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Journal of Applied Physics

Universal magnetic fluctuations in the two-dimensional XY model

P. Archambault, S. T. Bramwell, Jean-Yves Fortin, P. C. W. Holdsworth, S. Peysson, Jean-François Pinton

Published Jun 1, 1998• 21 citations
Publisher

Abstract

We discuss the probability distribution function for the magnetic order parameter M, in the low temperature phase of the two-dimensional XY model. In this phase the system is critical over the whole range of temperature. The thermally averaged value of the order parameter 〈M〉, which is zero in the thermodynamic limit, has abnormally large finite size corrections. An exact result, within a spin wave calculation gives 〈M〉=(1/2N)T/8πJ, where J is the magnetic exchange constant and N the number of spins. We show, using Monte Carlo simulation, that the distribution function, Q(y−〈y〉), y=T−1LT/4πJM, is an asymmetric universal function. Using a diagramatic technique, we show that the asymmetry comes from three-spin and higher correlations. If only two-spin correlations are considered, the distribution is Gaussian. However, as there are contributions from two-spin terms separated by all distances, the distribution remains broad and is consistent with a divergent susceptibility.

DOI
10.1063/1.367855

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We discuss the probability distribution function for the magnetic order parameter M, in the low temperature phase of the two-dimensional XY model. In this phase the system is critical over the whole range of temperature. The thermally averaged value of the order parameter 〈M〉, which is zero in the thermodynamic limit, has abnormally large finite size corrections. An exact result, within a spin wave calculation gives 〈M〉=(1/2N)T/8πJ, where J is the magnetic exchange constant and N the number of spins. We show, using Monte Carlo simulation, that the distribution function, Q(y−〈y〉), y=T−1LT/4πJM, is an asymmetric universal function. Using a diagramatic technique, we show that the asymmetry comes from three-spin and higher correlations. If only two-spin correlations are considered, the distribution is Gaussian. However, as there are contributions from two-spin terms separated by all distances, the distribution remains broad and is consistent with a divergent susceptibility.
We calculate the two-dimensional probability for the magnetization in a two-dimensional XY model of finite size. We show that, for arbitrarily large N, there is a topological difference between the distributions in the low-temperature spin wave regime and in the high-temperature paramagnetic regime. In the low-temperature phase is a well-defined ring function and we calculate an upper bound for . Even so, this is consistent with the susceptibility per spin, , being divergent. We show further that the distribution function Q(M) for the scalar magnetization has a universal form, scaling with the single variable , where L is the system size and J is the coupling constant. We show that has considerable structure, with two terms of order N, one due to the spin waves and the other due to vortices. This leads to a peak in at the Kosterlitz - Thouless - Berezinskii transition.
We present a detailed investigation of the probability density function (PDF) of order parameter fluctuations in the finite two-dimensional XY (2d XY ) model. In the low-temperature critical phase of this model, the PDF approaches a universal non-Gaussian limit distribution in the limit T → 0. Our analysis resolves the question of temperature dependence of the PDF in this regime, for which conflicting results have been reported. We show analytically that a weak temperature dependence results from the inclusion of multiple loop graphs in a previously derived graphical expansion. This is confirmed by numerical simulations on two controlled approximations to the 2d XY model: the harmonic and ‘harmonic XY ’ models. The harmonic model has no Kosterlitz–Thouless–Berezinskiĭ (KTB) transition and the PDF becomes progressively less skewed with increasing temperature until it closely approximates a Gaussian function above T ≈ 4π. Near to that temperature, we find some evidence of a phase transition, although our observations appear to exclude a thermodynamic singularity.
We calculate the probability density function for the order-parameter fluctuations in the low-temperature phase of the two-dimensional XY\mathrm{XY}XY model of magnetism near the line of critical points. A finite correlation length $\ensuremath{\xi},$ is introduced with a small magnetic field h, and an expression for $\ensuremath{\xi}(h)$ is developed by treating nonlinear contributions to the field energy using a Hartree approximation. We find analytically a series of universal non-Gaussian distributions of the finite-size scaling form $P(m,L,\ensuremath{\xi})\ensuremath{\sim}{L}^{\ensuremath{\beta}/\ensuremath{\nu}}{P}_{L}{(mL}^{\ensuremath{\beta}/\ensuremath{\nu}},\ensuremath{\xi}/L)$ and present a function of the form $P(x)\ensuremath{\sim}{\mathrm{exp}[x\ensuremath{-}\mathrm{exp}(x)]{}}^{a(h)}$ that gives the probability density functions to an excellent approximation. We propose a(h)a(h)a(h) as an indirect measure of the length scale of correlations in a wide range of complex systems.
We calculate the high-temperature series of the magnetic susceptibility and the second and fourth moments of the correlation function for the XY model on the square lattice to order beta;{33} by applying the improved algorithm of the finite lattice method. The long series allow us to estimate the inverse critical temperature as beta_{c}=1.1200(1) , which is consistent with the most precise value given previously by the Monte Carlo simulation. The critical exponent for the multiplicative logarithmic correction is evaluated to be theta=0.054(10) , which is consistent with the renormalization group prediction of theta=1/16 .
We study the square-lattice XY model in the presence of random phase shifts. We consider two different disorder distributions with zero average shift and investigate the low-temperature quasi-long-range order phase which occurs for sufficiently low disorder. By means of Monte Carlo simulations we determine several universal quantities which are then compared with the analytic predictions of the random spin-wave theory. We observe a very good agreement which indicates that the universal long-distance behavior in the whole low-disorder low-temperature phase is fully described by the random spin-wave theory.
This thesis addresses the nature of global (many body) fluctuations in highly correlated systems. We begin with the question of temperature dependence in finite two dimensional XY (2dXY) model magnets. Such systems have a fully critical low temperature phase. It is shown analytically, backed up by extensive Monte Carlo simulations, that the non-Gaussian distribution of order parameter fluctuations is not strictly universal but has an explicit temperature dependence - contrary to previous findings. The temperature dependence is used to explain why past studies derived the same distribution for fluctuations of the full order parameter and an approximate linearized form. The appearance of spin vortices in the related Harmonic model is discussed and an argument is presented for why these defects must always appear as bound pairs. The linearized order parameter of the 2dXY model leads to a family of dimensionally dependent models defined in reciprocal space. An argument is presented for the interpretation of these systems as being critical and a direct space Hamiltonian is derived for the one dimensional case. This model has order parameter fluctuations distributed according to the Fisher-Tippett-Gumbel distribution from extreme value statistics (EVS). The link between criticality and EVS is investigated, as are the origins of the non-Gaussianity. The ability to distinguish between critical distributions is discussed. It is seen that for one, two and three dimensions the critical models presented lead to functionally similar fluctuation distributions. A previously reported link between EVS and l/f noise is investigated. Our one dimensional critical model is mapped onto the action used to generate l/f signals and we propose an alternative interpretation of the link in the context of a 1/q dispersion of spatial normal modes. The experimental observability of the FTG distribution in l/f signals is considered with emphasis on imperfections in the noise. A physically relevant method of generating l/f noise from the superposition of random telegraph signals is also examined.