scholarly journals Free energy in a magnetic field and the universal scaling equation of state for the three-dimensional Ising model

2011 ◽  
Vol 83 (5) ◽  
Author(s):  
P. Butera ◽  
M. Pernici
Keyword(s):  
Magnetic Field ◽  
Free Energy ◽  
Ising Model ◽  
Equation Of State ◽  
Three Dimensional ◽  
Universal Scaling ◽  
Scaling Equation

Extended high temperature low field expansions for the Ising model

1979 ◽  
Vol 57 (8) ◽  
pp. 1239-1245 ◽  
Author(s):  
S. McKenzie
Keyword(s):  
Free Energy ◽  
High Temperature ◽  
Ising Model ◽  
Three Dimensional ◽  
Low Field ◽  
Hypercubic Lattice

High temperature low field expansions are derived from the free energy of the Ising model for several two- and three-dimensional lattices. These represent a considerable advance on earlier work. Expansions for the four-dimensional hypercubic lattice are also presented.

ISING MODEL IN THE MAGNETIC FIELD iπkT/2

1988 ◽  
Vol 02 (03n04) ◽  
pp. 471-481 ◽  
Author(s):  
K. Y. LIN ◽  
F. Y. WU
Keyword(s):  
Magnetic Field ◽  
Free Energy ◽  
Ising Model ◽  
Zero Field ◽  
Two Dimensional ◽  
Anisotropic Interactions ◽  
The Magnetic Field ◽  
Explicit Expressions

It is shown that the free energy and the magnetization of an Ising model in the magnetic field H = iπkT/2 can be obtained directly from corresponding expressions of these quantities in zero field, provided that the latter are known for sufficiently anisotropic interactions. Using this approach we derive explicit expressions of the free energy and the magnetization at H = iπkT/2 for a number of two-dimensional lattices.

scholarly journals A Method of the Riemann–Hilbert Problem for Zhang’s Conjecture 2 in a Ferromagnetic 3D Ising Model: Topological Phases

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2936
Author(s):  
Zhidong Zhang ◽  
Osamu Suzuki
Keyword(s):  
Magnetic Field ◽  
Ising Model ◽  
External Magnetic Field ◽  
Hilbert Problem ◽  
Three Dimensional ◽  
Preceding Paper ◽  
Mathematical Structure ◽  
Topological Phases ◽  
Riemann Hilbert Problem ◽  
3D Ising Model

A method of the Riemann–Hilbert problem is employed for Zhang’s conjecture 2 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in a zero external magnetic field. In this work, we first prove that the 3D Ising model in the zero external magnetic field can be mapped to either a (3 + 1)-dimensional ((3 + 1)D) Ising spin lattice or a trivialized topological structure in the (3 + 1)D or four-dimensional (4D) space (Theorem 1). Following the procedures of realizing the representation of knots on the Riemann surface and formulating the Riemann–Hilbert problem in our preceding paper [O. Suzuki and Z.D. Zhang, Mathematics 9 (2021) 776], we introduce vertex operators of knot types and a flat vector bundle for the ferromagnetic 3D Ising model (Theorems 2 and 3). By applying the monoidal transforms to trivialize the knots/links in a 4D Riemann manifold and obtain new trivial knots, we proceed to renormalize the ferromagnetic 3D Ising model in the zero external magnetic field by use of the derivation of Gauss–Bonnet–Chern formula (Theorem 4). The ferromagnetic 3D Ising model with nontrivial topological structures can be realized as a trivial model on a nontrivial topological manifold. The topological phases generalized on wavevectors are determined by the Gauss–Bonnet–Chern formula, in consideration of the mathematical structure of the 3D Ising model. Hence we prove the Zhang’s conjecture 2 (main theorem). Finally, we utilize the ferromagnetic 3D Ising model as a platform for describing a sensible interplay between the physical properties of many-body interacting systems, algebra, topology, and geometry.

ISING MODEL ON A LAYERED SQUARE LATTICE

1989 ◽  
Vol 03 (07) ◽  
pp. 1119-1128
Author(s):  
K.Y. LIN ◽  
K.J. HSU
Keyword(s):  
Magnetic Field ◽  
Free Energy ◽  
Ising Model ◽  
Transfer Matrix ◽  
Square Lattice ◽  
The Magnetic Field

We have considered the Ising model on a layered square lattice where each layer has a different set of horizontal and vertical interactions. The free energy is determined exactly by the method of Pfaffian at two values of the magnetic field, H=0 and H=iπkT/2. The free energy at H=0 was first derived by Wolff et al. using the method of transfer matrix.

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