We introduce a one dimensional Ising model with two competing interactions: nearest neighbor random couplings +/-J with equal probability and a positive infinite range coupling Lambda. At low temperature T the model exhibits a first order phase transition between a ferromagnetic state (with magnetization m(1) = 1 at T = 0) and a << ferrimagnetic >> state (with m(2) = 2/3 at T = 0), when the disorder strength J/Lambda is increased. For 5/12 < J/Lambda < 1, a whole spectrum of ferrimagnetic ground states with magnetization m(n) = 2/(n + 1) (n = 2, ..., infinity) is present while for J/lambda > 1 the ground state is given by a trivial one dimensional spin glass with m = 0. The main qualitative features of the model can be described by a simplified annealed model where the random couplings can arrange themselves to minimize free energy with the constraint that the number of positive couplings is fixed by the law of large numbers in the thermodynamic limit. This model is exactly solved at all temperatures and the diagram of phase is calculated.
We introduce a one dimensional Ising model with two competing interactions: nearest neighbor random couplings +/-J with equal probability and a positive infinite range coupling Lambda. At low temperature T the model exhibits a first order phase transition between a ferromagnetic state (with magnetization m(1) = 1 at T = 0) and a << ferrimagnetic >> state (with m(2) = 2/3 at T = 0), when the disorder strength J/Lambda is increased. For 5/12 < J/Lambda < 1, a whole spectrum of ferrimagnetic ground states with magnetization m(n) = 2/(n + 1) (n = 2, ..., infinity) is present while for J/lambda > 1 the ground state is given by a trivial one dimensional spin glass with m = 0. The main qualitative features of the model can be described by a simplified annealed model where the random couplings can arrange themselves to minimize free energy with the constraint that the number of positive couplings is fixed by the law of large numbers in the thermodynamic limit. This model is exactly solved at all temperatures and the diagram of phase is calculated.
The d=1 Ising ferromagnet and spin glass with long-range power-law interactions J r^-a is studied for all interaction range exponents a by a renormalization-group transformation that simultaneously projects local ferromagnetism and antiferromagnetism. In the ferromagnetic case, J>0, a finite-temperature ferromagnetic phase occurs for interaction range 0.742, the phase transition temperature discontinuously drops to zero and for a>2 there is no ordered phase above zero temperature, also as predicted by rigorous results. At the other end, on approaching a=0.74 from above, namely increasing the range of the interaction, the phase transition temperature diverges to infinity, meaning that, at all non-infinite temperatures, the system is ferromagnetically ordered. Thus, the equivalent-neighbor interactions regime is entered before (a > 0) the neighbors become equivalent, namely before the interactions become equal for all separations. The critical exponents alpha,beta, gamma,delta,eta,nu are calculated, from a large recursion matrix, varying as function of a. For antiferromagnetic J<0, all triplets of spins at all ranges have competing interactions and this highly frustrated system does not have an ordered phase. In the spin-glass system, where all couplings for all separations are randomly ferromagnetic or antiferromagnetic (with probability p), a finite-temperatures spin-glass phase is obtained in the absence of antiferromagnetic phase. In the spin-glass phase, the signature chaotic behavior under scale change occurs in a richer version than previously: In the long-range interaction of this system, the interactions at every separation become chaotic, yielding a piecewise chaotic interaction function.
The one-dimensional Ising model in an external magnetic field with uniform long-range interactions and random short-range interactions satisfying bimodal annealed distributions is studied. This generalizes the random model discussed by Paladin et al. (J. Phys. I France 4, 1994, p. 1597). Exact results are obtained for the thermodynamic functions at arbitrary temperatures, and special attention is given to the induced and spontaneous magnetization. At low temperatures the system can exist in a “ferrimagnetic” phase with magnetization 0 < σ < 1, in addition to the usual paramagnetic, ferromagnetic and antiferromagnetic phases. For a fixed distribution of the random variables the system presents up to three tricritical points for different intensities of the long-range interactions. Field-temperature diagrams can present up to four critical points. 1
Thermodynamics of a layered Ising model with infinite-range ferromagnetic intralayer interaction and random nearest-neighbor interlayer coupling is considered. A detailed analysis of the model with vanishing average interlayer coupling is presented. The Gibbs free energy is found in the critical region, and the existence of many metastable states is demonstrated. Thermodynamic parameters of the system are found for periodic states. As the mean square interlayer coupling increases, the equilibrium state of the system undergoes an infinite sequence of first-order phase transitions, the number of magnetic planes and the distance between them change discontinuously, and so do both bulk magnetization and magnetic susceptibility.
At T = 0 and in a sufficiently large field, the nearest-neighbor antiferromagnetic Ising chain undergoes a first-order spin-flop transition into the ferromagnetic phase. We consider its smearing under the random-bond disorder such that all independent random bonds are antiferromagnetic (AF). It is shown that the ground-state thermodynamics of this random AF chain can be described exactly for an arbitrary distribution P(J) of AF bonds. Moreover, the site magnetizations of finite chains can be found analytically in this model. We consider a continuous P(J) that is zero above some −J 1 and behaves near it as (−J 1−J)λ, λ > −1. In this case, the ferromagnetic phase emerges continuously in a field H > H c = 2J 1. At 0 > λ > −1, it has the usual second-order anomalies near H c with the critical indices obeying the scaling relation and depending on λ. At λ > 0, higher-order transitions occur (third, fourth, etc.), marked by a divergence of the corresponding nonlinear susceptibilities. In the chains with an even number of spins, the intermediate “bow-tie” phase with linearly modulated AF order exists between the AF and ferromagnetic phases at J 1 < H < H c . Its origin can be traced to the infinite correlation length of the degenerate AF phase from which it emerges. This implies the existence of similar inhomogeneous phases with size- and form-dependent order in a number of other systems with infinite correlation length. The possibility to observe the signs of the “bow-tie” phase in low-T neutron diffraction experiments is discussed.
The Ising linear chain with nearest-neighbor interactions of strength J s and infinite long-range ferromagnetic exponential interactions of strength J l was solved exactly in the presence of the external field. The spontaneous magnetization and the susceptibility are obtained. The transition temperature depends on J s and J l . For J s >0, the model shows the second order phase transion. The specific heat shows both a broad peak and a sharp peak. The value of sharp peak of the specific heat depends on J s and J l . For J s <0, the properties of the transition agree with those of Nagle model with long-range equivalent-neighbor ferromagnetic interactions. The exact free energy and the long-range order of our model are identical with those of the variational theory of a quasi-one-dimensional model.
The infinite range Ising model is usually investigated in the weak coupling limit. Here the model is solved with ferromagnetic coupling at fixed and finite strength. Exact analytical expressions are found for the thermodynamic potentials as functions of enthalpy and external field. These results differ from the potentials for the weak coupling limit. The model shows a temperature driven first order phase transition from a paramagnetic phase at high temperatures into a low temperature phase from which thermal fluctuations are absent.