We propose the mapping of polynomial of degree 2S constructed as a linear combination of powers of spin-$S$ (for simplicity, we called as spin-$S$ polynomial) onto spin-crossover state. The spin-$S$ polynomial in general can be projected onto non-symmetric degenerated spin up (high-spin) and spin down (low-spin) momenta. The total number of mapping for each general spin-$S$ is given by $2(2^{2S}-1)$. As an application of this mapping, we consider a general non-bilinear spin-$S$ Ising model which can be transformed onto spin-crossover described by Wajnflasz model. Using a further transformation we obtain the partition function of the effective spin-1/2 Ising model, making a suitable mapping the non-symmetric contribution leads us to a spin-1/2 Ising model with a fixed external magnetic field, which in general cannot be solved exactly. However, for a particular case of non-bilinear spin-$S$ Ising model could become equivalent to an exactly solvable Ising model. The transformed Ising model exhibits a residual entropy, then it should be understood also as a frustrated spin model, due to competing parameters coupling of the non-bilinear spin-$S$ Ising model.
We construct a quantum Wajnflasz-Pick model that is a generalized quantum Ising model, and investigate a nature of quantum phase transitions of the model with infinite-range interactions. Quantum phase transition phenomena have drawn attention in the field of quantum computing as well as condensed matter physics, since the phenomena are closely related to the performance of quantum annealing (QA) and adiabatic quantum computation (AQC). We add a quantum driver Hamiltonian to the Hamiltonian of classical Wajnflasz-Pick model. The classical Wajnflasz-Pick model consists of two-level systems as w…
Competitive effects of so-called three-site four-spin interactions, single ion anisotropy and bilinear interactions is studied in the mixed spin-1/2 and spin-1 Ising model on a decorated square lattice. Exploring the decoration-iteration transformation, we have obtained exact closed-form expressions for the partition function and other thermodynamic quantities of the model. From these relations, we have numerically determined ground- state and finite-temperature phase diagrams of the system. We have also investigated temperature variations of the correlation functions, internal energy, entropy…
The spin-1 Ising model with bilinear and quadrupolar short-range interactions under magnetic field is investigated within the two-particle cluster approximation. It is shown that for those values of the quadrupolar interaction when at zero magnetic field the system undergoes a temperature phase transition between quadrupolar and paramagnetic phases, a triple point may exist in the temperature vs magnetic field phase diagrams, necessarily along with a critical point. It is also shown that the critical points in the temperature vs magnetic field phase diagrams of the investigated model can be of…
Networks of coupled parametric resonators (parametrons) hold promise for parallel computing architectures. En route to realizing complex networks, we report an experimental and theoretical analysis of two coupled parametrons. In contrast to previous studies, we explore the case of strong bilinear coupling between the parametrons, as well as the role of detuning. We show that the system can still operate as an Ising machine in this regime, even though careful calibration is necessary to ensure that the correct solution space is available. Apart from the formation of split normal modes, new stat…
We analyze the long range Ising spin glass in a transverse field by using Grassmann variables in a field theory where the spin operators are represented by bilinear combinations of fermionic fields. We compare the results of two fermionic models. In the four state (4S)-model the diagonal Sz operator has two vanishing eigenvalues, that are suppressed by a restraint in the two states (2S)-model. Within a replica symmetric theory and in the static approximation we obtain similar results for both models.
We consider two bidimensional classical Ising models, coupled by a weak interaction bilinear in the energy densities of the two systems; the model contains, as limiting cases, the Ashkin-Teller and the Eight-vertex models for certain values of their parameters. We write the energy-energy correlations and the specific heat as Grassman integrals formally describing Dirac 1+1 dimensional interacting massive fermions on a lattice, and an expansion based on Renormalization Group is written for them, convergent up to temperatures very close to the critical temperature for small coupling. The asympto…
A mixed spin-1/2 and spin-3/2 Ising model on a decorated square lattice with a nearest- neighbor interaction, next-nearest-neighbor bilinear interaction, three-site four-spin in- teraction and single-ion anisotropy is exactly investigated using a generalized decoration- iteration transformation, Callen-Suzuki identity and differential operator technique. The ground-state and finite-temperature phase boundaries are obtained by identifying all rel- evant phases corresponding to minimum internal or free energy of the system. The thermal dependencies of magnetization, correlation functions, entrop…
In this paper, we theoretically study the critical properties of the classical spin-1 Ising model using two approaches: 1) the analytical low-temperature series expansion and 2) the numerical Metropolis Monte Carlo technique. Within this analysis, we discuss the critical behavior of one-, two- and three-dimensional systems modeled by the first-neighbor spin-1 Ising model for different types of exchange interactions. The comparison of the results obtained according the Metropolis Monte Carlo simulations allows us to highlight the limits of the widely used mean-field theory approach. We also sho…
We study the spin-1 Ising model with bilinear and biquadratic exchange interactions and single-ion crystal field. In addition to the four usual phases: disordered DIS, ferromagnetic FER, antiquadrupolar AQU and ferrimagnetic FRI, we found three new phases in the case of a thin magnetic film, namely: the sublattice A non magnetic phase NMA, the sublattice B non magnetic phase NMB and the global non magnetic phase NMG. These phases are studied, for each layer of the film, either in the temperature-crystal field plane $(T,Δ)$ or in the biquadratic coupling-crystal field plane $(d,Δ)$, for diferen…
The 2-spin Ising model in statistical mechanics and the 2x2 normal form game in game theory are compared. All configurations allowed by the second are recovered by the first when the only concern is about Nash equilibria. But it holds no longer when Pareto optimum condiderations are introduced like in the prisoner's dilemma. This gap can nevertheless be filled by adding a new coupling term to the Ising model, even if that term has up to now no physical meaning. An individual complete bilinear objective function is thus found to be sufficient to reproduce all possible configurations of a 2x2 ga…
We study a fermionic infinited-ranged Ising spin glass with a real space BCS interaction in the presence of an applied transverse field. The problem is formulated in the integral functional formalism where the SU(2) spins are given in terms of bilinear combinations of Grassmann fields. The problem is solved within static approximation and the replica symmetry ansatz combined with previous approaches used to study the critical behavior of the quantum Ising spin glass in a transverse field and the spin glass Heisenberg model with BCS pairing. Our results show that the transverse field has strong…
The mixed spin-(1,1/2) Ising-Heisenberg double sawtooth ladder containing mixture of both spin-1 and spin-1/2 nodal atoms, and the spin-1/2 interstitial dimers is approximately solved by the transfer-matrix method. Here, we study in detail the ground-state phase diagrams, also influences of the bilinear exchange coupling on the rungs and cyclic four-spin exchange interaction in square plaquette of each block on the magnetization and magnetic susceptibility of the suggested ladder at low temperature. Such a double sawtooth ladder may be found in a Shastry-Sutherland Lattice-type. In spite of od…
An exactly solvable variant of mixed spin-(1/2,1) Ising-Heisenberg diamond chain is considered. Vertical spin-1 dimers are taken as quantum ones with Heisenberg bilinear and biquadratic interactions and with single-ion anisotropy, while all interactions between spin-1 and spin-1/2 residing on the intermediate sites are taken in the Ising form. The detailed analysis of the $T=0$ ground state phase diagram is presented. The phase diagrams have shown to be rather rich, demonstrating large variety of ground states: saturated one, three ferrimagnetic with magnetization equal to 3/5 and another four…
We have examined the role of the BCS pairing mechanism in the formation of the magnetic moment and henceforth a spin glass (SG) phase by studying a fermionic Sherrington-Kirkpatrick model with a local BCS coupling between the fermions. This model is obtained by using perturbation theory to trace out the conduction electrons degrees of freedom in conventional superconducting alloys. The model is formulated in the path integral formalism where the spin operators are represented by bilinear combinations of Grassmann fields and it reduces to a single site problem that can be solved within the stat…
The spin-3/2 Blume-Emery-Griffiths model on a honeycomb lattice is studied by Monte Carlo simulations with the goal to determine phase diagrams for a range of the model parameters and to investigate the nature of the phase transitions between the respective phases. For positive values of the biquadratic to bilinear interaction ratio $α$, we find two ferromagnetically ordered phases, F1 and F2, with the sublattice magnetizations (1/2,1/2) and (3/2,3/2), respectively, and our results confirm the discontinuous character of the order-disorder critical line as a function of the single-ion anisotrop…
We discuss an universal relation $ε(iκ)=0$ with ${\rm Re} κ=1/ξ$ in 1D quantum spin systems with an excitation gap, where $ε(k)$ is the dispersion curve of the low-energy excitation and $ξ$ is the correlation length of the ground-state. We first discuss this relation for integrable models such as the Ising model in a transverse filed and the XYZ model. We secondly make a derivation of the relation for general cases, in connection with the equilibrium crystal shape in the corresponding 2D classical system. We finally verify the relation for the S=1 bilinear-biquadratic spin chain and $S=1/2$ zi…
