## SciPost Submission Page

# From "Weak" to "Strong" Hole Confinement in a Mott Insulator

### by Krzysztof Bieniasz, Piotr Wrzosek, Andrzej M. Oles, Krzysztof Wohlfeld

#### This is not the current version.

### Submission summary

As Contributors: | Andrzej M. Oles · Krzysztof Wohlfeld |

Arxiv Link: | https://arxiv.org/abs/1809.07120v5 |

Date submitted: | 2019-10-14 |

Submitted by: | Wohlfeld, Krzysztof |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | Condensed Matter Physics - Theory |

Approach: | Theoretical |

### Abstract

We study the problem of a single hole in an Ising antiferromagnet and, using the magnon expansion and analytical methods, determine the expansion coefficients of its wave function in the magnon basis. In the 1D case, the hole is "weakly" confined in a potential well and the magnon coefficients decay exponentially in the absence of a string potential. This behavior is in sharp contrast to the 2D square plane where the hole is "strongly" confined by a string potential and the magnon coefficients decay superexponentially. The latter is identified here to be a fingerprint of the strings in doped antiferromagnets that can be recognized in the numerical or cold atom simulations of the 2D doped Hubbard model. Finally, we attribute the differences between the 1D and 2D cases to the magnon-magnon interactions being crucially important in a 1D spin system.

###### Current status:

### Author comments upon resubmission

Reply to “Anonymous Report 2 on 2019-7-30 Invited Report”

**********************************************************

We would like to thank the Referee for such an extensive report and

for suggesting so many useful changes---we genuinely appreciate that effort.

Our reply consists of two parts. First, we comment on

the general statements / impressions written by the Referee in the

first page of their report (see section “General statements” below).

Second, we address the specific comments 1--9 (section “Specific statements”).

“General statements”

As already stated, we really appreciate the Referee’s detailed reading of

the manuscript and their very insightful comments. We have identified three main

general requests voiced by the Referee, which we shall address one by one:

(1) “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].”

REPLY:

We must confess that this is the *only* statement by the Referee that

we do not fully agree with.

Let us first state two (arguably) subjective remarks:

— Our point of view is that neglecting magnon-magnon interactions

should not be regarded as an artifact of SCBA but is rather due to

the linear spin wave approximation (which typically is a part of the

SCBA, but it is the linear spin wave approximation which comes

“first” and triggers the whole problem). Since the linear spin wave

theory is a relatively widely used and established approximation,

we would not say that we are “hostages of (...) [our] own approach”.

— We do not find neglecting magnon interactions as “academic

(i.e., not corresponding to any physical limit)” as the Referee would

suggest. We note that also the t-Jz model, and sometimes even the t-J model,

could be regarded just as “academic” a model (e.g., from the DFT

perspective).

However, what we find more important (than the above rather

subjective remarks) is that using the magnon language and

switching on and off the magnon interactions can have several

advantages:

— To start with, we consider using the same “magnon” language

both for the 1D as well as the 2D solution to be more elegant and

more didactic, than using different approaches: the “spinon”

domain wall language in 1D and the magnon models in 2D. Of course, as

suggested by the Referee, one could just as well talk about different

forms of the potentials in 1D (delta function) and in 2D (linear

potential). However, such approach is then approximate in 2D, as

on the square lattice the “Trugman loops et al.” make the problem

more complex. Moreover, and maybe more crucially, our numerical method

is expressed in the magnon-hole basis.

— Then, when the same magnon language is used in the 1D and 2D cases, it is

natural to ask the question: what is the origin of such a different

solution of the 1D and 2D problem? It turns out that this stems from

the magnon interactions. While in 2D they are only quantitatively

important, in 1D they qualitatively alter the spectral function / the

probability distribution P_n. Personally, we find this result, even if

“academic” and perhaps “to be expected”, interesting and really worth

presenting.

— Finally, the 1D case without magnon interactions allows for studying

a problem that is exactly solvable analytically and hence can be used both

for benchmarking the ME numerical approach (perfect fit) as well as

for deriving the exact expression for P_n in terms of the Bessel

functions [Eq. (9)]. The experience gained in 1D can then be used to firstly derive

the exact result for the Bethe lattice with the coordination number z>2 [Eq. (36)] and then postulate a

quantitatively approximate, but qualitatively exact, expression for the

“superexponential decay” in two dimensional square lattice [Eq. (11)], assuming the

same functional dependence in 2D (with and without magnon interactions)

as the exactly solvable 1D model without magnon interactions [Eq. (9)].

— As a side remark, we also believe that the discussion of the magnon

interactions in 2D is quite interesting. While the Referee

is of course completely right that the seminal paper by Bulaevskii,

Khomskii et al. from 1968 is still qualitatively correct, we find it

important to explicitly show to what extent the magnon-magnon

interactions alter that picture quantitatively, as already suggested

in Ref. [31].

We believe that the above arguments, while not well-presented in the

former version of the paper would, at least partially, convince the

Referee that such a result is interesting. In the hope that this

would be the case, we have updated the manuscript accordingly in order to explain

and highlight the above-mentioned points in a clear manner in the

current version of the manuscript; for details please see the response

to the “Specific statements” as well as the Summary of changes below.

(2) The authors should concentrate in their paper on

“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” where (a) stands for “present(ing) a new numerical

treatment of the problem (coined magnon-expansion) in the Ising limit

of spins” and (b) for “mak(ing) a potentially interesting observation

about “superexponential” distribution of the hole density along the

“strings” in the 2D case.”

REPLY:

We completely agree with the above statements and are grateful to the

Referee for suggesting which parts of the manuscript they find the

most interesting. Thus, we have fully followed the Referee’s recommendation

and in the current version of the manuscript we now:

— Describe the numerical method and its advantages in far more detail.

— Discuss the origin as well as the implications of the “superexponential”

distribution in a far more extensive way.

For details please see the response to the “Specific statements” as well

as the Summary of changes below.

(3) “(...) the implied parallels to the problem of localization are uncalled for (...)”

REPLY:

We also agree that using the word “localization” might be misleading

[due to the association with the many-body localization phenomenon;

however, note that the term “localization” is used in, e.g., Phys. Rev. B

57, 6444 (1998)] and thank the Referee for this important remark.

Therefore, following the Referee’s recommendation, we have now avoided

using this word as much as possible. Instead, whenever appropriate, we

now have replaced the term “electron localization” (and various versions

of this term) with the term “hole confinement”. Indeed, the latter describes

the observed phenomenon far better. We have also updated accordingly

the title, the abstract, as well as the introduction and conclusion

sections. In particular, we have added this crucial distinction between

the words “confinement” and “localization” by writing in the introductory

paragraph:

“Therefore, here we concentrate on perhaps the simplest, though still

nontrivial and realistic,\footnote{See the concluding section for a

detailed discussion.} problems of electron localisation in the Mott

insulator---the problem of the confinement of a particle by an effective potential

that takes place when a single hole is added to the ordered ground state

of the half-filled $t$-$J_z$ model~[24-38].”

We also followed the Referee's suggestion and decided to remove the last

paragraph of the concluding section, to avoid misunderstanding as

indeed it might have sounded misleading.

For details please see the Summary of changes below.

“Specific statements”

“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. “

REPLY:

Naturally, we agree that the 1D case can be easily solved using the

“spinon language”. Thus, stimulated by the Referee's comment we have

heavily revised the discussion of the 1D case by explicitly mentioning

that this case can be easily understood in terms of a bound state of a

hole and a spinon (see Summary of changes).

“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.”

REPLY:

In the reply to the “general statements” we have explained in some detail

why we believe that the use of the magnon language is also useful in 1D

case. Nevertheless, we completely agree with the Referee that the

previous version of the paper might have left a wrong impression that the

case without magnon interaction is “physical” and should be taken on

equal footing with the fully interacting case. Thus, in the current

version of the paper, we now stress that “Interestingly, especially when

contrasted with the 2D studies below, the above exponential decay is

\emph{not} obtained when the magnon-magnon interactions are not correctly

taken into account in the magnon language expression for the 1D $t$--$J_z$

model (…)”.

Moreover, when discussing the 2D case, we now stress that “The approximate

expression for the probability distribution $\{P_n\}$ in a square lattice

2D model [Eq. (11)] is motivated by the analytically exact expression obtained in the 1D

case without the magnon-magnon interactions, see Eq. (9).”

We have also performed several other changes in the “Results” section which

hopefully makes our case more transparent to the Reader

(see Summary of changes).

“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.”

REPLY:

Besides adding the spinon discussion (see above), we have now mentioned

in the text that “the decay length $l$ [is] given by the inverse of the

logarithm of the ground state energy of the Ising AF chain with a single

hole ($\varepsilon_{\rm GS}$)---see Appendix~\ref{sec:appendix} for the

exact expressions of $\{A, l, \varepsilon_{\rm GS} \}$.”

(cf. Summary of changes)

“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. “

REPLY:

First, we would like to emphasize that there exist no differences

(beyond the numerical errors) between “the exact result [36] and the

present method (red curves)” — as discussed in Sec. 4.3.:

“ taking advantage of an exact analytical result for the $t$--$J_z$ model

obtained in the spinon language and using the continued fractions

[36-38], we observe that the \textit{same} spectrum is

obtained as that of the full model in the ME method, see Fig. 2(b)”.

We agree with the Referee that the “ladder spectrum” is unphysical in

one dimension and one should be clear about that — we have revised the

appropriate parts of Sec. 4.3. (see Summary of changes).

Nevertheless, we still think that showing this “unphysical result” is

of interest to the Reader. Inter alia, it shows the role played by the

constraints C1 and C2 (not that important, cf. the \alpha=0 1D results

obtained using ME and SCBA) and that they are far less important than

the Trugman processes in 2D (cf. \alpha=0 2D results obtained using ME

and SCBA) — we have added this discussion to the current version of the

paper (see Summary of changes).

“I thus suggest rewriting the 1D part in order to focus on physics and

new physical insights, not on artificial problems.”

REPLY:

Indeed following the Referee's comment, we have heavily rewritten the 1D

part, cf. Summary of changes.

“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.”

REPLY:

We fully agree with the above comment and believe that this point of

view is also reflected in the paper.

5)“(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. (8) 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.”

REPLY:

These are very important points and we are grateful to the Referee

for raising them. Let us answer them one by one:

— Large-n asymptote: The answer here is a bit subtle, for the large-n

asymptote of the Gamma function is “almost” like exp(-A*n*ln(n)), i.e.:

(i) the large-n asymptote of ln[P(n)] ~ -2n ln(n) and is given by Eq.

(10) [for 1D and (12) for 2D] in the current version of the paper,

(ii) however, (i) does not imply that P(n) ~ exp[-2n ln (n)].

(iii) the proper large-n asymptote of P(n) is given in Appendix

[Eq. (30) or (37)] of the current version of the paper.

— “Super” stands for a decay which is faster than an exponential, i.e., that

ln P(n) decreases with n faster than -n. In “our” case this can be

observed either by looking at the asymptotic behavior or by

approximating P_n by a Gamma function---the latter one (which is a

continuous version of the factorial) “decays faster” than an

exponential. We now add this discussion to the current version of the

paper (see Summary of changes).

— Agreement with “Airy”: The large-n asymptote of the logarithm of the

Airy function is ~ -2/3 n^{3/2} and is different than the large-n

asymptote of the P(n). Thus, the continuum solution has a different

asymptote than the discrete case. Nevertheless, in both cases the

logarithms of both asymptotes “decay faster” than a linear function

and are thus understood as “superexponential”. This discussion is

also now included in the paper (see Summary of changes—note that this

discussion is added in the end of the current Sec. 4.1.).

— Plotting the asymptotes in Fig. 1: While plotting the asymptotes

might indeed be a good idea, in our opinion this would make the

already busy plots of Fig. 1 even less legible. This is because we

already compare the numerical results against the analytical formulae

for ln P(n): while in 1D the agreement is perfect,

in 2D there is some small discrepancy between the two approaches.

Moreover, we believe that by comparing the top panels of Fig. 1 one can

immediately observe that there is a contrast between the well-known

exponential decay in the 1D (alpha=1) case (linear behavior on the

ln P(n) plot) and all other cases for which one observes a faster than

exponential decay, i.e. a “superexponential” one. We find the latter

statement to be more important than the precise form of the asymptotic

behavior of ln P(n). Nevertheless, as mentioned above, in the current

version of the paper that precise form of the asymptotic behaviour is

also clearly mentioned in the form of Eq. (10) or (12).

5)“(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.”

REPLY:

Yes, this is a very good remark and we have implemented it in the new

version of the paper (see Summary of changes---note that this discussion

is added in the end of the current Sec. 4.1.; cf. first paragraph of

the Conclusion section).

5)“(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.”

REPLY:

This is also a very good remark and we have implemented it in the new

version of the paper (see Summary of changes---note that this discussion

is added in the end of the current Sec. 4.1.; cf. first paragraph

of the Conclusion section).

“6) The straightforward SCBA-like, or string approximation are known

to give qualitatively correct, 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.”

REPLY:

We fully agree with the above comment and believe that this point of

view is also reflected in the revised paper.

6)”(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].”

REPLY:

First of all let us stress that, just as in the 1D case, we believe

that it is worth to look at the case without magnon-magnon interactions

(see reasoning above). Nevertheless, we agree with the Referee that one

should better differentiate between the “physical” (interactions

included) and “approximate” (interactions neglected) cases---

which we hope we have done better in the newer version of the paper

(see Summary of changes).

Coming then to the main question posed above, we answer as follows: Yes, indeed,

in our opinion, the difference between the ground states is due to the onset

of the closed (Trugman) loops — since this, to the best of our understanding, is

the only difference between the SCBA and the ME method. We have checked that

indeed such a difference is slightly smaller at ${\bf k} = (\pi/ 2, \pi / 2)$ point,

though it is still extremely well visible. We included the above discussion

in the current version of the paper (see Summary of changes). Note that we have also

decided to show the spectral function for ${\bf k} = (\pi/ 2, \pi / 2)$ in Fig. 3(a),

so that the role of the Trugman loops is suppressed as much as possible.

Following the Referee's comment mentioned in the “[]” brackets we have

also expanded the Methods section by including an additional paragraph

which in far more detail describes the magnon expansion method.

6)”(b) There is more structure to the higher peaks in the ME approach.

Can one clarify the physical reason(s) for that?”

REPLY:

A comparison between the top and bottom panels of Fig. 3 suggests that

the origin of the onset of the more structure to the higher peaks in

the ME approach lies in the closed (Trugman) loops. While a detailed

study is needed to fully understand such behaviour (which is beyond

the scope of this paper), in our opinion this can be understood as a

result of the hole “cutting the strings” through the closed loops and

thus “disrupting” the string potential and “destroying” the “ladder”

spectrum. We added a comment on this problem in the new text of the

paper (see Summary of changes).

6)”(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.”

REPLY:

Yes, the closed loops are included in the ME approach---this method is

a numerically exact method of calculating the Green's function of the

polaronic model and is applied here on a “true” hypercubic lattice (1D,

2D square). I.e., it is *not* applied here to a Bethe lattice. We have

added an explicit statement on this problem in the current version of

the paper (both in the Methods section as well as in Sec. 4.3.).

Let us also note here that, as already discussed above, indeed we have

basically avoided using the word “localization”. However, strictly

speaking also using the word "confined" might be questionable here.

Nevertheless, we observe that on the practical level the effect of the

Trugman loops on the hole “deconfinement” (i.e., hole leaving the linear

potential in this case) in the ground state is negligible: the probability

of finding a hole is still adequately described in the 2D case by a function which

describes the “superexponential” decay and the hole confinement

in a linear potential.

6)“(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.”

REPLY:

In the new version of the paper we refer the reader to Ref. [29] when

we mention that the recurrence relation can be expressed by introducing

the Bessel functions (see Summary of changes).

“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?”

REPLY:

To answer the above remarks, it is important to state that our way of

reasoning is actually a bit different (unfortunately this was not

optimally presented in the former version of the paper---there was

just a short comment in the former footnote #3):

First, this work shows that in the 2D t-Jz model (or the 1D t-Jz

without magnon interactions) the “P_n” has a superexponential

functional dependence. Second, in such a t-Jz model the “string picture”

(i.e., the one which describes the hole moving in an AF as in a string

potential) is well-defined. Combining these two observations we

conclude that the “P_n” showing a superexponential decay may be

regarded as a signature of the “string picture” in any model. Thus, our

argument is that: if “P_n” on t-J or Hubbard models (with / without

longer-range hoppings, etc.) showed a superexponential decay, then this

would strongly indicate that a linear string potential indeed plays a

dominant role in the hole motion in the 2D doped Hubbard (or t-J)

models.

Of course, it is expected (as the Referee writes) that in the Hubbard

or t-J models there will be strong deviations from the “string picture”.

In fact, we *completely* agree with the Referee that adding the spin

flip terms and / or longer range hoppings (correlated or not) would

strongly alter the way the hole moves in the (doped) antiferromagnets.

However, the question we ask here is: how large these deviations are

(especially on the qualitative level)? We stress that it is not the

purpose of this work to answer this question. We only would like to

provide a criterion which can tell us whether the “string picture”

can be observed in the Hubbard or t-J models. It is then up to those

performing the optical lattice experiments or to the large-scale

simulations of the Hubbard / t-J models to apply this criterion and

check the validity of the string picture in the Hubbard / t-J cases.

In order to make the above way of reasoning clear we have rewritten the

respective parts of Sec. 4.2 as well as the Conclusion section., cf.

Summary of changes below.

“8) Since the authors implement the hard-core constraint (C1) right

away, there is no need for roots in their Eq. (2).”

REPLY:

Indeed such a change might make the presentation more clear. We have

followed the Referee’s suggestion and updated the text accordingly,

cf. Summary of changes below.

“9) Neel AF —> Ising AF. For most, Neel implies Heisenberg model.”

Following the Referee’s suggestion we have substituted everywhere

“Neel—>Ising”, cf. Summary of changes below.

**********************************************************

Reply to “Anonymous Report 1 on 2019-7-29 Invited Report”

**********************************************************

We are grateful to the Referee for providing such a kind and useful

report, in particular by appreciating that “the distribution of magnon

away from the hole in two dimensions decays superexponentially is

informative”.

Below we reply to the specific points raised by the Referee:

“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?”

REPLY:

Indeed, we agree with the Referee that this sentence sounds strange and

is not very informative. The current version of this sentence, in which

we do not use the word “peculiar” at all, reads:

“Finally, we attribute the differences between the 1D and 2D cases to

the magnon-magnon interactions being crucially important in a 1D spin

system.”

“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.”

REPLY:

We thank the Referee for pointing out this logical inconsistency.

The current version of the last part of the Introduction reads:

“(...) we unambiguously show that a single hole:

(i)~in 2D or higher dimensional models is ``strongly'' confined in the

ground state and its wave-function coefficients decay

\emph{superexponentially}, i.e., much faster than in the textbook case

of a single finite potential well,

(ii)~in a 1D chain experiences ``weak'' confinement, i.e., has the

wave-function coefficients decaying \emph{exponentially}, just as in a

potential well. Interestingly, we show below that these differences

between the 1D and higher-dimensional cases can be easily understood in

the magnon language, as originating from the crucial role played by the

magnon-magnon interactions in a 1D spin system. Altogether, this means

that lowering dimensionality and adding interactions may in fact remove

``strong confinement'' in favor of ``weak confinement'' in a strongly

correlated system.”

“3) Figure caption in Fig. 1 looks inconsistent with the figures.”

REPLY:

We thank the Referee for spotting these very unfortunate and confusing

typos in the figure caption. We have swapped the panels of the figure

so that the caption is consistent with the figure.

“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?”

REPLY:

We are grateful to the Referee for this very important comment.

We have verified that the critical value of $\alpha=1$, i.e., the

``strong’’ confinement happens for $\alpha\in [0,1)$, since any

$\alpha<1$ cannot compensate the string potential felt by the mobile

hole. Thus, we have updated the final sentence of the conclusion

section and also added Eq. (26) in Appendix A which shows that the

magnon coefficients decay superexponentially for $\alpha<1$.

“In fact, in the 1D $t$-$J_z$ model with a single hole a first order

quantum phase transition is observed when magnon-magnon interaction

can no longer compensate the string potential felt by the mobile hole

\footnote{This happens for $\alpha \in [0, 1)$, see Eq. (26) in Appendix

\ref{sec:appendix}}.”

### List of changes

(0) We have changed the title to better reflect the content of the paper

(we have replaced the word “localisation” to “confinement”).

(1) The abstract and introduction were changed to reflect the general concerns raised by the Referees.

(2) Eq. (2) (the Holstein-Primakoff transformation) was changed to its linearized form, following the Referee's suggestion that the effect of the square root is imposed implicitly. A remark clarifying that point was also added in the paragraph following that equation.

(3) At the end of the Methods section we have added three paragraphs that briefly explain the formalism of the ME method, without however going into all the intricate details of computing the free Green's functions and the technical task of generating the equations of motion. We believe that this formulation is now detailed enough to give an unfamiliar reader a good idea of how the procedure is executed, without overwhelming them with unnecessary technicalities.

(4) The Results section was significantly revised and extended to accommodate the Referee's comments. Subsections were introduced to keep the results organised and easier to navigate. Eqs (11) and (12) were added to improve the discussion.

(5) The discussion of the 1D system was supplemented with a description of the spinon language picture for completeness, following the Referee's suggestion. A more detailed discussion of the (super)exponential probability decay was added to clarify the presentation.

(6) The section 3.2 was added to extend the focus on the 2D Ising system, which should constitute the main focus of the paper.

(7) Section 4 "Discussion" was extended and reorganized. A new subsection 4.1 was added to contain the more intuitive points of discussion based on "cartoons". This was mostly carved out of the former introductory statements to section 4. However, two new paragraphs were added to extend the point about the asymptotic behavior of the potentials in 1D and 2D.

(8) Subsection 4.2 was extended with an additional paragraph. The previous paragraphs were slightly altered for readability and to incorporate the Referee's comments.

(9) The discussion in section 4.3 was substantially altered and extended to accommodate the Referee's objections.

(10) The Conclusions section was altered to reflect the new key points of the article.

(11) The Appendix was extended to include more information on the derivation of the 2D case. Subsections were introduced to better organize the discussion.

(12) The order of panels in Figure 1 was changed to better reflect the logic of the current discussion.

(13) In Figs 2 and 3 labels ("a" and "b") have been added.

(14) In Fig. 3 the ME data was changed and currently shows the results for k=(pi/2, pi/2). This way the influence of the Trugman loops on the spectrum is “reduced” (as requested by the Referee).

(15) Numerous small changes in text and presentation have been introduced in order to improve the readability of the article.

### Submission & Refereeing History

## Reports on this Submission

### Anonymous Report 1 on 2019-10-30 Invited Report

### Report

I, basically, accept all the changes in the resubmitted version as a very good to satisfactory response to all my prior comments. Some of the points that were not changed can be considered a matter of style and I should let the authors express themselves the way they prefer to do so.

However, there is one remaining point of concern. In Fig. 3 there is a noticeable disagreement between the red curves in (a) and (b) parts, with the reference to Ref. [31] in (b). The problem is that the results of Ref. [31] are essentially identical to the Exact Diagonalization (ED) results. The argument of the authors is that their curve in (a) includes Trugman paths, while [31]/(b) does not. However, ED does include Trugman paths and for that value of J/t (=0.4), the finite-size effects are minimal. I thus believe the authors are using the wrong expression for the (modified) SCBA self-energy.

The correct one is in Ref. [31], Eq.(15). I checked the Appendix A.3, Eq. (34) and the indices of the Bessel functions come out wrong, i.e., not equal to Ref.[31]. Since it is the results of Ref. [31] that the current method should be benchmarked against, the authors should correct their calculations, results, and the ensuing discussion.