# The BKT Universality Class in the Presence of Correlated Disorder

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Scaling from $T\to {T}_{BKT}^{-}$

#### 2.2. Scaling from $T>{T}_{BKT}$

## 3. Methods

## 4. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**(

**a**) Rescaled curves of the superfluid stiffness ${J}_{s}\left(T\right)$ by its zero-temperature value ${J}_{s}(T=0)$ for the clean case, the uncorrelated ${P}_{eff}$ and correlated random transverse field (RTF) disordered case at $W/J=10$. The temperature axis has also been rescaled by the value of the Berezinskii-Kosterlitz-Thouless (BKT) critical temperature obtained from the intersection between the critical line $2T/\pi $ and the superfluid-stiffness itself [34]. Despite the strong disorder, the ${P}_{eff}$ curve shows only a small finite-size effect above ${T}_{c}$, while the RTF stiffness is dramatically modified above and below the transition. (

**b**) Maps of the couplings ${J}_{i,i+x}$ for both the spatially uncorrelated (${P}_{eff}$) and correlated (RTF) disorder. The disorder level has been fixed to $W/J=10$ while the linear size of the system is $L=128$.

**Figure 2.**Temperature dependence of the superfluid stiffness for different values of the linear sizes L. The panel (

**a**) corresponds to the homogeneous case, while panel. (

**b**) to the disordered $RTF$ case at $W/J=10$. The solid black line in both the panels is the critical line $2T/\pi $ whose intersection with ${J}_{s}\left(T\right)$ would correspond to the critical point where the superfluid-stiffness jump is expected to occur.

**Figure 3.**Rescaling of the superfluid stiffness curves by means of Equation (9) both for (

**a**) the clean case, (

**b**) the $RTF$ disordered case with $W/J=10$. In the presence of disorder, for a better comparison with the clean case, one can rescale both the superfluid stiffness and the temperature by ${J}_{eff}={J}_{s}(T=0)$.

**Figure 4.**Superfluid stiffness curves of different linear size L, renormalised and collapsed on the same universal curve relative to the high temperature regime: $T>{T}_{BKT}$. (

**a**) Clean case (

**b**) RTF disordered case with $W/J=10$.

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Maccari, I.; Benfatto, L.; Castellani, C.
The BKT Universality Class in the Presence of Correlated Disorder. *Condens. Matter* **2018**, *3*, 8.
https://doi.org/10.3390/condmat3010008

**AMA Style**

Maccari I, Benfatto L, Castellani C.
The BKT Universality Class in the Presence of Correlated Disorder. *Condensed Matter*. 2018; 3(1):8.
https://doi.org/10.3390/condmat3010008

**Chicago/Turabian Style**

Maccari, Ilaria, Lara Benfatto, and Claudio Castellani.
2018. "The BKT Universality Class in the Presence of Correlated Disorder" *Condensed Matter* 3, no. 1: 8.
https://doi.org/10.3390/condmat3010008