## SciPost Submission Page

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

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

#### - Published as SciPost Phys. 7, 066 (2019)

### Submission summary

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

Arxiv Link: | https://arxiv.org/abs/1809.07120v6 |

Date accepted: | 2019-11-22 |

Date submitted: | 2019-11-12 |

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.

### Ontology / Topics

See full Ontology or Topics database.Published as SciPost Phys. 7, 066 (2019)

### Author comments upon resubmission

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/

### List of changes

(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.