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Journal of Physics A Mathematical and General

Planar model correlation functions

Albert B. Zisook, Leo P. Kadanoff

Published Oct 1, 1980• 5 citations
Publisher

Abstract

Correlation functions are calculated on the critical line of the two-dimensional planar or XY model. A crossover is found between simple scaling behaviour for 2 pi K eff >4, and more complex scaling behaviour for 2 pi K eff =4. At the Kosterlitz-Thouless critical point, 2 pi K eff =4, logarithmic factors appear in the correlation functions. These corrections to simple scaling behaviour are calculated for many correlation functions, including correlation functions not previously studied.

DOI
10.1088/0305-4470/13/10/007

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Correlation functions are calculated on the critical line of the two-dimensional planar or XY model. A crossover is found between simple scaling behaviour for 2 pi K eff >4, and more complex scaling behaviour for 2 pi K eff =4. At the Kosterlitz-Thouless critical point, 2 pi K eff =4, logarithmic factors appear in the correlation functions. These corrections to simple scaling behaviour are calculated for many correlation functions, including correlation functions not previously studied.
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.
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.
Using two sets of high-precision Monte Carlo data for the two-dimensional XY model in the Villain formulation on square L\ifmmode\times\else\texttimes\fi{}L lattices, the scaling behavior of the susceptibility \ensuremath{\chi} and correlation length \ensuremath{\xi} at the Kosterlitz-Thouless phase transition is analyzed with emphasis on multiplicative logarithmic corrections (ln L${)}^{\mathrm{\ensuremath{-}}2\mathrm{r}}$ in the finite-size scaling region and (ln \ensuremath{\xi}${)}^{\mathrm{\ensuremath{-}}2\mathrm{r}}$ in the high-temperature phase near criticality, respectively. By analyzing the susceptibility at criticality on lattices of size up to 5122{512}^{2}5122 we obtain r=-0.0270(10), in agreement with recent work of Kenna and Irving on the finite-size scaling of Lee-Yang zeros in the cosine formulation of the XY model. By studying susceptibilities and correlation lengths up to \ensuremath{\xi}\ensuremath{\approx}140 in the high-temperature phase, however, we arrive at quite a different estimate of r=0.0560(17), which is in good agreement with recent analyses of thermodynamic Monte Carlo data and high-temperature series expansions of the cosine formulation.
A number of two-dimensional (2D) critical phenomena can be described in terms of the 2D sine-Gordon model. With bosonization, several 1D quantum systems can be transformed to the same model. However, the transition of the 2D sine-Gordon model, the Berezinskii-Kosterlitz-Thouless (BKT) transition, is essentially different from a second-order transition. The divergence of the correlation length is more rapid than any power law, and there are logarithmic corrections. These pathological features make it difficult to determine the BKT transition point and critical indices from finite-size calculations. In this paper we calculate correlation functions of this model using a real-space renormalization technique. It is found that several correlation functions, or eigenvalues of the corresponding transfer matrix for a finite system, become degenerate on the BKT line, including the logarithmic corrections. By the use of this degeneracy, which reflects the hidden SU(2) symmetry on the BKT line, it is possible to determine the BKT critical line with high precision from a small amount of data and to identify the universality class. In addition, new universal relations are found. These results shed light on the relation between Abelian and non-Abelian bosonization.
We have analyzed new twelfth-order high-temperature series for the susceptibility and correlation length of classical planar models on the triangular lattice using an nnn-fit method of analysis tailored to the form of the singularity $A\mathrm{exp}(b{t}^{\ensuremath{-}\ensuremath{\nu}})$ predicted by Kosterlitz and Thouless. Test-function analysis shows that the nnn-fit method is significantly more reliable in treating a number of possible corrections to the leading singularity than is the DDD log Pad\'e analysis of the logarithm and logarithmic derivative used in earlier series work on tenth-order series. Our nnn-fit analysis leads to the results $\ensuremath{\nu}=0.5\ifmmode\pm\else\textpm\fi{}0.1$ and $\ensuremath{\eta}=0.27\ifmmode\pm\else\textpm\fi{}0.03$ in good agreement with the Kosterlitz-Thouless predictions $\ensuremath{\nu}=\frac{1}{2}$ and $\ensuremath{\eta}=\frac{1}{4}$.