Finite-size corrections to scaling of the magnetization distribution in the two-dimensional XY model at zero temperature

摘要

The zero-temperature, classical XY model on an L × 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→∞)(y) = Φ_0(y) is obtained with high precision. The two leading finite-size corrections Φ_L(y) ? Φ_0(y) ≈ 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) ]'. Thus, Φ_1(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 a_2(L) ∝ 1/L~2 and one finds that a_2 Φ_2(y) 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.