The zero-temperature, classical $XY$-model on an $L \times L$ square-latticeis studied by exploring the distribution $\Phi_L(y)$ of its centered andnormalized magnetization $y$ in the large $L$ limit. An integral representationof the cumulant generating function, known from earlier works, is used for thenumerical evaluation of $\Phi_L(y)$, and the limit distribution $\Phi_{L\rightarrow \infty} (y) = \Phi_0(y)$ is obtained with high precision. The twoleading 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 fromanalytic 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 beexpressed through the low-order derivatives of the limit distribution, $\Phi_1(y) = [\,y\, \Phi_0 (y) + \Phi'_0 (y)\,]'$. The second finite-size correctionhas an amplitude $a_2(L)\propto 1/L^2$ and one finds that $a_2\,\Phi_2(y) \lla_1 \,\Phi_1(y)$ already for small system size ($L> 10$). We illustrate thefeasibility of observing the calculated finite-size corrections by performingsimulations of the $XY$-model at low temperatures, including $T = 0$.
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