Diffusion in regular and disordered lattices

We present a new method for rigorously proving the existence of phase transitions. In particular, we prove that phase transitions occur in (φ·φ) 3 2 quantum field theories and classical, isotropic Heisenberg models in 3 or more dimensions. The central element of the proof is that for fixed ferromagnetic nearest neighbor coupling, the absolutely continuous part of the two point function ink space is bounded by 0(k−2). When applicable, our results can be fairly accurate numerically. For example, our lower bounds on the critical temperature in the three dimensional Ising (resp. classical Heisenberg) model agrees with that obtained by high temperature expansions to within 14% (resp. a factor of 9%).

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Infrared Bounds, Phase Transitions and Continuous Symmetry Breaking

The present situation in theoretical and experimental studies on one-dimensional magnetic systems is fully discussed. Equal-time as well as the dynamic properties are included with an emphasis on the latter. Four model systems are examined in detail: TMMC (Heisenberg antiferromagnet), CsNiF3 (planar ferromagnet), CoCl2. 2NC5H5 (Ising ferromagnet), CuCl2. 2NC5H5 (Heisenberg antiferromagnet with S = ½). The equal-time properties are quite well understood in theory and in experiment but the dynamical properties much less so. The open questions and possible investigations for the future are discussed.

Theoretical and experimental studies on one-dimensional magnetic systems

The resistance between two arbitrary nodes in a resistor network is obtained in terms of the eigenvalues and eigenfunctions of the Laplacian matrix associated with the network. Explicit formulae for two-point resistances are deduced for regular lattices in one, two and three dimensions under various boundary conditions including that of a Mobius strip and a Klein bottle. The emphasis is on lattices of finite sizes. We also deduce summation and product identities which can be used to analyse large-size expansions in two and higher dimensions.

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Theory of resistor networks: the two-point resistance

The resistance between two arbitrary grid points of several infinite lattice structures of resistors is calculated by using lattice Green’s functions. The resistance for d dimensional hypercubic, rectangular, triangular, and honeycomb lattices of resistors is discussed in detail. Recurrence formulas for the resistance between arbitrary lattice points of the square lattice are given. For large separation between nodes the asymptotic form of the resistance for a square lattice and the finite limiting value of the resistance for a simple cubic lattice are calculated. The relation between the resistance of the lattice and the van Hove singularity of the tight-binding Hamiltonian is given. The Green’s function method used in this paper can be applied in a straightforward manner to other types of lattice structures and can be useful didactically for introducing many concepts used in condensed matter physics.

Application of the lattice Green's function for calculating the resistance of an infinite network of resistors

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

Lattice Green's Function. Introduction

The lattice Green's function for the simple cubic lattice I(a)=1π3[triple intergral operator]0πdx dy dza+ie−cos x−cos y−cos z for a > 3 is expressed as a Mellin‐Barnes type integral. The analytic continuation gives simple and useful expressions in series for the numerical calculation of the real part IR(a) and the imaginary part II(a) of the integral for 0 < a < 1 and 1 < a < 3. The values at a = 1, a = 0, and a=√5 are obtained exactly: IR(1) = (π/2)[Γ(5/8)Γ(7/8)]−2, II(1)/(−i)=(√2)IR(1), and II(0) = 3·2−11/3π−4[(Γ(1/3)]6.

Lattice Green's function for the simple cubic lattice in terms of a Mellin-Barnes type integral. II

Lattice Green's Function for the Simple Cubic Lattice in Terms of a Mellin-Barnes Type Integral (II) (数値計算のアルゴリズムの研究)

To study the relations between Yang-Lee's theory of the phase transitions and the distribution of the zeros of the partition functions in the antiferromagnetic case, the partition functions of finite (4×4 and 4×6) Ising model with the nearest and the next nearest neighbor interactions are calculated. The distributions of the zeros in the fugacity z plane of the partition function for various combinations of the values of J / k T - J '/ k T or tanh ( J /2 k T )-tanh ( J '/2 k T ) are investigated. The patterns of the distribution of the zeros show several types corresponding to ferromagnetic state, antiferromagnetic state and superantiferromagnetic state. In the antiferromagnetic ordered state the locus of the zeros is found to be nearly two concentric circles and to cross the positive real axis at two points z c and 1/ z c when the next neighbor ferromagnetic interaction is introduced. This means that our system shows the phase transitions at finite critical fields H c and - H c . The locus of zeros of Fi...

Yoshihiko Abe

Papers

Lattice Green's Function. Introduction

Lattice Green's function for the simple cubic lattice in terms of a Mellin-Barnes type integral. II

Lattice Green's Function for the Simple Cubic Lattice in Terms of a Mellin-Barnes Type Integral (II) (数値計算のアルゴリズムの研究)

Distribution of Zeros of the Partition Function of the Ising Model

Lattice Green's function for the simple cubic and tetragonal lattices at arbitrary points