The partition function of the one-dimensional Heisenberg model is considered. Hamiltonian of the system H=−12 ∑ [J⊥(σlxσl+1x+σlyσl+1y)+J∥σlzσl+1z]−mH ∑ σlzis expressed in terms of Fermi operators. The term which contains J∥, the quartic term and a part of quadratic term in Fermi operators, have been regarded as perturbation, keeping the symmetry with respect to the magnetic field. Linked-cluster expansion in an appropriate form for this case has been developed and the partition function has been obtained up to the third order in J∥. Numerical values of energy, specific heat, and susceptibility up to second order in J∥ are shown. The ground-state energy is EN|J⊥|=−2π−2π2(J∥|J⊥|)−16π3(16−π2144)(J∥J⊥)2+O[(J∥J⊥)3].E/N|J∥| for the antiferromagnetic case J∥ = −|J∥| = J is −0.8899. Agreement with the exact value, −0.8863, is quite satisfactory.
A theorem is given for calculating the partition function of quantum mechanical systems. Using this theorem, it is shown that some sort of long-range interaction is equivalent to an effective field. The exact solution of linear anisotropic Heisenberg model with staggered fields, and consequently the zero field staggered susceptibility are given. Finally, it is shown that due to the effect of the transverse components, the value of the Curie point decreases both in the ferromagnetic and antiferromagnetic cases. The rate of decreases in the antiferromagnetic case is smaller than in the ferromagnetic case.
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.
We consider a “generalized Heisenberg model”, with a Hamiltonian given by ℋ︁(D) = –JΣij S(D)j, where the quantities Si are isotropically-interacting D-dimensional classical spins and the summation is restricted to nearest-neighbor pairs of sites <i,j>. Thus when D = 1, 2, and 3, this model reduces to the S = 1/2 Ising, plane rotator, and classical Heisenberg models which are useful for describing phase transitions in, respectively, liquid-gas systems, Bose fluids, and magnetic materials. Unfortunately this Hamiltonian is exactly soluble only in the limit D → ∞, in which case the solution corresponds to the spherical model of Berlin and Kac. For finite D, we must consider approximation procedures in order to obtain any useful information. In this paper we develop a diagrammatic representation, valid for arbitrary D, for successive terms in the high-temperature expansions of the free energy and the spin-spin correlation function—from which we can obtain series expansions for all the thermodynamic properties of the system.
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
z1 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
vij is the exchange potential. This is the molecular field value for the Curie point
Tc. 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(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.
We have considered a classical spin system, consisting of 3-component unit vectors, associated with a two-dimensional lattice {uk, k ∈ Z2}, and interacting via a translationally and rotationally invariant pair potential, of the long-range ferromagnetic form [Formula: see text] Here ∊ is a positive constant setting energy and temperature scales (i.e. T* = k B T/∊), and xj denotes dimensionless coordinates of the lattice sites. This potential model is known rigorously to possess an ordering transition at finite temperature, and has been characterized quantitatively by Monte Carlo simulation, whose results suggest a continuous transition taking place at [Formula: see text]; comparison is also reported with other theoretical treatments, such as Spherical Model, Molecular Field and Two Site Cluster approximations.