The random-field Ising model with asymmetric bimodal probability distribution

The Ising model, in the presence of a random field, is investigated within the mean-field approximation based on Landau expansion. The random field is drawn from the bimodal probability distribution P(h)=pδ(h−h₀)+(1−p)δ(h+h₀), where the probability p assumes any value within the interval [0,1], asymmetric distribution. The prevailing transitions are of second-order but, for some values of p and h₀, first-order phase transitions take place for smaller temperatures and higher h₀, thus confirming the existence of a tricritical point. Also, the possible reentrant phenomena in the phase diagram (T−h₀ plane) occur for appropriate values of p and h₀. Using the variational principle, we determine the equilibrium equation for magnetization and solve it for both transitions and at the tricritical point in order to determine the magnetization profile with respect to h₀.     

The random field Ising model with an asymmetric and anisotropic trimodal probability distribution

The Ising model in the presence of a random field, drawn from the asymmetric and anisotropic trimodal probability distribution P(_hi)=pδ(_hi−_h0)+qδ(_hi+λ∗_h0)+rδ(_hi)$P\left(h_i\right)=p\delta (h_i-h_0)+q\delta (h_i+\lambda \ast h_0)+r\delta \left(h_i\right)$, is investigated. The partial probabilities p,q,r$p,q,r$ take on values within the interval [0,1]$[0,1]$ consistent with the constraint p+q+r=1$p+q+r=1$; asymmetric distribution, _hi$h_i$ is the random field variable with basic absolute value _h0$h_0$ (strength); λ$\lambda$ is the competition parameter, which is the ratio between the respective strength of the random magnetic field in the two principal directions (+z)$(+z)$ and (−z)$(-z)$ and is positive so that the random fields are competing, anisotropic distribution. This probability distribution is an extension of the bimodal one allowing for the existence in the lattice of non magnetic particles or vacant sites. The current random field Ising system displays mainly second order phase transitions, which, for some values of p,q$p,q$ and _h0$h_0$, are followed by first order phase transitions joined smoothly by a tricritical point; occasionally, two tricritical points appear implying another second order phase transition. In addition to these points, re-entrant phenomena can be seen for appropriate ranges of the temperature and random field for specific values of λ$\lambda$, p$p$ and q$q$. Using the variational principle, we write down the equilibrium equation for the magnetization and solve it for both phase transitions and at the tricritical point in order to determine the magnetization profile with respect to _h0$h_0$, considered as an independent variable in addition to the temperature.     

Possible nonanalytic behavior of Ising models in random fields

The magnetization, M, of dilute Ising ferromagnets with quenched random fields is shown to have accumulation points of poles in each of these fields h_i(at h_i → 0) and in the external uniform field H (at H → -h_i). This occurs for all temperatures for which the nonrandom (zero-field) system has H ≠ 0. If both h_i and -h_i are possible then M is probably nonanalyti c for many values of h_i. This casts doubts on expansions in the random fields.     

Groundstate threshold in disordered anisotropic +/−J Ising models

In disordered anisotropic square +/− J Ising models SQ(p, q) at groundstates we investigate the pairs (p_c, q_c) of critical concentrations of antiferromagnetic bonds with concentrations p,q, respectively in orthogonal coordinate directions. We are led to p_c(q) ≈ π(q) with π(q) from the so-called adjoined problem. This approach is well supported by simulations for different values of q on the basis of minimal matchings of frustrated plaquettes. In particular, p_c(0) ≈ 0.21 from simulations and π(0) = 0.2113248 …, with the conjecture that p_c(0) = π(0). The concept of the adjoined problem is extended to d-dimensional (hyper-) cubic lattices. We hereby obtain for p_c,d especially in the sotropic case: p_c,3 ≈ 0.154, p_c,4 ≈ 0.170, p_c,5 ≈ 0.178, p_c,6 ≈ 0.182. Moreover, in analogy to SQ(p,q) we used the approach also for honeycomb Ising models HC(p,q,r) with no antiferromagnetic bonds in the third plaquette direction (r = 0).     

Random field effects on the phase diagrams of spin-1/2 Ising model on a honeycomb lattice

The Ising model with quenched random magnetic fields is examined for single Gaussian, bimodal and double Gaussian random field distributions by introducing an effective field approximation that takes into account the correlations between different spins that emerge when expanding the identities. Random field distribution shape dependence of the phase diagrams, magnetization and internal energy is investigated for a honeycomb lattice with a coordination number q=3. The conditions for the occurrence of reentrant behavior and tricritical points on the system are also discussed in detail.     

Glassy behaviour of random field Ising spins on Bethe lattice in external magnetic field

The thermodynamics and the phase diagram of random field Ising model (RFIM) on Bethe lattice are studied by using a replica trick. This lattice is placed in an external magnetic field (B). A Gaussian distribution of random field (hi) with zero mean and variance hi2 = HR2F is considered. The free-energy (F ), the magnetization (M) and the order parameter (q) are investigated for several values of coordination number (z). The phase diagram shows several interesting behaviours and presents tricritical point at critical temperature TC = J/k and when HRF = 0 for finite z. The free-energy (F) values increase as T increases for different intensities of random field (HRF) and finite z. The internal energy (U) has a similar behaviour to that obtained from the Monte Carlo simulations. The ground state of magnetization decreases as the intensity of random field HRF increases. The ferromagnetic (FM)-paramagnetic (PM) phase boundary is clearly observed only when z →∞. While FM-PM-spin glass (SG) phase boundaries are present for finite z. The magnetic susceptibility (χ) shows a sharp cusp at TC in a small random field for finite z and rounded different peaks on increasing HRF.     

Some properties of random Ising models

We consider an Ising model with random magnetic fieldh _i and random nearest-neighbor couplingsJ _ij . The random variablesh _i andJ _ij are independent and identically distributed with a nice enough distribution, e.g., Gaussian. We will prove that (i) at high temperature, infinite volume correlation functions are independent on the boundary conditions and decay exponentially fast with probability 1 and (ii) for any temperature with sufficiently strong magnetic field the correlation functions are again independent on the boundary conditions and decay exponentially fast with probability 1. We also prove that the averaged magnetization of the ground state configuration of the one-dimensional Ising model with random magnetic field is zero, no matter how small is the variance of theh _i .     

The transverse spin-1 Ising model with random interactions

The phase diagrams of the transverse spin-1 Ising model with random interactions are investigated using a new technique in the effective field theory that employs a probability distribution within the framework of the single-site cluster theory based on the use of exact Ising spin identities. A model is adopted in which the nearest-neighbor exchange couplings are independent random variables distributed according to the law P(J_ij)=pδ(J_ij−J)+(1−p)δ(J_ij−αJ). General formulae, applicable to lattices with coordination number N, are given. Numerical results are presented for a simple cubic lattice. The possible reentrant phenomenon displayed by the system due to the competitive effects between exchange interactions occurs for the appropriate range of the parameter α.     

Dynamics of the random field Ising model

Dynamics of the kinetic Ising model in the presence of static random fields is investigated using a self-consistent method. It is shown that if the interface fluctuations of the low temperature phase are small the system at low temperatures stays in a state without long range order. For this state the spin correlation function 〈S_q(t)S_?q(O)> averaged over all configurations of random fields decays exponentially in time with a single wavevector dependent relaxation time which is finite at the transition temperature T₀ and remains very long below T₀. In the mean field approximation the correlation time at the magnetic Bragg peak and at T₀ scales with the magnitude of the random field as τ∞h^?z_h with z_h = 1 for d = 2 and $\text{z}{}_{\mathrm{h}}\text{=}\text{4}\text{3}$ for d = 3, respectively.     

Phase diagrams of a nonequilibrium mixed spin-3/2 and spin-2 Ising system in an oscillating magnetic field

The phase diagrams of the nonequilibrium mixed spin-3/2 and spin-2 Ising ferrimagnetic system on square lattice under a time-dependent external magnetic field are presented by using the Glauber-type stochastic dynamics. The model system consists of two interpenetrating sublattices of spins σ=3/2 and S=2, and we take only nearest-neighbor interactions between pairs of spins. The system is in contact with a heat bath at absolute temperature T_abs and the exchange of energy with the heat bath occurs via one-spin flip of the Glauber dynamics. First, we investigate the time variations of average order parameters to find the phases in the system and then the thermal behavior of the dynamic order parameters to obtain the dynamic phase transition (DPT) points as well as to characterize the nature (first- or second-order) phase transitions. The dynamic phase diagrams are presented in two different planes. Phase diagrams contain paramagnetic (p), ferrimagnetic (i₁, i₂, i₃) phases, and three coexistence or mixed phase regions, namely i₁+p, i₂+p and i₃+p mixed phases that strongly depend on interaction parameters.     

On the Deser,Gilbert, Sudarshan representation for functions Wi(ν, q2)(i = 1, 2) describing the electroproduction process

The Deser, Gilbert, Sudarshan representation (D.G.S.R.) for the functions W_i(ν, q²) (i = 1,2) is considered as equations determining spectral functions h_i(a, α) via the values W_i(ν, q²) in the physical region of the electroproduction channel. It is shown that if W_i(ν, q²) obey the microcausality and spectrality conditions, then the equations for h_i(a, α) have solutions in the class of Schwartz temperated distributions and thereby the D.G.S.R. is proved. Formulae are obtained expressing spectral functions in the D.G.S.R. through the values of functions W_i(ν, q²) in the physical region of the electroproduction channel.     

Discontinuities in disordered systems

The entropyS _T (j) of a two-dimensional Ising spin glass with an independent distribution of the random couplingp(J)=x·δ(J+1)+(1-x)δ(J-j) is discontinuous for temperatureT=0 and rationalj>0 and continuous elsewhere. The integrated density of frequenciesk _M (ω ²) of an one-dimensional chain of coupled oscillators with an independent distribution of the random massesp(m)=x·δ(m-1)+(1-x)δ(m-M) has the same behaviour, whereω ² corresponds toj andM to 1/T. The discontinuity points for infiniteM are, for sufficiently large but finiteM, special, frequencies, wherek _M (ω ²) has a Lifshitz singularity.     

Exact results for the diluted Ising model with competing interactions

We consider the random-bond Ising model with the exchange integrals J > 0, ?J and 0 with the respective probabilities p, q and r, where p + q + r = 1. We give the exact value of the averaged internal energy and an exact upper bound to the averaged specific heat at temperature T determined by $\text{k}{}_{\mathrm{B}}\text{T =}\text{2J}\text{In[}\text{p}\text{(1 ? p ? r)}\text{]}\text{,}$ where k_B is the Boltsmann constant. We show that all the averaged correlation functions of even spins are non-negative at this temperature.     

On spectral representations for the functions Wi(v, q2) (i = 1, 2) describing electroproduction process

It is shown that the Deser, Gilbert, Sudarshan representation (DGSR) does not follow from microcausality and spectrality only. Examples of the functions W_i(v, q²) satisfying the microcausality and spectrality conditions are given which cannot be written as the DGSR with spectral function h(a, α) that is a temperature distribution. Instead of the DGSR the spectral representation for W_i(v, q²) has been proved (eq. (3)) which follows only from microcausality and spectrality.     

Multipole mixing ratios of transitions in Te

Gamma-gamma directional correlation measurements were made on nine transitions in ¹²⁴Te with a NaI(Tl)-Ge(Li) detector arrangement and multichannel analysis. The multipole mixing ratios obtained were δ(646) = 0.000±0.001, δ(714) = 1.5_−0.3^+0.6, δ(723) = −3.3±0.2, δ(1437) = 3.7_−2.0^+2.7, δ(1489) = −3.4_−1.5^+0.9, δ(968) = −0.03_−0.05^+0.06, δ(1368) = −0.045±0.090, δ(1045) = 0.041_−0.041^+0.047, δ(1691) = −0.02±0.01, and δ(2091) = 0.00_−0.03^+0.02. The first δ is M3/E2, the next three are E2/M1, and the last five are M2/E1. The retardation (a factor of approximately 50) of the crossover to cascade transitions from the 2039 keV, third 2⁺ level to the second and first 2⁺ levels is essentially the same for both the M1 and E2 components. In addition, spin and parity assignments of 2⁺ were made for the 2039 and 2092 keV levels.     

Metastable Behavior for Bootstrap Percolation on Regular Trees

We examine bootstrap percolation on a regular (b+1)-ary tree with initial law given by Bernoulli(p). The sites are updated according to the usual rule: a vacant site becomes occupied if it has at least θ occupied neighbors, occupied sites remain occupied forever. It is known that, when b>θ≥2, the limiting density q=q(p) of occupied sites exhibits a jump at some p _T=p _T(b,θ)∈(0,1) from q _T:=q(p _T)<1 to q(p)=1 when p>p _T. We investigate the metastable behavior associated with this transition. Explicitly, we pick p=p _T+h with h>0 and show that, as h ↓0, the system lingers around the “critical” state for time order h ^−1/2 and then passes to fully occupied state in time O(1). The law of the entire configuration observed when the occupation density is q∈(q _T,1) converges, as h ↓0, to a well-defined measure.     

Properties of the Ising magnet confined in a corner geometry

The properties of Ising square lattices with nearest neighbor ferromagnetic exchange confined in a corner geometry, are studied by means of Monte Carlo simulations. Free boundary conditions at which boundary magnetic fields ±h are applied, i.e., at the two boundary rows ending at the lower left corner a field +h acts, while at the two boundary rows ending at the upper right corner a field −h acts. For temperatures T less than the critical temperature T_c of the bulk, this boundary condition leads to the formation of two domains with opposite orientation of the magnetization direction, separated by an interface which for T larger than the filling transition temperature T_f(h) runs from the upper left corner to the lower right corner, while for T<T_f(h) this interface is localized either close to the lower left corner or close to the upper right corner. It is shown that for T=T_f(h) the magnetization profile m(z) in the z-direction normal to the interface simply is linear and the interfacial width scales as w∝L, while for T>T_f(h) it scales as . The distribution P(?) of the interface position ? (measured along the z-direction from the corners) decays exponentially for T<T_f(h) from either corner, is essentially flat for T=T_f(h), and is a Gaussian centered at the middle of the diagonal for T>T_f(h). Unlike the findings for critical wetting in the thin film geometry of the Ising model, the Monte Carlo results for corner wetting are in very good agreement with the theoretical predictions.     

Glassy phase and thermodynamics for random field Ising model on spherical lattice in magnetic field

Phase diagram and thermodynamic parameters of the random field Ising model (RFIM) on spherical lattice are studied by using mean field theory. This lattice is placed in an external magnetic field (B). The random field (hi) is assumed to be Gaussian distributed with zero mean and a variance