The equilibrium properties of the two-dimensional Ising model with nearest-neighbour interactions uniformly distributed between -J and J have been determined using a numerical transfer matrix method. The calculations are performed on a long strip of width L(3<or=L<or=11) with periodic boundary conditions and finite-size scaling arguments are used to extrapolate to L= infinity . The spin-glass correlation length diverges as T - nu at low temperature with nu =2.96+or-0.22. The correlation length scaling function differs from that for the random Ising model with interactions of magnitude J and random sign showing that the two models with different distribution functions belong to different universality classes. The heat capacity is linear at low temperature and can be interpreted using a continuous distribution of two-level states which is finite at zero energy and increases approximately linearly with energy. At zero temperature the entropy is zero and the internal energy extrapolated to L= infinity is -(0.7943+or-0.0010)J.
The equilibrium properties of the two-dimensional Ising model with nearest-neighbour interactions uniformly distributed between -J and J have been determined using a numerical transfer matrix method. The calculations are performed on a long strip of width L(3<or=L<or=11) with periodic boundary conditions and finite-size scaling arguments are used to extrapolate to L= infinity . The spin-glass correlation length diverges as T - nu at low temperature with nu =2.96+or-0.22. The correlation length scaling function differs from that for the random Ising model with interactions of magnitude J and random sign showing that the two models with different distribution functions belong to different universality classes. The heat capacity is linear at low temperature and can be interpreted using a continuous distribution of two-level states which is finite at zero energy and increases approximately linearly with energy. At zero temperature the entropy is zero and the internal energy extrapolated to L= infinity is -(0.7943+or-0.0010)J.
The thermodynamic properties of the two-dimensional Ising model with random (+or-J) nearest-neighbour interactions are determined using a numerical transfer matrix method. The calculations are performed on a long strip of width L(3<or=L<or=11) with periodic boundary conditions and finite-size scaling arguments are used to extrapolate to L= infinity . The central result is that the spin-glass correlation length diverges as T- nu at low temperature T with nu =2.59+or-0.13. The heat capacity is similar to the Schottky anomaly. At zero temperature the entropy per spin is 0.0701+or-0.0005 and the internal energy per spin is -(1.4024+or-0.0012)J.
Transfer-matrix methods are used to calculate spin-spin correlation functions (G), Helmholtz free energies (f) and magnetizations (m) in the two-dimensional random-field Ising model close to the zero-field bulk critical temperature T(c 0), on long strips of width L=3-18 sites, for binary field distributions. Analysis of the probability distributions of G for varying spin-spin distances R shows that describing the decay of their averaged values by effective correlation lengths is a valid procedure only for not very large R. Connections between field and correlation function distributions at high temperatures are established, yielding approximate analytical expressions for the latter, which are used for computation of the corresponding structure factor. It is shown that, for fixed R/L, the fractional widths of correlation-function distributions saturate asymptotically with L-2.2. Considering an added uniform applied field h, a connection between f(h), m(h), the Gibbs free energy g(m) and the distribution function for the uniform magnetization in a zero uniform field, P0(m), is derived and first illustrated for pure systems, and then applied for nonzero random field. From finite-size scaling and crossover arguments, coupled with numerical data, it is found that the width of P0(m) varies against (nonvanishing, but small) random-field intensity H0 as H(-3/7)(0).
Transfer-matrix methods are used to study the probability distributions of spin-spin correlation functions G in the two-dimensional random-field Ising model, on long strips of width L=3-15 sites, for binary field distributions at generic distance R, temperature T, and field intensity h(0). For moderately high T, and h(0) of the order of magnitude used in most experiments, the distributions are singly peaked, though rather asymmetric. For low temperatures the single-peaked shape deteriorates, crossing over towards a double-delta ground-state structure. A connection is obtained between the probability distribution for correlation functions and the underlying distribution of accumulated field fluctuations. Analytical expressions are in good agreement with numerical results for R/L > or approximately 1, low T, h(0) not too small, and near G=1. From a finite-size ansatz at T=T(c)(h(0)=0), h(0)-->0, averaged correlation functions are predicted to scale with L(y)h(0), y=7/8. From numerical data we estimate y=0.875+/-0.025, in excellent agreement with theory. In the same region, the rms relative width W of the probability distributions varies for fixed R/L=1 as W approximately h(kappa)(0) f(L h(u)(0)) with kappa approximately 0.45, u approximately 0.8; f(x) appears to saturate when x-->infinity, thus implying W approximately h(kappa)(0) in d=2.
We have calculated finite-size correlation lengths for strips of the square-lattice Ising spin glass using the transfer-matrix method. In order to minimize sampling errors we study strip lengths up to 5\ifmmode\times\else\texttimes\fi{}106 lattice spacings. A phenomenological renormalization-group analysis indicates that there are strong corrections to simple power-law scaling near the zero-temperature critical point, as is to be expected near a lower critical dimension. We examine models with Gaussian, exp(-J2/2), and exponential, exp(-\ensuremath{\Vert}J\ensuremath{\Vert}), distributions of couplings; the Gaussian distribution shows stronger finite-size corrections. The correlation-length exponent is estimated to be \ensuremath{\nu}=4.2\ifmmode\pm\else\textpm\fi{}0.5, although we do not want to rule out the possibility that \ensuremath{\nu} is significantly larger than this.
The ± J Ising model with asymmetric probability weight of ± bonds in two dimensions is studied in the p - T plane by the numerical transfer-matrix method. We calculate finite-width susceptibilities and finite-width correlation lengths of long strips of various sizes up to 14×10 5 . Finite-size scaling analysis suggests that the random antiphase state exists adjacent to the ferromagnetic phase. The boundary separating these phases is vertical (parallel to the temperature axis) in the phase diagram. The weak universality seems to hold along the ferromagnetic critical line.
We analyze changes in the thermodynamic properties of a spin system when it passes from the classical two-dimensional Ising model to the spin glass model, where spin-spin interactions are random in their values and signs. Formally, the transition reduces to a gradual change in the amplitude of the multiplicative noise (distributed uniformly with a mean equal to one) superimposed over the initial Ising matrix of interacting spins. Considering the noise, we obtain analytical expressions that are valid for lattices of finite sizes. We compare our results with the results of computer simulations performed for square N = L × L lattices with linear dimensions L = 50 ÷ 1000. We find experimentally the dependencies of the critical values (the critical temperature, the internal energy, entropy and the specific heat) as well as the dependencies of the energy of the ground state and its magnetization on the amplitude of the noise. We show that when the variance of the noise reaches one, there is a jump of the ground state from the fully correlated state to an uncorrelated state and its magnetization jumps from 1 to 0. In the same time, a phase transition that is present at a lower level of the noise disappears.