Clusters in the three-dimensional Ising model with a magnetic field

1989 ◽  
Vol 161 (2) ◽  
pp. 249-268 ◽  
Author(s):  
Jian-Sheng Wang
Keyword(s):  
Magnetic Field ◽  
Ising Model ◽  
Three Dimensional

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.

Close-packed anisotropic antiferromagnetic Ising lattices. I

1969 ◽  
Vol 47 (23) ◽  
pp. 2621-2631 ◽  
Author(s):  
John Stephenson
Keyword(s):  
Magnetic Field ◽  
Experimental Data ◽  
Ising Model ◽  
Lattice Structure ◽  
Integral Formula ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Ising Lattice ◽  
Ising Lattices ◽  
Comparison With Experimental Data

A detailed account of the nonmagnetic thermodynamic properties of the two-dimensional, anisotropic, antiferromagnetic, triangular Ising lattice, in the absence of a magnetic field, is presented. The notion of a disorder temperature, TD, is introduced via the exact integral formula for the logarithm of the partition function per spin, and the energy and entropy per spin are evaluated explicitly at TD. Study of the theoretical specific heat curves, and of graphs of the percentage of energy and entropy below the Néel point, shows that the close-packed antiferromagnetic triangular lattice has a qualitatively different behavior from two-dimensional loose-packed lattices. It is suggested that a similar situation will occur in three-dimensional close-packed lattices, and therefore that it is important to use a close-packed Ising model for comparison with experimental data on antiferromagnets which have close-packed lattice structure, or loose-packed lattice structure with higher neighbor interactions.

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