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arXiv:1604.00948 · Physical review. E

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

Guillermo Palma, Ferenc Niedermayer A, Zoltàn Ràcz, A. Riveros, D. Zambrano

Published Aug 29, 2016• 4 citations
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Abstract

The zero-temperature, classical $XY$ model on an $L\ifmmode\times\else\texttimes\fi{}L$ square lattice is studied by exploring the distribution ${\mathrm{\ensuremath{\Phi}}}_{L}(y)$ 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 ${\mathrm{\ensuremath{\Phi}}}_{L}(y)$, and the limit distribution ${\mathrm{\ensuremath{\Phi}}}_{L\ensuremath{\rightarrow}\ensuremath{\infty}}(y)={\mathrm{\ensuremath{\Phi}}}_{0}(y)$ is obtained with high precision. The two leading finite-size corrections ${\mathrm{\ensuremath{\Phi}}}_{L}(y)\ensuremath{-}{\mathrm{\ensuremath{\Phi}}}_{0}(y)\ensuremath{\approx}{a}_{1}(L)\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Phi}}}_{1}(y)+{a}_{2}(L)\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Phi}}}_{2}(y)$ 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 ${\mathrm{\ensuremath{\Phi}}}_{1}(y)$ can be expressed through the low-order derivatives of the limit distribution, ${\mathrm{\ensuremath{\Phi}}}_{1}(y)={[\phantom{\rule{0.16em}{0ex}}y\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Phi}}}_{0}(y)+{\mathrm{\ensuremath{\Phi}}}_{0}^{\ensuremath{'}}(y)\phantom{\rule{0.16em}{0ex}}]}^{\ensuremath{'}}$. Thus, ${\mathrm{\ensuremath{\Phi}}}_{1}(y)$ 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 ${a}_{2}(L)\ensuremath{\propto}1/{L}^{2}$ and one finds that ${a}_{2}\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Phi}}}_{2}(y)\ensuremath{\ll}{a}_{1}\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Phi}}}_{1}(y)$ 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 $T=0$.

DOI
10.1103/physreve.94.022145

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The zero-temperature, classical XYXYXY model on an L\ifmmode×\else\texttimes\fiLL\ifmmode\times\else\texttimes\fi{}LL\ifmmode×\else\texttimes\fiL square lattice is studied by exploring the distribution \ensuremathΦL(y){\mathrm{\ensuremath{\Phi}}}_{L}(y)\ensuremathΦL​(y) of its centered and normalized magnetization yyy in the large-LLL limit. An integral representation of the cumulant generating function, known from earlier works, is used for the numerical evaluation of \ensuremathΦL(y){\mathrm{\ensuremath{\Phi}}}_{L}(y)\ensuremathΦL​(y), and the limit distribution \ensuremathΦL\ensuremath→\ensuremath∞(y)=\ensuremathΦ0(y){\mathrm{\ensuremath{\Phi}}}_{L\ensuremath{\rightarrow}\ensuremath{\infty}}(y)={\mathrm{\ensuremath{\Phi}}}_{0}(y)\ensuremathΦL\ensuremath→\ensuremath∞​(y)=\ensuremathΦ0​(y) is obtained with high precision. The two leading finite-size corrections \ensuremathΦL(y)\ensuremath−\ensuremathΦ0(y)\ensuremath≈a1(L)\ensuremathΦ1(y)+a2(L)\ensuremathΦ2(y){\mathrm{\ensuremath{\Phi}}}_{L}(y)\ensuremath{-}{\mathrm{\ensuremath{\Phi}}}_{0}(y)\ensuremath{\approx}{a}_{1}(L)\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Phi}}}_{1}(y)+{a}_{2}(L)\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Phi}}}_{2}(y)\ensuremathΦL​(y)\ensuremath−\ensuremathΦ0​(y)\ensuremath≈a1​(L)\ensuremathΦ1​(y)+a2​(L)\ensuremathΦ2​(y) are also extracted both from numerics and from analytic calculations. We find that the amplitude a1(L){a}_{1}(L)a1​(L) scales as ln(L/L0)/L2ln(L/{L}_{0})/{L}^{2}ln(L/L0​)/L2 and the shape correction function \ensuremathΦ1(y){\mathrm{\ensuremath{\Phi}}}_{1}(y)\ensuremathΦ1​(y) can be expressed through the low-order derivatives of the limit distribution, \ensuremathΦ1(y)=[y\ensuremathΦ0(y)+\ensuremathΦ0\ensuremath′(y)]\ensuremath′{\mathrm{\ensuremath{\Phi}}}_{1}(y)={[\phantom{\rule{0.16em}{0ex}}y\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Phi}}}_{0}(y)+{\mathrm{\ensuremath{\Phi}}}_{0}^{\ensuremath{'}}(y)\phantom{\rule{0.16em}{0ex}}]}^{\ensuremath{'}}\ensuremathΦ1​(y)=[y\ensuremathΦ0​(y)+\ensuremathΦ0\ensuremath′​(y)]\ensuremath′. Thus, \ensuremathΦ1(y){\mathrm{\ensuremath{\Phi}}}_{1}(y)\ensuremathΦ1​(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)\ensuremath∝1/L2{a}_{2}(L)\ensuremath{\propto}1/{L}^{2}a2​(L)\ensuremath∝1/L2 and one finds that a2\ensuremathΦ2(y)\ensuremath≪a1\ensuremathΦ1(y){a}_{2}\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Phi}}}_{2}(y)\ensuremath{\ll}{a}_{1}\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Phi}}}_{1}(y)a2​\ensuremathΦ2​(y)\ensuremath≪a1​\ensuremathΦ1​(y) already for small system size (L>10L>10L>10). We illustrate the feasibility of observing the calculated finite-size corrections by performing simulations of the XYXYXY model at low temperatures, including T=0T=0T=0.
In recent works [S. T. Bramwell, P. C. W. Holdsworth, and J.-F. Pinton, Nature (London) 396, 552 (1998); S. T. Bramwell et al., Phys. Rev. Lett. 84, 3744 (2000)], a generalized universality has been proposed, linking phenomena as dissimilar as two-dimensional (2D) magnetism and turbulence. To test these ideas, we performed Monte Carlo simulations of the 2D XY model. We found that the shape of the probability distribution function for the magnetization M is non-Gaussian and independent of the system size-in the range of the lattice sizes studied-below the Kosterlitz-Thoules temperature. However, our results suggest that in the full 2D XY model the shape of these distributions has a slight dependence on temperature-for finite volume-below the lattice-shifted critical temperature T*(L). This behavior can be explained by using renormalization group arguments and an extended finite-size scaling analysis, and by the existence of bounds for M.
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The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction ~1/L^0.25 of a very small magnitude also has been detected, as it can be expected from our recently developed GFD (grouping of Feynman diagrams) theory. Monte Carlo simulations of the squared magnetization of 3D Ising model have been performed by Wolff's algorithm in the range of the reduced temperatures t =< 0.000086 and system sizes L =< 410. The effective critical exponent beta_eff(t) tends to increase above the currently accepted numerical values. The critical coupling K_c=0.22165386(51) has been extracted from the Binder cumulant data within 96 =< L =< 384. The critical exponent 1/nu, estimated from the finite-size scaling of the derivatives of the Binder cumulant, tends to decrease slightly below the RG value 1.587 for the largest system sizes. The finite-size scaling of accurately simulated maximal values of the specific heat C_v in 3D Ising model confirms a logarithmic rather than power-like critical singularity of C_v.
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