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arXiv:cond-mat/0507424 · Journal of Physics A Mathematical and General

Temperature-dependent fluctuations in the two-dimensionalXYmodel

Simon T. Banks, S. T. Bramwell

Published Jun 8, 2005• 15 citations
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Abstract

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.

DOI
10.1088/0305-4470/38/25/001

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•Jan 1, 2005•0 citations
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 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.
Monte Carlo simulations of the two-dimensional XY model are performed in a square geometry with fixed boundary conditions. Using a conformal mapping, it is very easy to deduce the exponent η σ ( T ) of the order parameter correlation function at any temperature in the critical phase of the model. The temperature behaviour of η σ ( T ) is obtained numerically with a good accuracy up to the Kosterlitz-Thouless transition temperature. At very low temperatures, a good agreement is found with Berezinskii's harmonic approximation. Surprisingly, we show some evidence that there are no logarithmic corrections to the behaviour of the order parameter density profile (with symmetry-breaking surface fields) at the Kosterlitz-Thouless transition temperature.
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.
Monte Carlo simulations of the two-dimensional XY model are performed in a square geometry with various boundary conditions (BC). Using conformal mappings we deduce the exponent η σ ( T ) of the order parameter correlation function and its surface analogue η ∥ ( T ) as a function of the temperature in the critical (low-temperature) phase of the model. The temperature dependence of both exponents is obtained numerically with a good accuracy up to the Kosterlitz–Thouless transition temperature. The bulk exponent follows from simulations of correlation functions with periodic boundary conditions or order parameter profiles with open boundary conditions and with symmetry breaking surface fields. At very low temperatures, η σ ( T ) is found in pretty good agreement with the linear temperature-dependence of Berezinskii's spin-wave approximation. We also show some evidence that there are no noticeable logarithmic corrections to the behaviour of the order parameter density profile at the Kosterlitz–Thouless (KT) transition temperature, while these corrections exist for the correlation function. At the KT transition the value η σ ( T KT ) = 1/4 is accurately recovered. The exponent associated with the surface correlations is similarly obtained after a slight modification of the boundary conditions: the correlation function is computed with free BC, and the profile with mixed fixed–free BC. It exhibits a monotonic behaviour with temperature, starting linearly according to the spin-wave approximation and increasing up to a value η ∥ ( T KT ) ≃ 1/2 at the Kosterlitz–Thouless transition temperature. The thermal exponent η ε ( T ) is also computed and we give some evidence that it keeps a constant value in agreement with the marginality condition of the temperature field below the KT transition.
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 ${T}_{c}=1.07\ifmmode\pm\else\textpm\fi{}0.01.$
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.