Effective Hamiltonian for the Two-Dimensional Arbitrary Spin Ising Model

Ken‐Yu Lin

J. Szṅajd

Abstract A method of the reduction of the generalized arbitrary spin two‐dimensional Ising model to the spin‐half Ising model is presented. The method is demonstrated in detail by calculating the effective interaction constants to the third order in a cumulant expansion for the triangular spin‐one Ising model (the Blume‐Emery‐Griffiths model).

No author provided

The Hamiltonian of the Ising model in d dimensions has been analysed using unitary transformations and combinatorics. We have been able to obtain closed formulas for the eigenvalues of the Ising Hamiltonian in an arbitrary number of dimensions. Furthermore, generalisations to a higher number of spin components on each site are possible within this method. We have made numerical comparisons with the partition function from the earlier analytical expressions known in the literature for one and two-dimensional cases and found complete agreement. (C) 2001 Elsevier Science B.V. All rights reserved.

Adam Lipowski, Masuo Suzuki

Andrzej Komoda, Andrzej Pękalski

Shaomeng Wang, Wenjia Zhang, Xin Ye +1 more

The spatial photonic Ising machine has achieved remarkable advancements in solving combinatorial optimization problems. However, it still remains a huge challenge to flexibly map an arbitrary problem to the Ising model. In this paper, we propose a general spatial photonic Ising machine based on the interaction matrix eigendecomposition method. The arbitrary interaction matrix can be configured in the two-dimensional Fourier transformation based spatial photonic Ising model by using values generated by matrix eigendecomposition. The error in the structural representation of the Hamiltonian decreases substantially with the growing number of eigenvalues utilized to form the Ising machine. In combination with the optimization algorithm, as low as ∼65% of the eigenvalues are required by intensity modulation to guarantee the best probability of optimal solution for a 20-vertex graph Max-cut problem, and this percentage decreases to below ∼20% for near-zero probability. The 4-spin experiments and error analysis demonstrate the Hamiltonian linear mapping and ergodic optimization. Our work provides a viable approach for spatial photonic Ising machines to solve arbitrary combinatorial optimization problems with the help of the multi-dimensional optical property.