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Progress of Theoretical Physics

Singularity of Specific Heat in the Second Order Phase Transition

Ryuzo Abe

Published Aug 1, 1967• 58 citations
Publisher

Abstract

On the basis of Fisher's method, singularity of specific heat near the transition point is studied for the Ising or Heisenberg model with an arbitrary value of spin and range of interaction. The following results are obtained for a reasonable distribution of zeros of canonical partition function. (a) If the specific heat has a singularity C+(T-Tc)-α above the transition point Tc, it should have a singularity C-(Tc-T)-α below Tc, C- being not necessarily equal to C+. (b) If the specific heat has a logarithmic singularity -Aln(T-Tc)+B+ above Tc, it should have a singularity -Aln(Tc-T)+B- below Tc, B- being not necessarily equal to B+. Some theoretical and experimental works so far reported are discussed in the light of these results.

DOI
10.1143/ptp.38.322

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On the basis of Fisher's method, singularity of specific heat near the transition point is studied for the Ising or Heisenberg model with an arbitrary value of spin and range of interaction. The following results are obtained for a reasonable distribution of zeros of canonical partition function. (a) If the specific heat has a singularity C+(T-Tc)-α above the transition point Tc, it should have a singularity C-(Tc-T)-α below Tc, C- being not necessarily equal to C+. (b) If the specific heat has a logarithmic singularity -Aln(T-Tc)+B+ above Tc, it should have a singularity -Aln(Tc-T)+B- below Tc, B- being not necessarily equal to B+. Some theoretical and experimental works so far reported are discussed in the light of these results.
We study the energy fluctuations in 3D Ising model near the phase transition point. Specific heat is a relevant quantity which is directly related to the mean squared amplitude of the energy fluctuations in the system. We have made extensive Monte Carlo simulations in 3D Ising model to clarify the character of the singularity of the specific heat Cv based on the finite-size scaling of its maximal values Cvmax depending on the linear size of the lattice L. An original iterative method has been used which automatically finds the pseudocritical temperature corresponding to the maximum of Cv. The simulations made up to L ≤ 128 with application of the Wolff's cluster algorithm allowed us to verify the possible power-like as well as logarithmic singularity of the specific heat predicted by different theoretical treatments. The most challenging and interesting result we have obtained is that the finite-size scaling of Cvmax in 3D Ising model is well described by a logarithmic rather than power-like ansatz, just like in 2D case. Another modification of our iterative method has been considered to estimate the critical coupling of 3D Ising model from the Binder cumulant data within L ε [96; 384]. Furthermore, the critical exponent β has been evaluated from the simulated magnetization data within the range of reduced temperatures t ≥ 0.000086 and system sizes L ≤ 410.
We present an interpolation method for the specific heat cv(T){c}_{v}(T)cv​(T), when there is a phase transition with a logarithmic singularity in cv{c}_{v}cv​ at a critical temperature T=TcT={T}_{c}T=Tc​. The method uses the fact that cv{c}_{v}cv​ is constrained both by its high temperature series expansion and just above Tc{T}_{c}Tc​ by the type of singularity. We test our method on the ferro- and antiferromagnetic Ising models on the two-dimensional square, triangular, honeycomb, and kagome lattices, where we find an excellent agreement with the exact solutions. We then explore the XXZ Heisenberg model, for which no exact results are available.
A linked cluster expansion for a system of Ising spins is rearranged into a form suitable for studying the phase transition of the system. A function I(q) analogous to proper self-energy part in quantum theory is introduced and shown to play an important role to determine both the transition temperature and the singularity of thermodynamic functions at the transition point. The case above the transition temperature and without magnetic field is considered, and it is proved that diagrams for I(q) dominant near the transition point have precisely the same structure as in the system of Bose fluid. Thus the method developed by Patashinsky and Pokrovsky for the latter case is taken over to the present problem and leads to a logarithmic singularity of specific heat.
The partition functions of the finite Ising models of spin S =1, 3/2, 2, 5/2 and 3 have been calculated by a high speed computer. The periodicity condition at the boundaries has been imposed, and only the nearest neighbor interaction has been assumed. The specific heats of these finite systems have been calculated as a function of temperature. The maxima of the specific heat increase proportionally to long N in the two-dimensional Ising model with spin S =1, N being the number of lattice points, which indicates that the specific heat for the above lattice has a singularity such as C v / k ∼ A log | T - T c |+ B ± : A ∼0.10 and B - - B + =0.24. The difference ( B + - B - ) has been estimated from the asymmetric distribution of the zeros of the partition function in the complex temperature plane.