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Selfconsistent theory of manybody localisation in a quantum spin chain with longrange interactions
by Sthitadhi Roy, David E. Logan
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Sthitadhi Roy 
Submission information  

Preprint Link:  https://arxiv.org/abs/1903.04851v2 (pdf) 
Date accepted:  20191001 
Date submitted:  20190730 02:00 
Submitted by:  Roy, Sthitadhi 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Manybody localisation is studied in a disordered quantum spin1/2 chain with longranged powerlaw interactions, and distinct powerlaw exponents for interactions between longitudinal and transverse spin components. Using a selfconsistent meanfield theory centring on the local propagator in Fock space and its associated selfenergy, a localisation phase diagram is obtained as a function of the powerlaw exponents and the disorder strength of the random fields acting on longitudinal spincomponents. Analytical results are corroborated using the wellstudied and complementary numerical diagnostics of level statistics, entanglement entropy, and participation entropy, obtained via exact diagonalisation. We find that increasing the range of interactions between transverse spin components hinders localisation and enhances the critical disorder strength. In marked contrast, increasing the interaction range between longitudinal spin components is found to enhance localisation and lower the critical disorder.
Author comments upon resubmission
Dear Editorincharge,
We thank you for your editorial recommendation, and the Referee for his/her comments and questions, on our submission entitled "Selfconsistent theory of manybody localisation in a quantum spin chain with longrange interactions". We are pleased to see the strengths of the manuscript appreciated by the Referee, namely concrete analytical results corroborated by numerical results, and clear explanations of the phenomena predicted. We would like to resubmit our modified manuscript, where we believe the points raised by the Referee have been addressed and the text suitably edited in the light of the Referee's helpful report. Before we respond in detail to the Referee's specific technical questions, a few general remarks and responses are in order.
The general concern of the Referee relates to the "the validity of the main results". In order to address this, let us restate the central result of the paper: in a disordered spinchain with longranged interactions, making the longitudinal interactions longer ranged favours manybody localisation (manifest by a parametric lowering of the critical disorder), whereas making the transverse interactions longer ranged has exactly the opposite effect. We present an analytical theory, of a meanfield nature and on the Fock space. The qualitative predictions of the theory are corroborated by the numerics. The numerical results are wholly independent of, and complementary to, the analytical predictions, and that they qualitatively match is we believe a compelling argument for the validity of the main results. Indeed, the Referee also makes a comment concomitant with our results that ``this behaviour can be somehow expected by general argument''. However, our theory goes well beyond the common lore that longranged interactions generally disfavour localisation. As such, we believe that a concrete theory highlighting this qualitative phenomenon is of significance.
Let us now comment briefly on the Referee's points about the \textit{critical} values of $\alpha$ and $\beta$. We would like to clarify that the lines $\alpha=1/2$ and $\beta=1/2$ are not critical lines, but simply represent boundaries of nogo regions for the localised or delocalised phases. The meanfield prediction for the critical surface in the $\alpha$$\beta$$W$ parameter space is described by Eq. (42) (and illustrated in Fig. 2b). For any particular choice of disorder and interaction strengths, the critical boundaries in the $\alpha\beta$ plane lie away from these lines (see e.g. panel c1 of Fig. 2). A more precise statement is that the aforementioned lines are limiting values on the critical $\alpha$ and $\beta$ obtained from the relative scaling with system size of effective disorder and connectivities on the Fock space. Of course, these limiting values are from our meanfield treatment of the problem. However, what stands firm is that there exists a region in $\alpha$$\beta$$W$ parameter space where the infinitetemperature states are localised for infinitesimal disorder and, similarly, delocalised even for arbitrarily strong disorder. This is consonant with longerranged longitudinal and transverse interactions respectively favouring and disfavouring localisation; which is the main result of the work. We have modified the text in the manuscript as well as the caption to Fig. 2 to highlight this clarification.
The Referee also raises questions regarding our ``appeal to continuity'' in the Discussion section (Sec. 6). We point out respectfully that we don't in fact appeal to continuity from any noninteracting model to actually derive our results. Rather, we use it simply to connect to previous results, and to provide an explanation for the apparent discrepancy at small values of $\beta$ in Fig. 4 between the numerical results and the meanfield predictions. The Referee asks why a freefermionic Anderson localised system should remain localised upon adding infinitesimally small but longranged interactions. This is indeed a nontrivial question, but comes under the very umbrella of questions that our manuscript seeks to answer, namely the phase diagram of the model; which we obtain from both the meanfield analysis and the numerical results. As we have emphasised in the manuscript, both methods used have their pros and cons; while one is an analytic but approximate calculation in the thermodynamic limit, the other is exact but affected by finitesize effects. Hence, as is often the case in studies of MBL, while the two sets of results agree qualitatively there can be some tension regarding quantitative specifics. In such a case it is natural to invoke arguments based on physical grounds. Let us briefly clarify the argument in this case.
For simplicity, consider here $\alpha\to\infty$ (extension to finite $\alpha$ is elaborated on in our Response below). It is clear that in the limit of $\beta\to 0$, the system is effectively a disordered freefermionic system with nearestneighbour hoppings, and hence is Anderson localised with vanishing critical disorder strength (in 1D). This is confirmation of the fact that the apparent finite critical disorder strength in Fig. 4 at the smallest value of $\beta$ is a numerical finitesize effect. Second, the rather good agreement between the two sets of results at $\beta \gtrsim 1$ (Fig. 4) suggests that there does exist a localised phase in this regime, with the critical disorder growing with $\beta$ but saturating to a finite value as $\beta\to\infty$. So we are left with the contentious regime of small but finite $\beta$. Since we find no evidence of any nonmonotonicity as a function of disorder strength and $\beta$, one can reasonably speculate only two scenarios: (i) the critical disorder vanishes at a finite value of $\beta$; (ii) the critical disorder vanishes as $\beta \to 0$. The meanfield treatment predicts the former; although the precise value of $\beta$ at which the critical disorder vanishes is an open question that naturally invites future work. We have now significantly edited the text in the Discussion section to clarify these points.
We reply below to the technical questions of the Referee pointwise. We thank the Referee for his/her comments, which undoubtedly have improved the manuscript. We hope that we have addressed the points satisfactorily, and that the modified manuscript is suitable for publication in SciPost Physics.
Yours sincerely, S. Roy and D.E. Logan
Response to Referee's specific questions

The model considered in the work, described by the Hamiltonian Eq. (1), conserves total magnetisation, $M_z=\sum_\ell\sigma^z_\ell$. Hence, each of the sectors with fixed $M_z$ can be considered independently. We choose to work in the $M_z=0$ sector as it has the largest Fockspace dimension and dominates the entire Fock space made up of all $M_z$ sectors in the thermodynamic limit. We would like to mention that this is very much the conventional case considered in the literature on manybody localisation in systems with conserved magnetisation (or equivalently particle number). While we have not shown it explicitly, we add that the scaling with system size of effective disorder and connectivities on the Fock space, stay the same for any $M_z$ sector whose Fockspace dimension is exponentially large in the system size. We have now mentioned this explicitly right after introducing the model.

In the shortranged limit of $\alpha\to\infty$ and $\beta\to\infty$, the result in Eq. (42) does indeed reproduce the results of the previous meanfield analysis of a shortranged model presented in Ref. [29]. The critical disorder for the shortranged case obtained from this meanfield analysis is somewhat lower than that obtained in largescale exact diagonalisation studies (c.f. Ref [17]), which is quite expected from a meanfield treatment. The quantitative comparison between the meanfield predictions and exact diagonalisation results presented in the present work is in our view remarkably good for intermediate values of $\alpha$ and $\beta$. The reason for this can be argued to be the much higher connectivity (both in real space and Fock space) of the longranged models compared to nearestneighbour models, which makes the former more suitable for a meanfield analysis.

The model considered here does indeed have a freefermionic description for (i) $\alpha\to\infty$ and $J_z=0$ for any value of $\beta$ and $J$ (ii) $\alpha\to\infty$ and $\beta\to0$ for any value of $J_z$ and $J$ Note that the second of the two cases was used to obtain the freefermionic representation in Eq. (48).

Let us now return to the issue of continuity in the phase diagram briefly mentioned before, and elaborate on it with reference to a figure that can be viewed at https://bit.ly/2LOnDvX
We first consider the $\beta=0$ line. In this limit, the longitudinal interaction drops out of the Hamiltonian as a constant owing to magnetisation conservation. On this line, $\alpha\to\infty$ is trivially localised due to Anderson localisation, but that localisation persists for finite values of $\alpha$. This is due to the subtle fact that the effective hoppings, although longranged in real space, have no inherent randomness. Such systems are known (see Ref. [6366]) to exhibit Anderson localisation except for a measure zero set of singleparticle eigenstates at the edges of the spectrum. Anderson localisation at $\beta=0$ is hence established and is denoted by the red line in the figure linked above. Second, the rather good match between the meanfield predictions and the numerical results at higher values of $\beta$ (manuscript Fig. 4), suggest that the critical disorder grows with $\beta$, eventually saturating as $\beta\to\infty$ (note that the presence of a manybody localisation transition in nearestneighbour models is well established). This yields a partial phase boundary shown by the solid black curve. Thus, for small but finite values of $\beta$, the two likely possibilities are that the critical disorder either goes to zero at a finite $\beta$ (the blue dashed line), or it goes to zero only as $\beta\to 0$ (green dotted line). The precise critical value of $\beta$ may or may not be captured accurately by the meanfield treatment and constitutes a topic of future work. However, what still holds is that increasing the range of longitudinal interactions (decreasing $\beta$) favours localisation, as manifest in the decrease of the critical disorder strength.
List of changes
 The text following Eq.(1) has been modified to clarify that total magnetisation is conserved in the model and that we work in the zeromagnetisation sector. Infinite temperature traces are then defined accordingly.
 The caption to Figure. 2 and the pertinent text towards the end of Section. 4 has been edited to better explain the meaning of the $\alpha=1/2$ and $\beta=1/2$ lines in the phase diagram. Emphasis has been laid on clarifying that the lines do not represent critical lines, and boundaries of nogo regions for localised or delocalised phases is an accurate description of them.
 Section 6 (Discussion) has been substantially revised in the light of the Referees's concerns about our appeal to continuity, and his/her question about why localisation is expected to persist upon addition of weak longranged interactions to an Anderson localised system. In particular, the text has been modified
(i) to clarify that the noninteracting fermion limit of the problem has longranged but "nonrandom" hoppings, and its consequences have been discussed
(ii) to explain better why the $\beta=0$ line is localised for all $\alpha$ and $W$ in 1D.
(iii) to add a new discussion on various quantitatively different scenarios that can arise in the $\beta$$W$ plane, and how they are the same qualitatively.
 References have been updated, and a couple of new references have been added.
Published as SciPost Phys. 7, 042 (2019)
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The authors replied to all my questions and fully addressed the points that were not completely clear. Clearly the validity of mean field approach and finite size numerics is always a problematic issue but this study satisfies all scientific criteria. I recommend publication as it stands.