The Ising Magnetic Systems

Roberto Zivieri

In this work, the critical properties of the classical spin-1 Ising Hamiltonian applied to magnetic systems characterized by the first-neighbors biquadratic exchange, the anisotropy and the external magnetic field contributions are theoretically investigated. The first-neighbors bilinear exchange interaction is set equal to zero. For magnetic systems the bicubic exchange interaction must be set equal to zero as it would break the time-reversal invariance of the exchange Hamiltonian. To determine the critical behavior, the spin-1 Ising Hamiltonian is mapped onto the spin-1/2 Ising Hamiltonian by using the Griffith’s variable transformation. The critical surface of a 2D square magnetic lattice is determined in the parameter space as a function of the magnetic parameters and the phase transition occurring across it is quantitatively discussed by calculating, for each spin, the free energy and the magnetization. The free energy of the 2D square magnetic lattice, described via the three-state spin-1 Ising model, is obtained from an empirical expression of the partition function recently proposed for a spin-1/2 Ising model in an external magnetic field and applied to a 2D magnetic lattice. These results could pave the way to numerical simulations and to measurements able to confirm the analytical predictions.

Subir Sachdev A

T. Morita

The concept of the irreducible block is introduced for the one-dimensional Ising system with an arbitrary interaction of finite range. It is proved that the ground state energy of the system occurs for a regular chain of one of the irreducible blocks or for coexistence of the regular chains of the irreducible blocks. It is shown that the ground state energy of the linear Ising magnet with pair interactions up to third neighbours is effected by the seven spin orderings recently given for the case of spin 1 / 2 by Katsura and Narita (1973), and that the ground state energy of the Ising magnet of spin larger than 1 / 2 is effected by the same set of orderings as for the corresponding Ising magnet of spin 1 / 2 .

Gesualdo Delfino A, Giuseppe Mussardo

Andrzej Pękalski

Wenhan Chen, Haodi Tang, Yu Wang +4 more

With its unique computer paradigm, the Ising annealing machine has become an emerging research direction. The Ising annealing system is highly effective at addressing combinatorial optimization (CO) problems that are difficult for conventional computers to tackle. However, Ising spins, which comprise the Ising system, are difficult to implement in high-performance physical circuits. We propose a novel type of Ising spin based on an electrically-controlled magnetic tunnel junction (MTJ). Electrical operation imparts true randomness, great stability, precise control, compact size, and easy integration to the MTJ-based spin. In addition, simulations demonstrate that the frequency of electrically-controlled stochastic Ising spin (E-spin) is 50 times that of the thermal disturbance MTJ-based spin (p-bit). To develop a large-scale Ising annealing system, up to 64 E-spins are implemented. Our Ising annealing system demonstrates factorization of integers up to 264 with a temporal complexity of around O(n). The proposed E-spin shows superiority in constructing large-scale Ising annealing systems and solving CO problems.