LATTICE ALGEBRAS AND THE HIDDEN SYMMETRY OF THE 2D ISING MODEL IN A MAGNETIC FIELD

1991 ◽  
Vol 06 (28) ◽  
pp. 5127-5153 ◽  
Author(s):  
DAN LEVY
Keyword(s):  
Magnetic Field ◽  
Ising Model ◽  
External Magnetic Field ◽  
Lattice Models ◽  
Natural Generalization ◽  
Xxz Model ◽  
Affine Hecke Algebra ◽  
Finite Dimensional ◽  
Hidden Symmetry ◽  
Boundary Terms

Lattice algebras are defined and used to study the symmetries of 2D lattice models. New and interesting examples of such algebras are provided by the affine Hecke algebra, owing to the possibility of constructing braid generators out of its generators. I propose an Ansatz for the braid generators and derive some solutions. A particular finite-dimensional quotient is shown to be a natural generalization of the Temperley-Lieb-Jones algebra. It is used to give a unified picture of known and unknown symmetries of the spin-½ xxz model with boundary terms. The Ising model in an external magnetic field is also a representation of this quotient.

THE GROUND-STATE PHASE DIAGRAMS OF THE SPIN-2 ISING MODEL

2007 ◽  
Vol 21 (31) ◽  
pp. 5265-5274 ◽  
Author(s):  
AHMET ERDİNÇ
Keyword(s):  
Magnetic Field ◽  
Ising Model ◽  
External Magnetic Field ◽  
Phase Diagrams ◽  
Crystal Field ◽  
Ground State ◽  
Exchange Interactions ◽  
Nearest Neighbors ◽  
Biquadratic Exchange ◽  
Model Hamiltonian

The ground-state phase diagrams are obtained for the spin-2 Ising model Hamiltonian with bilinear and biquadratic exchange interactions and a single-ion crystal field. The interactions are assumed to be only between nearest-neighbors. Obtained phase diagrams are presented in the (Δ,J), (K,J), (Δ/J,K/J), (Δ/|J|,K/|J|), (Δ/|K|,J/|K|), (H/J,Δ/J), (H/|J|,Δ/|J|), (H/J,K/J), and (H/|J|,K/|J|) planes where J, K, Δ, and H are the bilinear, biquadratic exchange interactions, the single-ion crystal field, and the external magnetic field, respectively. The influence of the external magnetic field on the spin configurations is investigated.

scholarly journals Entanglement in a two-qubit Ising model under a site-dependent external magnetic field

2004 ◽  
Vol 333 (5-6) ◽  
pp. 438-445 ◽  
Author(s):  
Andreas F. Terzis ◽  
Emmanuel Paspalakis
Keyword(s):  
Magnetic Field ◽  
Ising Model ◽  
External Magnetic Field ◽  
A Site

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.

Analogue Optical Simulation of the 2D Ising Model in an External Magnetic field

2021 ◽  
Author(s):  
Aneek Biswas ◽  
Tommaso McPhee ◽  
Mohammad-Ali Miri ◽  
Kevin Cognee ◽  
Vinod Menon
Keyword(s):  
Magnetic Field ◽  
Ising Model ◽  
External Magnetic Field ◽  
Optical Simulation
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