# Many-body localization in the Fock space of natural orbitals

### Submission summary

 As Contributors: Wouter Buijsman Arxiv Link: http://arxiv.org/abs/1712.06892v2 Date submitted: 2018-04-12 Submitted by: Buijsman, Wouter Submitted to: SciPost Physics Domain(s): Theor. & Comp. Subject area: Quantum Physics

### Abstract

We study the eigenstates of a paradigmatic model of many-body localization in the Fock basis constructed out of the natural orbitals. By numerically studying the participation ratio, we identify a sharp crossover marking the onset of eigenstate localization at a disorder strength significantly below the many-body localization transition. We repeat the analysis in the conventionally used computational basis, and show that many-body localized eigenstates are much stronger localized in the Fock basis constructed out of the natural orbitals than in the computational basis.

###### Current status:
Has been resubmitted

### List of changes

Besides the changes mentioned in the replies to the referee reports, we have made the following changes:

- We have re-organized Section 4 in view of the new content added.
- We have updated Ref. [26] of the revised manuscript from the arXiv to the published version, added Ref. [30] of the revised manuscript, and removed Ref. [21] of the original manuscript as the relevant content is also covered in Ref. [7] of the original manuscript.
- We made several minor changes in the text and graphics with the aim of increasing clarity and/or readibility.

### Submission & Refereeing History

Resubmission 1712.06892v3 (31 May 2018)
Resubmission 1712.06892v2 (12 April 2018)
Submission 1712.06892v1 (20 December 2017)

## Invited Reports on this Submission

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### Strengths

1- Interesting and timely subject
2- interesting approach to Fock space localization

### Weaknesses

1- The authors addressed in their new version the comments of the referees. In
particular, they included more details on the construction of the Fock space,
although I think it would be beneficial for the reader to include a definition
of $|\psi_i^{(n)}\rangle$ in terms of an equation, to clarify that the creation and
annihilation operators $d_i^{(\dagger)}$ are used. Also a clearer definition of
$|\psi^{(0)}\rangle$ would be useful.

2- I think it should in general be expected that the quantity $P^{(n)}$ does not
show a sharp signature of the MBL transition, since Slater determinants in terms
of natural orbitals only become good approximations of eigenvectors in the limit
of large disorder, where one expects $P^{(0)}=1$.

3- Fig. 4 shows not the ratio of the typical PR with the dimension of the Hilbert
space, illustrating strong localization in the Fock basis at strong disorder.
The authors observe a crossing of curves corresponding to different system
sizes and interpret this as an indication for a lower critical disorder
(compared to $W_c\approx 3.6$ in the literature). However, this interpretation
ignores a very strong drift of the crossings of consecutive system sizes towards
higher disorder and I think this statement should therefore be removed. One can
not seriously conclude a smaller critical disorder for $L\to \infty$ from the presented data.

4- Minor remark: Just below Eq. (1), "commutator" should be changed to "anticommutator".

### Report

1- The authors now added errorbars to all results based on a jackknife resampling,
showing that fluctuations in particular for $L=16$ in Fig. 3 are of statistical
nature and of the order of the size of the errorbars.

2- The Histograms for the $\log_{10}(PR)$ shown in Fig. 5 demonstrate that the peak in
the variance at intermediate disorder originates from broad distributions,
although the distributions are not bimodal. This is consistent with what is
observed e.g. in the entanglement entropy. Interestingly, the broadest
distributions seem to be at slightly different positions in the two bases (lower
disorder in the Fock basis). Maybe the authors could plot at least the L=16
result in Fig 3 for both bases in both panels for comparison.

### Requested changes

1- Add explicit definitions of $|\psi_i^{(n)}\rangle$ in terms of equations.
2- Remove statement that the data suggests a lower value of $W_c$, discuss instead drifts with system size.

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

### Anonymous on 2018-05-31

(in reply to Report 1 on 2018-05-24)

We are grateful to the referee for the positive report and the useful suggestions to improve the manuscript. We reply to the points raised below in the order in which they appear in the report.

${\textbf{Point 1}}$
==================================================
The authors addressed in their new version the comments of the referees. In particular, they included more details on the construction of the Fock space, although I think it would be beneficial for the reader to include a definition of $| \psi_i^{(n)} \rangle$ in terms of an equation, to clarify that the creation and annihilation operators $d_i^{(\dagger)}$ are used. Also a clearer definition of $|\psi^{(0)} \rangle$ would be useful.
==================================================

In the revised manuscript, we have followed this suggestion.

${\textbf{Point 2}}$
==================================================
I think it should in general be expected that the quantity $P^{(n)}$ does not show a sharp signature of the MBL transition, since Slater determinants in terms of natural orbitals only become good approximations of eigenvectors in the limit of large disorder, where one expects $P^{(0)} = 1$.
==================================================

To avoid the suggestion that $P^{(n)}$ should show a sharp signature of the MBL transition, we have replaced the sentence "Interestingly, no clear $\ldots$ the MBL transition'' by "No clear signatures of the MBL transition can be observed, and on average eigenstates seem to remain localized at disorder strengths even below the MBL transition.'' in the revised manuscript. However, we would like to stress that the succesful use of the occupation discontinuity $\Delta n$ as a probe for the MBL transition in $\it{e.g.}$ Ref. [23] suggests that the structure of eigenstates in the Fock basis changes qualitatively across the MBL transition, which one might expect to be reflected in $P^{(n)}$ in some (probably non-trivial) way.

${\textbf{Point 3}}$
==================================================
Fig. 4 shows not the ratio of the typical PR with the dimension of the Hilbert space, illustrating strong localization in the Fock basis at strong disorder. The authors observe a crossing of curves corresponding to different system sizes and interpret this as an indication for a lower critical disorder (compared to $W_c \approx 3.6$ in the literature). However, this interpretation ignores a very strong drift of the crossings of consecutive system sizes towards higher disorder and I think this statement should therefore be removed. One can not seriously conclude a smaller critical disorder for $L \to \infty$ from the presented data.
==================================================

We are not aiming to suggest that the critical disorder strength $W \approx 3.6$ is incorrect. In the revised manuscript, we have removed phrasings that might lead to this impression. Unofrtunately, we were not able to make conclusive statements about the $L \to \infty$ behaviour, which we mention in the revised manuscript.

${\textbf{Point 4}}$
==================================================
Minor remark: Just below Eq. (1), commutator'' should be changed to anticommutator''.
==================================================

We thank the referee for bringing this to our attention. We have fixed this in the revised manuscript.