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Journal of Physics Condensed Matter

Magnetization and universal sub-critical behaviour in two-dimensional XY magnets

S. T. Bramwell, P. C. W. Holdsworth

Published Jan 25, 1993• 226 citations
Publisher

Abstract

Layered magnets, considered to be experimental realizations of the 2D XY model, have a magnetization with a characteristic exponent beta =0.23. They show, using modified renormalization group equations, that this value of beta is a universal signature of finite-sized 2D XY behaviour. They present simulation data in agreement with both the calculation and the experimental observations.

DOI
10.1088/0953-8984/5/4/004

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Layered magnets, considered to be experimental realizations of the 2D XY model, have a magnetization with a characteristic exponent beta =0.23. They show, using modified renormalization group equations, that this value of beta is a universal signature of finite-sized 2D XY behaviour. They present simulation data in agreement with both the calculation and the experimental observations.
In the low-temperature phase of the two-dimensional XY model, spin-spin correlations decay so slowly with distance that the thermodynamic limit is inaccessible. As a result the Mermin-Wagner theorem is inapplicable to any realizable system, and all have a measurable finite-size magnetization. We show that there is a regime of universal behavior near to the Kosterlitz-Thouless-Berezinskii transition. In experiments, the regime is identified by a magnetization exponent \ensuremath{\beta}=0.23, which we have calculated by a renormalization-group analysis. Monte Carlo simulations of an harmonic XY model clearly locate the boundaries of the universal regime, and identify it with vortex renormalization.
We improve the theoretical estimates of the critical exponents for the three-dimensional XY universality class. We fine alpha=-0.0146(8), gamma =1.3177(5), nu =0.671 55(27), eta =0.0380(4), beta =0.3485(2), and delta =4.780(2). We observe a discrepancy with the most recent experimental estimate of alpha; this discrepancy calls for further theoretical and experimental investigations. Our results are obtained by combining Monte Carlo simulations based on finite-size scaling methods, and high-temperature expansions. Two improved models (with suppressed leading scaling corrections) are selected by Monte Carlo computation. The critical exponents are computed from high-temperature expansions specialized to these improved models. By the same technique we determine the coefficients of the small-magnetization expansion of the equation of state. This expansion is extended analytically by means of approximate parametric representations, obtaining the equation of state in the whole critical region. We also determine the specific-heat amplitude ratio.
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.
Two-dimensional condensed matter is realized in increasingly diverse forms that are accessible to experiment and of potential technological value. The properties of these systems are influenced by many length scales and reflect both generic physics and chemical detail. To unify their physical description is therefore a complex and important challenge. Here we investigate the distribution of experimentally estimated critical exponents, beta, that characterize the evolution of the order parameter through the ordering transition. The distribution is found to be bimodal and bounded within a window similar to 0.1 <= beta <= 0.25, facts that are only in partial agreement with the established theory of critical phenomena. In particular, the bounded nature of the distribution is impossible to reconcile with the existing theory for one of the major universality classes of two-dimensional behaviour - the XY model with four-fold crystal field - which predicts a spectrum of non-universal exponents bounded only from below. Through a combination of numerical and renormalization group arguments we resolve the contradiction between theory and experiment and demonstrate how the 'universal window' for critical exponents observed in experiment arises from a competition between marginal operators.
The thermodynamics of the two-dimensional (2D) XY model is formulated by a transfer-matrix method and analyzed by a density-matrix renormalization group. The finite-size scaling and the beta function of the model are studied by the Roomany-Wyld renormalization-group theory. It is found that the 2D XY model has an essential finite-size effect and the Berezinskii-Kosterlitz-Thouless transition with the critical temperature TBKT=0.892{T}_{\mathrm{BKT}}=0.892TBKT​=0.892 appears in a finite system of 2000--3000 spins as a massless to massive transition with the effective critical temperature ${T}_{c}=1.07\ifmmode\pm\else\textpm\fi{}0.01.$