Finite-Size Corrections to Scaling of the Magnetization Distribution in the Two-Dimensional Xy Model at Zero Temperature

The zero-temperature, classical XY model on an L×L square lattice is studied by exploring the distribution ΦL(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 ΦL(y), and the limit distribution ΦL→∞(y)=Φ0(y) is obtained with high precision. The two leading finite-size corrections ΦL(y)-Φ0(y)≈a1(L)Φ1(y)+a2(L)Φ2(y) are also extracted both from numerics and from analytic calculations. We find that the amplitude a1(L) scales as ln(L/L0)/L2 and the shape correction function Φ1(y) can be expressed through the low-order derivatives of the limit distribution, Φ1(y)=[yΦ0(y)+Φ0′(y)]′. Thus, Φ1(y) carries the same universal features as the limit distribution and can be used for consistency checks of universality claims based on finite-size systems. The second finite-size correction has an amplitude a2(L)∝1/L2 and one finds that a2Φ2(y)a1Φ1(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. © 2016 American Physical Society.