Devil's staircases of topological Peierls insulators and Peierls supersolids

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.