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Progress of Theoretical Physics

Statistical Mechanics of a System of Ising Spins near the Transition Point

Ryuzo Abe

Published Apr 1, 1965• 8 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.33.600

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Physical Review•Jan 15, 1963•130 citations
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.
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.
The critical phenomenon of the second order phase transition is considered for temperatures above the critical point. A general phenomenological approach is developed. The generalized free energy F[{η q }: T ] is introduced. It is a functional of the Fourier component η q of the local order parameter η( r ). A general property of the correlation function <η q η - q > is established rigorously. The equilibrium free energy and the wave-number dependent susceptibility χ q are expressed in terms of the linked clusters, large clusters corresponding to highr order fluctuation of the order parameter. The general rule for calculating the contribution of each cluster is given. The rule is shown to be very simple. It is found that the specific heat is expressed as a sum of the temperature derivatives of χ q 's. As an example of the application of the linked cluster method, the susceptibility is calculated in the vicinity of the critical point by summing up the clusters that are most divergent when approaching the criticat point. The linked cluster expansion developed here could be a clue to the whole mystery of the singularity at the critical point.
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 behavior of a three-dimensional Ising-like system in an external field near the critical point by using the non-Gaussian spin-density fluctuations, namely, the quartic measure density with the even and odd powers of the variable (the asymmetric ${\ensuremath{\rho}}^{4}$ model). The basic idea of the analytic method for deriving complete expressions of the thermodynamic characteristics (including the scaling functions) is described. The proposed method allows us to perform the calculations on the microscopic level without any adjustable parameters. Explicit expressions for the total free energy, order parameter, susceptibility, entropy, and specific heat of the system are obtained as functions of the temperature and field. The regions of the so-called weak and strong fields are considered for temperatures above and below Tc{T}_{c}Tc​ (Tc{T}_{c}Tc​ is the phase-transition temperature in the absence of an external field). The average spin moment and susceptibility, depending on the field variation and the proximity to Tc{T}_{c}Tc​, are investigated. It is confirmed that the temperature and field fluctuations for the order parameter play the leading roles in the weak and strong fields, respectively.
The problem of N interacting spins on a lattice is equivalent to one of N clusters linked in a specific manner. The energy of any configuration of spins can be expressed in terms of the energy levels of this cluster. A new expression is obtained for the probability of occurrence of any configuration. A closed form expression is obtained for the partition function per spin in terms of the energy levels of this cluster with the degeneracies being a function of temperature. On physical grounds it is suggested to be the form of the exact partition function per spin. The partition functions of Ising-like systems all have a common form. For the 3D Ising model seven functions need to be determined to describe the partition function completely. The key to understanding second order phase transitions and critical phenomena lies in the temperature dependence of various degeneracies. It is necessary to develop new techniques to determine the partition function that account for this temperature dependence, as it would represent the underlying physics correctly.
A general perturbation theory of the statistics of spin interactions is developed in the form of a linked cluster expansion with particular emphasis on the Ising model. The theory applies to the evaluation of the expectation value of arbitrary spin functions as well as of the free energy. The thermodynamical consistency of the perturbation expansion is shown to arise from (1) variational principles satisfied by the free energy after a "renormalization procedure" has been carried out and (2) a generalized "Ward Identity" between renormalized quantities. These results are used to discuss inconsistencies in recent high-density theories of ferromagnetism and an improved theory obtained by the summation of all the convolution diagrams (nodal expansion) is briefly presented. The applicability of the method to general quantum mechanical many-body problems, including the theory of the Heisenberg model of ferromagnetism, is shown.