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Journal of Physics A Mathematical and General

Series of effective-field approximations and coherent anomaly in Kosterlitz-Thouless transitions

Xiao Hu

Published Apr 7, 1994• 2 citations
Publisher

Abstract

A series of effective-field approximations are formulated for the Kosterlitz-Thouless transition in the sine-Gordon model by means of the cumulant expansion and the variational method. Effective-field essential singularities xi approximately exp( xi n (K c (n) -K)-1) are obtained for the correlation length as first derived by Saito in a single approximation. However, a systematic variance of the effective-field critical coefficient xi n approximately 2ln(J/2y 0 )/(n+1) pi is found when the order of approximation n increases. The true critical exponent nu of the Kosterlitz-Thouless transition is thus revealed to be less than the effective-field exponent nu 0 , nu < nu 0 =1, from Suzuki's coherent-anomaly method. The phase transition in the two-dimensional XY model is studied from its relation to the sine-Gordon model. The critical exponent eta c , of the spin-spin correlation function at the critical point is found to be eta c =1/(4+1/ pi 2 ).

DOI
10.1088/0305-4470/27/7/013

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A series of effective-field approximations are formulated for the Kosterlitz-Thouless transition in the sine-Gordon model by means of the cumulant expansion and the variational method. Effective-field essential singularities xi approximately exp( xi n (K c (n) -K)-1) are obtained for the correlation length as first derived by Saito in a single approximation. However, a systematic variance of the effective-field critical coefficient xi n approximately 2ln(J/2y 0 )/(n+1) pi is found when the order of approximation n increases. The true critical exponent nu of the Kosterlitz-Thouless transition is thus revealed to be less than the effective-field exponent nu 0 , nu < nu 0 =1, from Suzuki's coherent-anomaly method. The phase transition in the two-dimensional XY model is studied from its relation to the sine-Gordon model. The critical exponent eta c , of the spin-spin correlation function at the critical point is found to be eta c =1/(4+1/ pi 2 ).
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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 present detailed Monte Carlo results for the susceptibility \ensuremath{\chi}, correlation length \ensuremath{\xi}, and specific heat Cv{\mathit{C}}_{\mathit{v}}Cv​ for the XY model. The simulations are done on 642{64}^{2}642, 1282{128}^{2}1282, 2562{256}^{2}2562, and 5122{512}^{2}5122 lattices over the temperature range 0.98\ensuremath{\le}T\ensuremath{\le}1.43 corresponding to 270. Fits to \ensuremath{\chi} and \ensuremath{\xi} data favor a Kosterlitz-Thouless (KT) singularity over a second-order transition; however, unconstrained four-parameter KT fits do not confirm the predicted values \ensuremath{\nu}=0.5 and \ensuremath{\eta}=0.25. Our best estimate, Tc{\mathit{T}}_{\mathit{c}}Tc​=0.894(5), is obtained using KT fits with \ensuremath{\nu} fixed at 0.5. The exponent \ensuremath{\eta} is calculated as a function of temperature in the spin-wave phase using Monte Carlo renormalization-group and finite-size-scaling methods. Both methods give consistent results and we find \ensuremath{\eta}\ensuremath{\approxeq}0.235 at T=0.894. We also present results for the behavior of vortex density across the transition and exhibit how the dilute-gas approximation breaks down.
High-temperature series expansions of the spin-spin correlation function for the plane rotator (or XY) model on the square lattice are extended by three terms through order ${\mathrm{\ensuremath{\beta}}}^{20}$. Tables of the expansion coefficients are reported for the correlation function spherical moments of order l=0,1,2. The expansion coefficients through ${\mathrm{\ensuremath{\beta}}}^{15}$ for the vorticity are also tabulated. Our analysis of the series supports the Kosterlitz-Thouless predictions on the structure of the critical singularities and leads to fairly accurate estimates of the critical parameters.
A Kosterlitz-Thouless (KT)-type transition in the one-dimensional S =1/2 XXZ model at the ground state is analyzed using the double-cluster approximation (DCA) and the coherent-anomaly method (CAM). It is numerically shown that the periodic-boundary condition (PBC) is suitable for the construction of a systematic series of cluster approximations in quantum antiferromagnets. The transition point and the exponent of the essential singularity are estimated as g c * ≃0.9969 and σ≃0.527, which are in good agreement with the exact values, g c * =1 and σ=0.5, respectively. Such accuracy of estimation is due to the consideration of a power-law factor in front of the exponential singularity of \barχ, which had been neglected until now.