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Journal of Applied Physics

The specific heat of planar spin models

Martin V. Ferer, A. Hamid Aidinejad, Michael J. Velgakis

Published Mar 15, 1984• 3 citations
Publisher

Abstract

In an attempt to understand the discrepancy between renormalization group predictions of a rounded specific heat peak about 38% above the Kosterlitz–Thouless transition temperature Tc and Monte-Carlo results indicating a sharp peak about 7% above Tc, we have undertaken an analysis of our thirteenth-order, high-temperature series for the free energy of the XY model on the triangular lattice. We have performed N fits on our free energy series in order to estimate the dominant singularity. These results are not consistent with either F≂ξ−2 or a transition at temperatures above Tc. Using these estimates of the leading singularity, we have extrapolated partial sums of our specific heat series for temperatures above Tc, thereby locating the specific heat peak.

DOI
10.1063/1.333687

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In an attempt to understand the discrepancy between renormalization group predictions of a rounded specific heat peak about 38% above the Kosterlitz–Thouless transition temperature Tc and Monte-Carlo results indicating a sharp peak about 7% above Tc, we have undertaken an analysis of our thirteenth-order, high-temperature series for the free energy of the XY model on the triangular lattice. We have performed N fits on our free energy series in order to estimate the dominant singularity. These results are not consistent with either F≂ξ−2 or a transition at temperatures above Tc. Using these estimates of the leading singularity, we have extrapolated partial sums of our specific heat series for temperatures above Tc, thereby locating the specific heat peak.
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 $\ensuremath{\eta}=\frac{1}{4}$.
Using the tensor renormalization group method based on the higher-order singular value decomposition, we have studied the thermodynamic properties of the continuous XY model on the square lattice. The temperature dependence of the free energy, the internal energy, and the specific heat agree with the Monte Carlo calculations. From the field dependence of the magnetic susceptibility, we find the Kosterlitz-Thouless transition temperature to be 0.8921(19), consistent with the Monte Carlo as well as the high temperature series expansion results. At the transition temperature, the critical exponent δ is estimated as 14.5, close to the analytic value by Kosterlitz.
In this paper we report analyses of high temperature series expansions for the spin [Formula: see text] XY model on the triangular and square lattices. Quantities for which series are analyzed include the fluctuation in the transverse magnetization, fourth order fluctuations in the same quantity, second and fourth moments of the transverse spin–spin correlations, specific heat, and entropy. The evidence favours a phase transition at a finite temperature with conventional power law critical singularities. Scaling seems to hold but hyperscaling seems to be violated. Estimates for critical exponents include γ = 2.50 ± 0.3. Δ = 2.38 ± 0.2, and ν = 143 ± 0.10. The specific heat exhibits no singular behaviour at Tc.
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.
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