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

First of all we are grateful to the Referee for reading in detail our long
response and for suggesting that he / she “accept(s) all the changes
in the resubmitted version as a very good to satisfactory response to all my prior comments”.
We are also very thankful for spotting one more mistake in our
results which concerns the energy of the ground state in the 2D SCBA results
(with the magnon-magnon interactions included). To this end, we have verified that:

(1) “Our” SCBA equations for the 2D case with magnon-magnon interactions included
[e.g. Eq. (34)] are *identical* to Eqs. (21-23) of Ref. [31]. The only difference
is due to the different zero energy level (\delta \omega =2J), see discussion
in the Summary of changes [point (2)] below.

(2) The reason why the SCBA and ME results of Fig. 3 did not match
(i.e. the ground state energies were shifted) in the previous version was
due to an accidental shift by J/2 of the SCBA result. This probably must have occurred
when “playing around” with adding / removing the C1 and C2 constraints and shifting
the zero energy level. We have now corrected this mistake and, as the Referee can observe,
the ground state energies at k=(pi/2, \pi/2) point are basically the same in the SCBA
and in the ME method (in agreement with Ref. [31]).

We thank the Referee for spotting this (rather crucial) mistake!

(3) Altogether, this means that indeed the role of the Trugman loops in obtaining the
numerically exact ground state energy seems to be rather small (in agreement with [31]).
In order to account for this fact, we have modified the text of the manuscript in few places,
see Summary of changes [point (3)] below.

We have thoroughly read the latest version of the manuscript to correct for few other
small typos and errors. We believe that the submitted version can now be published
in SciPost Physics.

We would like to thank the Referee for such a careful reading of our manuscript
and for suggesting very important changes.

Sincerely,

Krzysztof Wohlfeld

/On behalf of all Authors/

(1) Following the Referee comments we have verified our results and updated Fig. 3(b)
by correcting the curve showing the SCBA \alpha = 1 results which was incorrectly shifted
by \delta \omega = J/2 in the previous version of the paper.

(2) Following the Referee comments we have updated the Appendix by adding the following
two sentences, which discuss the equality between the SCBA equations used here and in Ref. [31]:

“The above result, with $z=4$ and $\alpha=1$, is equal to the self-energy calculated using
Eqs.~(21-23) in Ref.~\cite{Che99}: one merely needs to substitute in Eqs.~(21-23)
$\varepsilon \rightarrow \varepsilon - 2J$. This change is due to the differently defined zero energy level:
in Ref.~\cite{Che99} the zero energy level corresponds to the Ising antiferromagnet with one hole
whereas in the present paper the zero energy level corresponds to the Ising antiferromagnet.“

(3) Following the Referee comments we have modified two sentences in Sec. 4.3 (as well as removed
one last sentence of that paragraph) regarding the role of the Trugman loops in obtaining the numerically
exact ground state energy. These sentences now read:

“(i) the higher energy peaks contain incoherent spectral weight in the
ME method whereas they are of delta--like (``quasiparticle'') character
on the SCBA level;
(ii) although the energy of the ground state in the ME method and
in the SCBA method (for $\alpha=1$ and the ``canonical'' value of $J=0.4t$) is basically the same
at $\vect{k}=(\pi/2,\pi/2)$ point ($E=-1.58t$),
there is a small difference between the two results at, e.g., $\vect{k}=(0,0)$ point
($\delta E= 0.05 t$, since according to the ME method the ground state energy reads then $E=-1.63t$; unshown),
in agreement with Ref.~\cite{Che99} which suggests a slight variance between the SCBA
and numerical methods once $\vect{k} \neq (\pi/2,\pi/2)$ and $J=0.4t$.”

(4) We have added one last sentence to the caption of Fig. 2 which clarifies
the inclusion of the C1 and C2 constraints:

“Note that in the SCBA calculations for $\alpha=0$ ($\alpha=1$)
constraints $C1, C2$ are excluded (included), respectively. ”

(5) We have corrected minor typos throughout the text.