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Physical Review

Statistical Mechanical Theory of Ferromagnetism. High Density Behavior

R. Brout

Published May 15, 1960• 181 citations
Publisher

Abstract

The partition function of the Ising model of ferromagnetism is examined in the limit of high density in the anticipation that in the limit of infinite density one recovers the Weiss molecular field. The formal parameter of expansion is $\frac{1}{z}$ where $z$ is the number of spins in the range of the exchange potential (not restricted to nearest neighbor interactions). In the absence of long-range order, only ring diagrams in the cluster expansion contribute. These give a divergence in the specific heat at $k{T}_{c}=\ensuremath{\Sigma}{j\ensuremath{\ne}i}^{}{v}_{\mathrm{ij}}$ where ${v}_{\mathrm{ij}}$ is the exchange potential. This is the molecular field value for the Curie point ${T}_{c}$. In the presence of a magnetic field the partition function is evaluated for fixed magnetic moment $M$ in the same approximation, $M$ being determined by minimization. This results in a susceptibility differing from the molecular field theory and hence an inconsistency in the theory.The inconsistency is traced back to the observation that the acceptance of ring diagrams is equivalent to the gaussian model of Kac and Berlin which violates the sum rule ${{\ensuremath{\Sigma}}_{i=1}}^{N}{{\ensuremath{\mu}}_{i}}^{2}=N$. Here ${\ensuremath{\mu}}_{i}$ is the "spin" per particle and $N$ is the total number of particles. This condition is reinstated by insuring the sum rule. The result leads to the spherical model. Thus, a consistent high density approximation to the Ising model is the spherical model. Below the Curie point or for fixed magnetic field, $M$ is again held fixed and only the Fourier components of the spin density with nonvanishing wave vector are "sphericalized." The result leads to a physically acceptable model which becomes the molecular field theory at low temperatures or high fields and deviates in $O(\frac{1}{z})$ in general. Formally, the results are simply expressed in terms of a temperature dependent Weiss field. These results differ from the ordinary spherical model which is physically unacceptable below the Curie point. However, a molecular field modification of the spherical model due to M. Lax yields the same result when properly interpreted.It is shown that the above results are also valid (to the same approximation) in the quantum mechanical Heisenberg model, for temperatures above the Curie point.

DOI
10.1103/physrev.118.1009

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A new and simple diagrammatic expansion is developed for the free energy of the Ising model with arbitrary spin and range of interaction. The contribution of each diagram is a product of (1) the reciprocal of the order of the symmetry group of the diagram, (2) a product of semi-invariants, with a factor ${M}_{n}$ for each spin in the diagram $n$ being the number of bonds joined to that spin) and (3) a sum of products $\ensuremath{\Sigma}{i,j,k,\ensuremath{\cdots}}^{}(2\ensuremath{\beta}{J}_{\mathrm{ij}})(2\ensuremath{\beta}{J}_{\mathrm{kl}})\ensuremath{\cdots}$, where $2\ensuremath{\beta}{J}_{\mathrm{ij}}$ corresponds to a bond between spins $i$ and $j$, and where the summation is carried out with no restrictions on the summation indices (i.e., no "excluded volume" corrections). The expansion variables are the spin deviation operators ${\ensuremath{\sigma}}_{i}\ensuremath{\equiv}\overline{S}\ensuremath{-}{S}_{\mathrm{iz}}$. The quantity $\overline{S}$ is chosen to eliminate a large class of diagrams; this choice also minimizes the corresponding free-energy contribution and implies $\overline{S}=〈{S}_{\mathrm{iz}}〉$. By further renormalization of the vertices all reducible (i.e., simply articulated) diagrams are eliminated. To zero order the molecular field solution is obtained. The next approximation consists of the summation of renormalized simple loop diagrams. The justification of this choice of diagrams rests, at low temperature, on the decrease of the value of the higher order semi-invariants, on the relationship of these loop diagrams to the random phase approximation, and on the agreement with the rigorous low-temperature series result. At the Curie temperature the same choice of diagrams is justified by a modification of Brout's $\frac{1}{z}$ criterion, so that the expansion can be viewed as an expansion in $\frac{1}{z}$, where $z$ is the effective number of spins interacting with a given spin. Finally the choice of loop diagrams is justified at high temperature by exact agreement to second order in $\frac{{T}_{c}}{T}$ (and by a very close agreement to fourth order) with the rigorous high-temperature series expansion. Thus the theory agrees with rigorous results in the low-temperature and paramagnetic regions and has a well-defined criterion of validity in the intermediate temperature region. For nearest-neighbor interactions and spin \textonehalf{} the specific heat is continuous through the Curie temperature, with an infinite slope on the low-temperature side of the transition.

Physical Review•Dec 15, 1961•174 citations
The partition function of the Ising model of ferromagnetism is examined in the limit of high density in the anticipation that in the limit of infinite density one recovers the Weiss molecular field. The formal parameter of expansion is 1z\frac{1}{z}z1​ where zzz is the number of spins in the range of the exchange potential (not restricted to nearest neighbor interactions). In the absence of long-range order, only ring diagrams in the cluster expansion contribute. These give a divergence in the specific heat at kTc=\ensuremathΣj\ensuremath≠ivijk{T}_{c}=\ensuremath{\Sigma}{j\ensuremath{\ne}i}^{}{v}_{\mathrm{ij}}kTc​=\ensuremathΣj\ensuremath=ivij​ where vij{v}_{\mathrm{ij}}vij​ is the exchange potential. This is the molecular field value for the Curie point Tc{T}_{c}Tc​. In the presence of a magnetic field the partition function is evaluated for fixed magnetic moment MMM in the same approximation, MMM being determined by minimization. This results in a susceptibility differing from the molecular field theory and hence an inconsistency in the theory.The inconsistency is traced back to the observation that the acceptance of ring diagrams is equivalent to the gaussian model of Kac and Berlin which violates the sum rule \ensuremathΣi=1N\ensuremathμi2=N{{\ensuremath{\Sigma}}_{i=1}}^{N}{{\ensuremath{\mu}}_{i}}^{2}=N\ensuremathΣi=1​N\ensuremathμi​2=N. Here \ensuremathμi{\ensuremath{\mu}}_{i}\ensuremathμi​ is the "spin" per particle and NNN is the total number of particles. This condition is reinstated by insuring the sum rule. The result leads to the spherical model. Thus, a consistent high density approximation to the Ising model is the spherical model. Below the Curie point or for fixed magnetic field, MMM is again held fixed and only the Fourier components of the spin density with nonvanishing wave vector are "sphericalized." The result leads to a physically acceptable model which becomes the molecular field theory at low temperatures or high fields and deviates in O(1z)O(\frac{1}{z})O(z1​) in general. Formally, the results are simply expressed in terms of a temperature dependent Weiss field. These results differ from the ordinary spherical model which is physically unacceptable below the Curie point. However, a molecular field modification of the spherical model due to M. Lax yields the same result when properly interpreted.It is shown that the above results are also valid (to the same approximation) in the quantum mechanical Heisenberg model, for temperatures above the Curie point.
It is shown that there is a class of graphs of the Ising model (or (Heisenberg model for T>TCurieT>{T}_{\mathrm{Curie}}T>TCurie​) which is comprised of cycle graphs plus some excluded volume effects which sum to the spherical model. The spherical model, suitably generalized for T<TCurieT<{T}_{\mathrm{Curie}}T<TCurie​, was conjectured in a previous work to be the high-density limit of the Ising model, correct to 1z\frac{1}{z}z1​, where zzz is the number of spins in the range of the exchange potential (not restricted to nearest neighbor interactions). z13{z}^{\frac{1}{3}}z31​ measures the range of the exchange potential. This is now proved by examining the omitted graphs. The error is shown to be O(1z2)O(\frac{1}{{z}^{2}})O(z21​).
The statistical thermodynamics of the quantum spin- s XXZ Heisenberg model in the Weiss (mean-field) limit rests on the outer multiplicities of the irreducible representations in the Clebsch–Gordan decomposition of the n -fold spin- s tensor product, which determine the energy-level degeneracies and hence the partition function. Using known closed forms for these multiplicities in terms of generalized binomial coefficients of order 2 s + 1 , we write the exact finite-size partition function of the effective collective Hamiltonian and evaluate its thermodynamic limit by a large-deviation (Laplace) analysis validated against direct summations for systems of up to n = 10 4 spins. The free energy then follows from a two-parameter variational principle in the total spin and the magnetization per site. The ferromagnetic ordering is driven by the rotationally invariant part of the Hamiltonian, so that the Curie temperature T c = 1 3 J s ( s + 1 ) is independent of the anisotropy Δ, and the transition is of second order for every spin s , with classical mean-field exponents. The anisotropy reshapes the phase diagram: for Δ < 1 the field-driven first-order line of the isotropic ( Δ = 1 ) model opens into a dome of easy-plane ferromagnetic order, bounded by a continuous transition that restores the residual U(1) symmetry, and no first-order transitions or metastability occur throughout − 1 ≤ Δ < 1 ; the isotropic Curie-Weiss ferromagnet is recovered as Δ → 1. In two appendices we detail the asymptotics of the outer multiplicities down to the Gaussian-fluctuation prefactor and discuss the q -number and Jacobi-theta structures underlying the model, including the partial-theta-type structure that obstructs a standard theta-function reduction of the residual total-spin sum.
This is a continuation of an effort to reduce to the solution of a characteristic value problem the rigorous calculation of thermodynamic properties of systems in which the intermolecular forces are sufficiently short ranged so that practically the entire potential energy of the system results from interactions between nearest neighbors. The partition function of such a system can be expressed in terms of the largest characteristic value of a linear operator equation and finally as the ratio of partition functions of systems with relatively few particles whose potential energy functions differ somewhat from those in the original system. A method of evaluating grand partition functions and one of introducing interactions between more distant neighbors are discussed. The general theory is applied to the calculation of magnetization, internal energy, and specific heat of two-dimensional ferromagnets on the basis of the Ising model. There seems to exist a λ-point phase transition in the change from ferromagnetic to nonferromagnetic states.
The partition function of the Ising model with NNN spins, written as a functional integral, is approximated by replacing the field ${\ensuremath{\psi}}_{l}$ at the lattice site lll by the root-mean-square value of these fields, and evaluating the integral by a saddle-point method. A phase transition, whose mechanism is mathematically similar to that of the spherical model, is predicted at a temperature lower than the Weiss critical temperature. After parametrizing the mean-square field, one may apply a general variational principle to choose the parameters which minimize the difference between the approximate free energy and the exact Ising-model free energy. This optimum choice leads to the Horwitz-Callen theory.
The behavior of the spherical model of a ferromagnet with an interaction energy between the magnetic spins which varies with distance as $\frac{1}{{r}^{d+\ensuremath{\sigma}}}$ (where ddd is the dimensionality of the lattice and $\ensuremath{\sigma}>0$) is analyzed. It is shown that the model exhibits a ferromagnetic transition in one and two dimensions, providing $0<\ensuremath{\sigma}<d$. (The usual spherical model with nearest-neighbor interactions does not have a transition in one and two dimensions.) The critical-point behavior is investigated. It is found that the singularities in the specific heat and susceptibility are dependent on $\ensuremath{\sigma}$ and ddd, but the behavior of the magnetization is independent of $\ensuremath{\sigma}$ and ddd. In three dimensions the susceptibility diverges as ${(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\gamma}}$, where $\ensuremath{\gamma}=1$ for $0<\ensuremath{\sigma}<\frac{3}{2}$, $\ensuremath{\gamma}=\frac{\ensuremath{\sigma}}{(3\ensuremath{-}\ensuremath{\sigma})}$ for $\frac{3}{2}<\ensuremath{\sigma}<2$ and $\ensuremath{\gamma}=2$ for $\ensuremath{\sigma}>2$. The asymptotic form of the spin-spin correlation function $\ensuremath{\Gamma}(\mathrm{r})$ is studied in the neighborhood of the critical temperature Tc{T}_{c}Tc​. At T=TcT={T}_{c}T=Tc​, $\ensuremath{\Gamma}(\mathrm{r})$ decays for large rrr as $\frac{1}{{r}^{d\ensuremath{-}\ensuremath{\sigma}}}$. Several two-dimensional models with long-range interactions falling off as 1r2\frac{1}{{r}^{2}}r21​ in certain directions only are also investigated.
A new and simple diagrammatic expansion is developed for the free energy of the Ising model with arbitrary spin and range of interaction. The contribution of each diagram is a product of (1) the reciprocal of the order of the symmetry group of the diagram, (2) a product of semi-invariants, with a factor Mn{M}_{n}Mn​ for each spin in the diagram nnn being the number of bonds joined to that spin) and (3) a sum of products $\ensuremath{\Sigma}{i,j,k,\ensuremath{\cdots}}^{}(2\ensuremath{\beta}{J}_{\mathrm{ij}})(2\ensuremath{\beta}{J}_{\mathrm{kl}})\ensuremath{\cdots}$, where $2\ensuremath{\beta}{J}_{\mathrm{ij}}$ corresponds to a bond between spins iii and jjj, and where the summation is carried out with no restrictions on the summation indices (i.e., no "excluded volume" corrections). The expansion variables are the spin deviation operators ${\ensuremath{\sigma}}_{i}\ensuremath{\equiv}\overline{S}\ensuremath{-}{S}_{\mathrm{iz}}$. The quantity S‾\overline{S}S is chosen to eliminate a large class of diagrams; this choice also minimizes the corresponding free-energy contribution and implies S‾=〈Siz〉\overline{S}=〈{S}_{\mathrm{iz}}〉S=〈Siz​〉. By further renormalization of the vertices all reducible (i.e., simply articulated) diagrams are eliminated. To zero order the molecular field solution is obtained. The next approximation consists of the summation of renormalized simple loop diagrams. The justification of this choice of diagrams rests, at low temperature, on the decrease of the value of the higher order semi-invariants, on the relationship of these loop diagrams to the random phase approximation, and on the agreement with the rigorous low-temperature series result. At the Curie temperature the same choice of diagrams is justified by a modification of Brout's 1z\frac{1}{z}z1​ criterion, so that the expansion can be viewed as an expansion in 1z\frac{1}{z}z1​, where zzz is the effective number of spins interacting with a given spin. Finally the choice of loop diagrams is justified at high temperature by exact agreement to second order in TcT\frac{{T}_{c}}{T}TTc​​ (and by a very close agreement to fourth order) with the rigorous high-temperature series expansion. Thus the theory agrees with rigorous results in the low-temperature and paramagnetic regions and has a well-defined criterion of validity in the intermediate temperature region. For nearest-neighbor interactions and spin \textonehalf{} the specific heat is continuous through the Curie temperature, with an infinite slope on the low-temperature side of the transition.