From "Weak" to "Strong" Electron Localization in a Mott Insulator

I have carefully read the manuscript and reviewed key references. I believe the results presented in this work are interesting and should be published in some form. However, while the authors clearly have something to say, they have also chosen a very confusing, if not a confused way of doing so. I thus suggest major revisions to their presentations before this manuscript can move any further.

The core of the present work is devoted to one of the most studied and if not well-understood, but at least better-understood problems in the strongly-correlated systems: single hole in an antiferromagnet. It has been my understanding that the problem is essentially solved, qualitatively AND quantitatively, in 1D, and qualitatively, with some good quantitative agreement with exact numerics, in 2D. The authors of the present work (a) present a new numerical treatment of the problem (coined magnon-expansion) in the Ising limit of spins, and (b) make a potentially interesting observation about “superexponential” distribution of the hole density along the “strings” in the 2D case.

However, instead of demonstrating what quantitative and qualitative advantages and insights their method (a) presents over previous results and focusing on what is the physics of the (b) feature, the author spent a significant portion of the presentation on a largely unphysical discussion of “strings in 1D” that appear “once the magnon-magnon interactions are neglected” (sic!). This discussion is largely void of a physical context and does not correspond to any physical limit of any model. It seems as if the authors are acting as hostages of their own approach, since the described setting is so clearly an artifact of the authors’ method (or SCBA) applied blindly to the problem, which, by the way, has been exactly solved more than 20 years ago, see Ref. [36].

I must also comment that the implied parallels to the problem of localization are uncalled for. In that field, such effects as interference and dephasing are important that have nothing to do with the manuscript, which simply discusses one-particle motion in two different confining potentials. Yet this name is featured in the title, abstract, and introduction, but in the end it deserves only a qualitative “looks the same”-type paragraph in the very end of the paper, from which it is clear that such a parallel is not only not instructive, but has little to no relevance to the problems at hand. As I point out in more detail below, there is more than one reason to avoid such a parallel altogether.

Let me proceed with the detailed comments/suggestions.
1) For the 1D problem of one hole in an AF Ising background, Ref. [36], the exact ground-state is the bound state of a fully mobile holon (hole on the AF domain wall) with a spinon (immobile AF domain wall). Somehow, a simple statement like that is avoided in text, with the only cryptic mentioning of the bound state in the caption of Fig. 2. Yet this picture would give the reader an immediate insight into the authors’ results concerning probabilities, because it maps the problem to the particle motion in a lattice-equivalent of a 1D delta-function (attractive) potential. The solution for the wave-function in this case is a textbook one, with a simple exponential decay away from the origin, and the length of such a decay given by the binding energy.
2) Instead, the reader is presented with a long-winded discussion of a dichotomy, posed by the authors as a “no-magnon-magnon” vs “magnon-magnon interactions”, which is purely artificial and is internal to the method they advocate.
3) I strongly suggest describing this picture above, listing the corresponding binding energy, which is known analytically [36], and its explicit relation to the decay length.
4) In Fig. 2, the only meaningful comparison is the one of the exact result [36] and the present method (red curves), with the explanation of any possible differences. One may mention (and may be show) that the result is unphysical (i.e., 2D-like) if one forgets about physics of the problem.

I thus suggest rewriting the 1D part in order to focus on physics and new physical insights, not on artificial problems.

5) For the 2D problem of one hole in the Ising background, regardless of the approximation, the qualitative (and qualitatively correct) picture is that of Ref. [25], more than 50 years ago. It maps the problem of the hole motion onto the motion in the linear (confining) potential. The continuum-limit solution for the wave function is that of Airy functions, I believe.
(a) I was genuinely surprised that with all the talk about “superexponential”, the authors have failed to derive a large-r (or large-n) asymptote of their probability distribution. Is it exp(-A*n*ln(n)) as it seems from Eq.(6) and from the asymptote of the Gamma-function? Basically, what does “super” stand for in this case?
Does the result agree with what the continuum solution of [25] would predict (Airy?)?
I suggest, once derived, plotting this asymptote in Fig. 1 to compare with the numerical results. This is, in my view, would be the main new result of the present study.
(b) I find it strange that, while implied, there is no direct and clear statement in text that the discussed behavior of probabilities should be a characteristic feature of a state in a linear confining potential.
(c) The main difference of the two type of the hole confinement, in 1D and in 2D problem, is between the confinement in a delta-functional and in the linear potential, respectively. This has to be said, loud and clear.

6) The straightforward SCBA-like, or string approximation are known to give qualitatively correct results, but quantitatively not-so-satisfactory agreement with, say, numerics. This has been understood as a result of several things. First is what authors refer to as to the effects of magnon-magnon interaction. Taking them into an account has lead to a modified SCBA-like approximation, Ref. [31], with much improved agreement with the available exact numerics. This approximation still neglects closed (or Trugman) loops [vertex corrections] as well as some subtler corrections due to crossed or tangential paths.
(a) The authors’ method (magnon expansion), [which, by the way, needs a slightly more than a brief description, not a just list of references] does, presumably, include all possible paths for the hole motion.
Then, the most important direct comparison needed in Fig. 3 is, again, between that of the ME method with that of Ref. [31] (red curves). There is an energy difference between the two approaches for the lowest peak (ground state). Is it due to Trugman loops?
Their effect can be largely avoided by moving k-vector to (pi/2,pi/2) point as discussed in Ref. [31].
(b) There is more structure to the higher peaks in the ME approach. Can one clarify the physical reason(s) for that?
(c) One needs an explicit statement in the text on whether the closed loops (Trugman paths) are included in the ME approach of the paper. The reason is that they are well-known to be delocalizing, thus making ANY parallels to the localization problem meaningless and self-contradictory. Or the authors are working on the Bethe lattice without ever mentioning it.
(d) The representation of the hole Green’s function in terms of the ratio of Bessel’s functions with the variable in the index was first found in Ref. [29]. This has to be mentioned explicitly.

Other comments.
7) I find that the discussion of the relevance of the current work to the interpretation of the optical experiments needs to address the following differences of the t-Jz model with the Hubbard or t-J model.
(a) It is well-known that the fluctuations in the more realistic t-J model erase strings and generate a coherent hole band of width ~2J. Is there a physical reason to expect that the strings longer than l=1 can be reliably observed? Will the peaks in the spectral function survive because of some fractional powers of J/t controlling peak separation?
(b) In the yet more realistic Hubbard model, the dispersion (and delocalization) is also provided by the effective next-neighbor hoppings (correlated 3-site terms). Same questions, are there any arguments for the survival of the string picture?

8) Since the authors implement the hard-core constraint (C1) right away, there is
no need for roots in their Eq. (2).
9) Neel AF —> Ising AF. For most, Neel implies Heisenberg model.

This paper examined a single hole motion of the t-Jz model in one- and two dimensions by using the magnon expansion technique. Since the underlying problem is quite old, it is important to clearly show what is new. I realized that the finding that the distribution of magnon away from the hole in two dimensions decays superexponentially is informative. Also the fact that the decay changes to exponentially in one dimension is interesting, though it is natural to see such a behavior in the presence of distance-independent string potential. Application of the results to optical lattice as well as the prediction to real materials are also important. The paper is well-written, however, there are several points to be improved.

1) In the last sentence of the abstract, it was mentioned that "Finally, we attribute ... to the peculiarities of the magnon-magnon interactions." How are the interactions peculiar?
2) In the Introduction section, it is written that "... that a single hole: ... (ii) experiences "weak" localization, ... but also certain crucial interactions present in the system are included." I could not find such an example, where crucial interactions are included. Note that this is not for the case of one dimension but more general statement.
3) Figure caption in Fig. 1 looks inconsistent with the figures.
4) In the Conclusion section, the first-order quantum phase transition was mentioned in connection with switching on and off of magnon-magnon interaction. The parameter \alpha was taken either one or zero. However, the parameter should be changed in between. What is the critical value of \alpha? What happens if you change \alpha continuously from zero to one?