that these terms play a relevant role when considering particle number fluctuations and therefore can not be neglected even when the interaction term is not too strong, at least in these finite sizes. Based on this, the Authors also claim: “as a consequence, it appears questionable if the microscopic model possessed an exponential number of exactly conserved local charges”. This implication is not clear to me. At the level of the local conserved quantities , the approximation scheme discussed in this work corresponds to projecting the conserved operators onto a small subspace, that is the kernel of the commutator . This is a quite substantial approximation to the “true” conserved quantities: its failure in reproducing certain features of the dynamics is in my view not sufficient to question overall the existence of local (non-approximated) conserved operators, and thus to question MBL integrability. Could the Authors make their argument more stringent?
6) In the Introduction, it is written “if one wants to take into account the normalisation of the orbitals [….], then one has to ensure that this renormalisation does not ultimately lead to delocalised orbitals”. Similarly, in the Conclusion the Authors write “The question then is, whether […] such a normalization ultimately leads to charges becoming non-local”. In fact, this question has been the subject of several previous works, cited in the introduction of the manuscript. In particular, Ref.  claimed a proof of the fact that the renormalisation procedure does not ultimately lead to delocalized orbitals, at least for a particular class of locally-interacting 1d spin models. The same is claimed based on approximate perturbative arguments for fermonic Hamiltonians of the type discussed in this manuscript. I believe that this previous work should be accounted for when raising this point, even more so if the Authors have reasons to challenge those results or interesting intuitions on possible loopholes in the proof or in the perturbative arguments: at the moment, the way the discussion is formulated gives the impression that this issue has remained completely unadressed in the literature.
7) I would also suggest to be more explicit one the fact that this manuscript is dealing with a specific approximation, the one leading to model (iii), and does not actually analyse particle number fluctuations for generic models of the form (1), introduced in Ref.  as effective models for MBL systems. In particular, on p. 3 the Authors write “we will argue that the qualitative findings are generic”: do they mean with this that the conclusions can be extended to a more general class of effective models? Could they point out where is this argued in the manuscript?
8) The interpretation of the finite-size data on the number entropy presented by the Authors in this manuscript and in the previous publications [28-31] has been the subject of debate recently, in particular for what concerns the implications for the thermodynamics limit. In [Luitz et al, PRB 102, 100202(R), 2020] it is claimed that the double-logarithmic behaviour of the averaged number entropy is a transient effect, and that the distribution of the number entropy at infinite-time has tails that decay exponentially fast with L. Could the Authors comment on this point?
I would ask to the Authors to address the points 3,5,6,7 above by adjusting the manuscript.
I would ask them to comment on points 1,2,4,8 in the manuscript, if appropriate.