We study critical hysteresis in the random-field Ising model on a two-dimensional periodic lattice with a variable coordination number
zeff in the range $3\ensuremath{\le}{z}_{\mathrm{eff}}\ensuremath{\le}6$. We find that the model supports critical behavior in the range $4<{z}_{\mathrm{eff}}\ensuremath{\le}6$, but the critical exponents are independent of
zeff. The result is discussed in the context of the universality of nonequilibrium critical phenomena and extant results in the field.
Exact analytic results are presented for the critical properties of the dilute Ising model on a Bethe lattice, where a fraction p of the sites are occupied at random by spins. It is found that the critical behaviour is the same as that of the pure system (p=1) except for transitions which occur by varying p at zero temperature, where the critical behaviour is different and corresponds to the percolation problem. For T to 0 and p to p c , the percolation concentration, the zero-field magnetization shows crossover behaviour of the form expected from scaling theories. Outside the critical region, a rapidly convergent numerical method is developed and used to calculate the spontaneous magnetization.
We present a numerical study of the critical behavior of the nonequilibrium zero-temperature random field Ising model in two dimensions on a triangular lattice. Our findings, based on the scaling analysis and collapse of data collected in extensive simulations of systems with linear sizes up to L=65536, show that the model is in a different universality class than the same model on a quadratic lattice, which is relevant for a better understanding of model universality and the analysis of experimental data.
We study an Ising model with nearest-neighbor antiferromagnetic interactions. It is placed on a triangular lattice, where each site is occupied by a spin with
x probability. There is no applied magnetic field. Randomness and frustration, two essential ingredients of spin-glasses, are present in this model. We study its critical properties here. The entropy is obtained by a transfer-matrix calculation as a function of
x at low temperature (
T=kB0.3J) for systems on a lattice of 10 \ifmmode\times\else\texttimes\fi{} 20 sites. A fairly shallow minimum appears near $x\ensuremath{\simeq}0.9$, which suggests that this case is the one most likely to show a transition into an ordered state at low temperature. We study the cases
x=1,0.9,and0.74, which is about half-way to the critical percolation. We simulate systems on lattices of 50 \ifmmode\times\else\texttimes\fi{} 50 sites and 30 \ifmmode\times\else\texttimes\fi{} 30 sites by the Monte Carlo method. The specific heat has a broad maximum at $T\ensuremath{\simeq}0.9$ for
x=0.74and0.9. ${\ensuremath{\chi}}_{\mathrm{SG}}$, defined by ${\ensuremath{\chi}}_{\mathrm{SG}}={N}^{\ensuremath{-}1}\ensuremath{\Sigma}{i,j}^{}{〈{\ensuremath{\sigma}}_{i}{\ensuremath{\sigma}}_{j}〉}^{2}$, and the relaxation time ($\ensuremath{\tau}$) are obtained for $T\ensuremath{\ge}\frac{0.6J}{{k}_{B}}$. Both quantities, $\ensuremath{\tau}$ and ${\ensuremath{\chi}}_{\mathrm{SG}}$, turn out to be proportional to $\mathrm{exp}[\frac{A}{{(T\ensuremath{-}{T}_{0})}^{c}}]$ and $0\ensuremath{\le}{T}_{0}\ensuremath{\lesssim}0.4$; a fit with
T0=0 yields $c\ensuremath{\approx}1$ for
x=1 but $c\ensuremath{\approx}2$ for
x=0.74and0.9.
The hypothesis of weak universality is tested for the moments of the zero-field correlation function and the magnetic-field derivatives of the free energy of dilute Ising models. The investigation, which is carried out for dilution of both the magnetic sites (site dilution) and interactions (bond dilution), is performed just above the critical temperature. In the case of bond dilution both the quenched and annealed distributions are considered. The quenched model is investigated by series methods for site and bond dilution on the face-centred cubic lattice and for bond dilution on the triangular lattice.
Exact analytical and numerical results for the site-diluted antiferromagnetic Ising model on the triangular lattice (AFIT) are presented. For infinitesimal dilution the change in the free energy of the system is related to the distribution of local fields, and it is shown that for a frustrated system such as the AFIT, dilution lowers the entropy per spin. For lattices of finite size and dilution the transfer matrix for the partition function is evaluated numerically. The entropy per spin shows a marked minimum near a concentration of spins x=0.70, in some disagreement with earlier transfer-matrix results.