The probability density function (PDF) of some global average quantity plays a fundamental role in critical and highly correlated systems. We explicitly compute this quantity as a function of the magnetization for the two dimensional XY model in its harmonic approximation. Numerical simulations and perturbative results have shown a Gumbel-like shape of the PDF, in spite of the fact that the average magnetization is not an extreme variable. Our analytical result allows to test both perturbative analytical expansions and also numerical computations performed previously. Perfect agreement is found for the first moments of the PDF. Also for large volume and in the high temperature limit the distribution becomes Gaussian, as it should be. In the low temperature regime its numerical evaluation is compatible with a Gumbel distribution.
The probability density function (PDF) of some global average quantity plays a fundamental role in critical and highly correlated systems. We explicitly compute this quantity as a function of the magnetization for the two dimensional XY model in its harmonic approximation. Numerical simulations and perturbative results have shown a Gumbel-like shape of the PDF, in spite of the fact that the average magnetization is not an extreme variable. Our analytical result allows to test both perturbative analytical expansions and also numerical computations performed previously. Perfect agreement is found for the first moments of the PDF. Also for large volume and in the high temperature limit the distribution becomes Gaussian, as it should be. In the low temperature regime its numerical evaluation is compatible with a Gumbel distribution.
We compute exact analytical expressions for the skewness and kurtosis in the two-dimensional harmonic XY model. These quantities correspond to the third and fourth normalized moments of the probability density function (PDF) of the magnetization of the model. From their behavior, we conclude that they depend explicitly on the system temperature even in the thermodynamic limit, and hence the PDF itself must depend on it. Our results correct the hypothesis called universal fluctuations, they confirm and extend previous results which showed a T dependence of the PDF, including perturbative expansions within the XY model up to first order in temperature.
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.
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.
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 propose an interpretation of previous experiments and numerical experiments showing that, for a large class of systems, distributions of global quantities are similar to a distribution originally obtained for the magnetization in the 2D-XY model (Bramwell S. T. et al. , Nature , 396 (1998) 512). This approach, developed for the Ising model, is based on previous numerical observations (Clusel M. et al. , Phys. Rev. E , 70 (2004) 046112). We obtain an effective action using a perturbative method, which successfully describes the order parameter fluctuations near the phase transition. This leads to a direct link between the D-dimensional Ising model and the XY model in the same dimension, which appears to be a generic feature of many equilibrium critical systems and which is at the heart of the above observations.
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.