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

The zero-temperature, classical $XY$-model on an $L \times L$ square-lattice is studied by exploring the distribution $Φ_L(y)$ 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 $Φ_L(y)$, and the limit distribution $Φ_{L \rightarrow \infty} (y) = Φ_0(y)$ is obtained with high precision. The two leading finite-size corrections $Φ_L (y) -Φ_0 (y) \approx a_1(L)\, Φ_1(y) + a_2(L)\,Φ_2(y)$ 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 $Φ_1 (y)$ can be expressed through the low-order derivatives of the limit distribution, $Φ_1 (y) = [\,y\, Φ_0 (y) + Φ'_0 (y)\,]'$. The second finite-size correction has an amplitude $a_2(L)\propto 1/L^2$ and one finds that $a_2\,Φ_2(y) \ll a_1 \,Φ_1(y)$ 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 $T = 0$.