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Devil's staircases of topological Peierls insulators and Peierls supersolids

by Titas Chanda, Daniel González-Cuadra, Maciej Lewenstein, Luca Tagliacozzo, Jakub Zakrzewski

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Submission summary

As Contributors: Titas Chanda · Jakub Zakrzewski
Preprint link: scipost_202012_00001v2
Date submitted: 2021-04-01 11:10
Submitted by: Chanda, Titas
Submitted to: SciPost Physics
Academic field: Physics
Specialties:
  • Atomic, Molecular and Optical Physics - Theory
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approach: Theoretical

Abstract

We consider a mixture of ultracold bosonic atoms on a one-dimensional lattice described by the XXZ-Bose-Hubbard model, where the tunneling of one species depends on the spin state of a second deeply trapped species. We show how the inclusion of antiferromagnetic interactions among the spin degrees of freedom generates a Devil's staircase of symmetry-protected topological phases for a wide parameter regime via a bosonic analog of the Peierls mechanism in electron-phonon systems. These topological Peierls insulators are examples of symmetry-breaking topological phases, where long-range order due to spontaneous symmetry breaking coexists with topological properties such as fractionalized edge states. Moreover, we identify a staircase of supersolid phases that do not require long-range interactions. They appear instead due to a Peierls incommensurability mechanism, where competing orders modify the underlying crystalline structure of Peierls insulators, becoming superfluid. Our work show the possibilities that ultracold atomic systems offer to investigate strongly-correlated topological phenomena beyond those found in natural materials.

Current status:
Has been resubmitted



Reports on this Submission

Anonymous Report 1 on 2021-8-12 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202012_00001v2, delivered 2021-08-12, doi: 10.21468/SciPost.Report.3377

Strengths

numerical calculations seem to valid and state of the art.

potentially interesting model with a rich phase diagram.

Weaknesses

This paper has a major weakness, and this is the analysis and the interpretation of the numerical results. My main objection is that the nature of many of the phases labelled as "ordered" in this work are not actually ordered according to the commonly accepted definitions in the field.

Report

The analysis is focused around the theme of a boson Peierls mechanism, which has been advocated by some of the present authors in a number of recent publications.

While I think that the model presents an interesting phase diagram I am puzzled by the broad use of the notion "ordered" and the notion of a "staircase" of solids and supersolids.

The current model put forward by the authors is different from previous works by some of the authors in that it puts an XXZ Hamiltonian at work for the spin bond variables, and then operates in a regime of the XXZ anisotropy Delta, where at zero magnetisation the XXZ chain alone exhibits true Neel order, a long standing result. However away from zero magnetization the XXZ chain exhibits Luttinger liquid physics, as is also text book knowledge. On the other hand the Bose Hubbard chain is also a Luttinger liquid away from the Mott insulating filling and interaction strength. So at some level the phase diagram studied here is an instance of two coupled Luttinger liquids, similar to earlier studies of Bose Bose mixtures.

The main problem I have with the analysis in the paper is that the authors almost automatically assume that a commensurate density in one or both channels implies long range order of some sort. It is at the heart of Luttinger liquid theory that these local observables fluctuate at large distances with a power law character which can depend on the interaction strength and the filling etc. In some cases these power law decays can be quite slow, and can be mistaken for true long range order. This aspect is not really taken into account in most of the
discussion about the phases which carry a "O" in their abbreviation.

This issue carries over the discussion of the various "staircases". If the different fillings actually form compressible (multi channel) Luttinger liquids, then I think it is not adequate to call this phenomenon a "staircase", since the belong to one phase with continuously changing fillings of the bosons or the spins (i.e. their magnetization) and continuously changing power law exponents.

Of course it also known that Luttinger liquids can be gapped out in the framework of a Sine-Gordon like mechanism, driven by commensurability effects of the lattice or the other channel, and then show true long range order. But the authors do not provide an analysis where these possibilities are addressed from the actual numerical data.

Requested changes

For a publication in SciPost Physics, the analysis of the nature or absence of true long range order in the density or bond energies has to be provided. This may then well impact the structure of the phase diagram, and also whether some scenarios regarding "symmetry breaking topological phases" then survive in the absence of actual long range order aka symmetry breaking.

  • validity: low
  • significance: ok
  • originality: ok
  • clarity: low
  • formatting: good
  • grammar: good

Author:  Titas Chanda  on 2021-09-03  [id 1728]

(in reply to Report 1 on 2021-08-12)

We would like to thank the Referee for their careful reading of our manuscript and providing us constructive criticisms and comments. Please find here below detailed answers to the raised comments. Quotes from the Referee are presented first, followed by our replies and the associated changes.

**The Referee writes:**

“This paper has a major weakness, and this is the analysis and the interpretation of the numerical results. My main objection is that the nature of many of the phases labelled as "ordered" in this work are not actually ordered according to the commonly accepted definitions in the field.”

**Our response:**

In this work, we have encountered mainly two types of ‘orders’ in the system, namely (1) diagonal long-range order that is presented by a finite peak in the spin structure factor $S_{\sigma}(k)$ that survives the thermodynamic limit, and (2) off-diagonal quasi-long-range order where off-diagonal correlations $\langle{\hat{b}^{\dagger}_j \hat{b}_{j+R}}\rangle$ decay algebraically with the distance $R$ following a power-law. While the diagonal long-range order is present both in the commensurate Peierls insulators and incommensurate Peierls supersolids, the off-diagonal quasi-long-range order only exists in the later case.

**The Referee writes:**

“The current model put forward by the authors is different from previous works by some of the authors in that it puts an XXZ Hamiltonian at work for the spin bond variables, and then operates in a regime of the XXZ anisotropy Delta, where at zero magnetisation the XXZ chain alone exhibits true Neel order, a long standing result. However away from zero magnetization the XXZ chain exhibits Luttinger liquid physics, as is also text book knowledge. On the other hand the Bose Hubbard chain is also a Luttinger liquid away from the Mott insulating filling and interaction strength. So at some level the phase diagram studied here is an instance of two coupled Luttinger liquids, similar to earlier studies of Bose Bose mixtures.” 

“The main problem I have with the analysis in the paper is that the authors almost automatically assume that a commensurate density in one or both channels implies long range order of some sort. It is at the heart of Luttinger liquid theory that these local observables fluctuate at large distances with a power law character which can depend on the interaction strength and the filling etc. In some cases these power law decays can be quite slow, and can be mistaken for true long range order. This aspect is not really taken into account in most of the discussion about the phases which carry a "O" in their abbreviation.”

**Our response:**

We thank the Referee for raising this issue in our analysis. We want to point out that only incommensurate Peierls supersolids are described by Luttinger liquid theory where off-diagonal quasi-long-range order exists, as shown in Fig. 7 (c) and (f), while Peierls insulators are not. To confirm this, we now show in Fig. 3 (a) that the off-diagonal correlations $\langle{\hat{b}^{\dagger}_j \hat{b}_{j+R}}\rangle$ decay exponentially with the distance $R$ that validates our claim that these commensurate phases are indeed insulators. Moreover, we also show that the peak of the spin structure factor $S_{\sigma}(k)$ remains finite in the thermodynamic limit by extrapolating the finite-size data in Fig. 3 (b). This confirms the existence of diagonal long-range order in the system.

**The Referee writes:**

“This issue carries over the discussion of the various "staircases". If the different fillings actually form compressible (multi channel) Luttinger liquids, then I think it is not adequate to call this phenomenon a "staircase", since the belong to one phase with continuously changing fillings of the bosons or the spins (i.e. their magnetization) and continuously changing power law exponents.”

**Our response:**

This is indeed true in the incommensurate Peierls supersolid region which is compressible in nature and thus the steps in terms of the bosonic density $\rho$ are not stable. We have already shown this in the previous version (Fig. 4 (c) in the present version). However, the steps seem to remain stable in terms of the spin degrees-of-freedom, e.g., $S_{\sigma}(k)$ as seen in Fig. 8 (b). Following the referee’s suggestion we do not use the term ‘staircase’ in the revised version when describing the supersolids as the staircase structure in densities vanishes in the thermodynamic limit in this case.

On the other hand, the staircase structure is indeed stable in terms of both the bosonic and the spin degrees-of-freedom for Peierls insulators. However, it is important to note that not every step is equally stable to quantum or thermal fluctuations, since the insulating gap is different for each of them. This has also been verified in Fig. 4 (c), where we show that some steps (corresponding to lower integer values of $q$ in $\rho = p/q$) are more stable compared to others.

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