Shigetoshi Katsura

Dynamical properties of the isotropic XY model

The dynamical longitudinal properties of the isotropic one-dimensional XY model are obtained by the two-time Green function method, which involves 《a+λaμ|a+νaσ》(7minus;), 《a+λaμ|a+νaσ》(+) and 《aμ|a+λaσa+λ》(+) defined by the anticommulator and the commutator. The frequency-and wave number- dependent susceptibility, the response function and the relaxation function (canonical) correlation function) are obtained and discussed for all temperatures. The imaginary part of the frequency- and wave number-dependent susceptibility diverges at ω=4J sin (κ2) and vanishes for ω >; 4J sin (κ2). The difference between isothermal and zero-frequency isolated susceptibilities is resolved by taking the limit κ→0 χzz(κ, 0). The contribution of the zero-frequency pole in the two-frequency pole in the two-time Green function is also discussed and clarified.

One‐Dimensional Ising Model with General Spin

The one‐dimensional Ising model with general spin S has been formulated as an eigenvalue problem of order 2S + 1. Two methods to reduce the order to [S + 1] have been developed for calculating the energy and the susceptibility at zero external field. Exact solutions for S = 32 and S = 1 have been obtained. Numerical calculations of S = 32, 1, and ½ have been compared.

Lattice Green's Function. Introduction

Physical, analytical, and numerical properties of the lattice Greens functions for the various lattices are described. Various methods of evaluating the Greens functions, which will be developed in the subsequent papers, are mentioned.

Lattice Green's Functions for the Rectangular and the Square Lattices at Arbitrary Points

The lattice Greens functions of the rectangular and the square lattices Irect(a;m,n;α,β)≡1π2[double integral operator]0πcosmxcosny dx dya−ie−αcosx−βcosy,Isq(a;m,n)≡Irect(a;m,n;1,1) are considered. The integral Irect(a, m, n; α, β) for a > α + β is evaluated and expressed in terms of the generalized hypergeometric function F4. Expressions of Isq(a; m, n) for a > 2, a < 2, and a ∼ 2, and Irect(a; m, m; α, β) in terms of pFp−1 are presented by the method of the analytic continuation using the Mellin‐Barnes type integral. They are useful for the understanding of the nature of the singularity and for numerical calculation. The behaviors of Isq(a; m, n) are shown in figures.

Spin glasses for the infinitely long ranged bond Ising model and for the short ranged binary bond Ising model without use of the replica method

The long ranged Gaussian random Ising bond model and short ranged binary random Ising bond model are discussed by the method of the pair approximation of the cluster variation and by the method of the integral equation for the distribution function of the effective fields. For the long ranged model, Sherrington and Kirkpatricks result is generalized and rederived without use of the replica method. For the short ranged model, i.e. a binary mixture of JA = -JB, the integral equation is solved exactly and the energy of the spin glass state is obtained at T = 0.

Exactly solvable models showing chaotic behavior II

Several examples of the exactly solvable two-dimensional mapping are obtained. The problem of the accuracy of the numerical calculation is demonstrated by showing examples.

Lattice Green's Function for the Body‐Centered Cubic Lattice

The lattice Greens function for the body‐centered cubic (bcc) lattice I(t)=1π3∫ ∫ 0π∫dx dy dzt±ie−cosxcosycosz is considered. With the use of the analytic continuation to complex value of t from Maradudins result for t > 1, the value of the real and imaginary parts of the integral I(t ± ie) for 0 < t < 1, e → 0, is obtained. The expressions valid for t → ∞, t≳1, t≲1, and t ∼ 0 are given. They are useful for analyzing the nature of the singularity and for carrying out numerical calculations in all regions of t.

Irreducible Cluster Integrals of Hard‐Sphere Gases. II

It is shown that a class of irreducible clusters of hard‐sphere gases can be evaluated by a multiple‐series method by changing the integration variable from position vectors to bond vectors and from bond vectors to face vectors using bond—face transformation. The class includes graphs which were intractable so far, for example, Nos. 9 and 10 clusters of the fifth virial coefficient. The value of the fifth virial coefficient E of the hard‐sphere gases is recalculated and is found to be 0.11040±0.00006, which has maximum accuracy among values so far obtained.

Linear Heisenberg Model of Ferro‐ and Antiferromagnetism

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 ∑ σlz is 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.

The energy of the spin-glass state of a binary mixture at T = 0 and its variational properties

In the random-bond model of Ising spins, the concept of a multiple-bond distribution of effective field was introduced in the pair approximation. The integral equation for a single-bond distribution was derived intuitively. The variational energy at T = 0 is expressed in terms of two parameters μ and η where μ is the probability of zero effective field in the single-bond distribution and η is the magnetization per spin. For η = 0, the energy of the spin-glass state corresponds to a local minimum as a function of μ, for an even z (number of the nearest neighbours) and to an inflection point for an odd z. It was shown that the spin-glass state corresponds to a local minimum with respect to μ and η for z = 4, to an inflection point with respect to μ and a local minimum with respect to η for z = 3. It is conjectured that the maximum of the energy of the spin-glass state of Sherrington and Kirkpatrick is attributed not to the replica method, but to the mean field approximation. Stationary properties of the energy as a function of both μ and η were examined in detail.