Devil's staircases of topological Peierls insulators and Peierls supersolids

The manuscript studies an interesting model of interacting bosons and spins in one-dimensions and also present a proposal for the experimental realization of the the model with cold atomic gases. The results are very interesting and a very rich phase diagram appears. While the experimental realization is very challenging, the manuscript is another example how interesting quantum phases can be realized with cold atomic gases. The method is based on a numerical state of the art analysis with matrix product states, which is the most suitable approach to address the problem. While the parameter space is very large, the authors fix the phase diagram to very specific values and just vary the chemical potential $\mu$ for the bosons as well as the anti-ferromagnetic coupling $J$.

Unfortunately, there are also some sever points of criticisms, which should be addressed before a decision about publication can be made:

(1) There are several short comings in the figures as well as the figures captions:
(i) In Figure 1, there are no axes labels. It is completely unclear from which range $J$ and $\mu$ are varied. This is a very poor style and unacceptable for any publication, even if the plot just shows the qualitative phase diagram. The figure has to be significantly improved.
(ii) In Figure 2, it would be important to add the values of the chemical potentials, and the referee was unable the find the value $J$ for the simulation.
(iii) It would help a lot if the two scans along the phase diagram in Fig. 3 are illustrated in Fig.1 .
(iv) In figure caption 1, the authors claim that each phase appears with a topological and a trivial sector. This is in contradiction to the text, where the authors state "we find (1) phases with commensurate long-range order that are topologically non-trivial and (2) ..."

(2) The notion of incommensurability is very unconventional and based on a relation, which is not explained: Below Eq. (6) the authors claim that the appearnce of a peak in the spin structure factor is completely determined by $m$. It might be that for the present parameters this indeed happens, but unfortunately, this statement is not further justified. Especially, the referee is convinced that in general this relation is not correct. Therefore, it is a very weak point, that the manuscript uses a model dependent parameter for the definition of a phase diagram. This in contrast to long standing history in condensed matter physics, where phases are labeled by observable signatures independent on the underlying problem. Especially, the notion of incommensurability is in general understood as compressible phase, where the density varies smoothly and independently on commensurability with the underlying lattice. This is obviously not the case in the present manuscript, as can be seen in Fig. 3a, where
the density exhibits clear incompressible plateaux.

The latter is also in contrast to the claim in figure caption 1, where the authors claim that the incommensurable phase is a superfluid: A superfluid is a compressible phase and can not exhibit plateaux as shown in Fig. 3a.

(3) There is also a contradiction in the summary of results, where the authors claim that the quasi superfluid phase extends to the MI at integer fillings.
" This occurs for any number of bosons N, which are then in a quasi-superfluid (qSF) phase except at integer fillings, where they form a Mott insulator (MI) (Fig. 1(b))." As it is clearly visible in Fig. 1, the qSF phase does not extend to the MI. This point should clearly be properly explained.

(4) Furthermore, the referee is extremely puzzled that the notion of Peierls phase: a characteristic property of the Peierls instability is, that it appears for arbitrary weak couplings. This would imply that the system is incompressible for any value of $J$ at half filling. In Figure 1, however, the authors claim that there is always a superfluid phase. At least for $U\rightarrow \infty$ this should not be correct due to the famous Peierls instability. The referee would imagine, that a variational approach for the spins with all spins down, but a small additional anti-ferromagnetic modulaton allows the hard-core bosons to open a gap and gain energy via the famous Peierls instability. For such a state, the averaged magnetization would be $ m \approx 1$, but the peak in the structure factor appears at 1/2. Such a state would be an example, which violates the above unmotivated definition. The authors should explain in detail and demonstrate, why in the present manuscript there is no suchPeierls instability, and why they believe the naming is still meaningful.

(5) The referee is very confused by the notion that, the authors observe a symmetry protected topological phases. The only signature are what they call edge states, which are protected by inversion symmetry. According the seminal work by F. Pollmann [PRB, 064439
(2010)] it is clearly stated that edge states are not a signature of a topological phase protected by inversion symmetry for bosons in one-dimension. This is also nicely confirmed by Ref. [68], where the edge states continuously disappear onto change of the parameters. Furthermore, the general classification of symmetry protected topological phases based on the seminal work above requires a fourfold ground state degeneracy in the thermodynamic limit. This property is not demonstrated in the present manuscript. The referee expects, that the authors rather find some phases which are characterized by some hidden string order as in Ref. [21 ] or a finite polarization is in Ref. [68]. In the referees opinion it is very important that the authors properly explain their results in the well established context of topological states of matter, and explain which topological signature is a true characteristic property of their system. If the authors are convinced that they observe a symmetry protected topological phase, they should clearly demonstrate the robust fourfold ground state degeneracy and explain the projective symmetry realized on the edge and easy accessible within matrix product states. At the moment, they only demonstrate a two-fold degeneracy, which would be fully consistent with the observations in Ref[21] and [68].

(6) The authors fail completely to provide any evidence of fractional excitations. Especially, the the extreme limit with $J\gg t$ and half filling, the spin pattern just provides a static back ground, and the model smoothly connects to the famous SSH model for $U \rightarrow \infty$ via a Jordan-Wigner string. It is well established that the SSH model does not give rise to fractional excitations. Therefore, the present model can also not give rise to fractional excitations. Furthermore, the explained signatures are fully consistent with an integer charge at the edge. A clear signature of fractionalization would be that the systemsexhibits two degenerate quantum many-body ground state, which differ by one particle, and the density between the two states exhibits a difference by 1/2 at each edge. However, such a signature is completely missing in the present manuscript, and therefore the referee is convinced that the notion of fractionalization is incorrect.

In summary, in the present form the manuscript can not be accepted for publication and significant improvement and clarifications are required before a final decision can be made.

See report for the required significant modifications.