Finite-size corrections to scaling of the magnetization distribution in the $2d$ $XY$-model at zero temperature

TL;DR

This paper analyzes finite-size effects on the magnetization distribution in the 2D XY-model at zero temperature, deriving precise corrections and demonstrating their observability through simulations.

Contribution

It provides the first detailed analytical and numerical characterization of finite-size corrections to the magnetization distribution in the 2D XY-model at zero temperature.

Findings

Finite-size correction amplitude a_1(L) scales as ln(L)/L^2.

Shape correction function Φ_1(y) relates to derivatives of the limit distribution.

Second correction a_2(L) is proportional to 1/L^2 and is smaller than the first correction for L > 10.

Abstract

The zero-temperature, classical $XY$-model on an $L \times L$ square-lattice is studied by exploring the distribution $\Phi_L(y)$ of its centered and normalized magnetization $y$ in the large $L$ limit. An integral representation of the cumulant generating function, known from earlier works, is used for the numerical evaluation of $\Phi_L(y)$, and the limit distribution $\Phi_{L \rightarrow \infty} (y) = \Phi_0(y)$ is obtained with high precision. The two leading finite-size corrections $\Phi_L (y) -\Phi_0 (y) \approx a_1(L)\, \Phi_1(y) + a_2(L)\,\Phi_2(y)$ are also extracted both from numerics and from analytic calculations. We find that the amplitude $a_1(L)$ scales as $\ln(L/L_0) /L^2$ and the shape correction function $\Phi_1 (y)$ can be expressed through the low-order derivatives of the limit distribution, $\Phi_1 (y) = [\,y\, \Phi_0 (y) + \Phi'_0 (y)\,]'$. The second finite-size correction has an amplitude $a_2(L)\propto 1/L^2$ and one finds that $a_2\,\Phi_2(y) \ll a_1 \,\Phi_1(y)$ already for small system size ($L> 10$). We illustrate the feasibility of observing the calculated finite-size corrections by performing simulations of the $XY$-model at low temperatures, including $T = 0$.