Perturbative analysis of disordered Ising models close to criticality (original) (raw)
Related papers
Strong disorder fixed points in the two-dimensional random-bond Ising model
Journal of Statistical Mechanics: Theory and Experiment, 2006
The random-bond Ising model on the square lattice has several disordered critical points, depending on the probability distribution of the bonds. There are a finite-temperature multicritical point, called Nishimori point, and a zerotemperature fixed point, for both a binary distribution where the coupling constants take the values ±J and a Gaussian disorder distribution. Inclusion of dilution in the ±J distribution (J = 0 for some bonds) gives rise to another zero-temperature fixed point which can be identified with percolation in the non-frustrated case (J ≥ 0). We study these fixed points using numerical (transfer matrix) methods. We determine the location, critical exponents, and central charge of the different fixed points and study the spin-spin correlation functions. Our main findings are the following: (1) We confirm that the Nishimori point is universal with respect to the type of disorder, i.e. we obtain the same central charge and critical exponents for the ±J and Gaussian distributions of disorder. (2) The Nishimori point, the zero-temperature fixed point for the ±J and Gaussian distributions of disorder, and the percolation point in the diluted case all belong to mutually distinct universality classes. (3) The paramagnetic phase is re-entrant below the Nishimori point, i.e. the zero-temperature fixed points are not located exactly below the Nishimori point, neither for the ±J distribution, nor for the Gaussian distribution.
Temperature-dependent criticality in random 2D Ising models
The European Physical Journal Plus, 2021
We consider 2D random Ising ferromagnetic models, where quenched disorder is represented either by random local magnetic fields (random-field Ising model) or by a random distribution of interaction couplings (random-bond Ising model). In both cases, we first perform zero- and finite-temperature Monte Carlo simulations to determine how the critical temperature depends on the disorder parameter. We then focus on the reversal transition triggered by an external field and study the associated Barkhausen noise. Our main result is that the critical exponents characterizing the power law associated with the Barkhausen noise exhibit a temperature dependence in line with existing experimental observations.
A note on the Ising model in high dimensions
Communications in Mathematical Physics, 1989
We consider thed-dimensional Ising model with a nearest neighbor ferromagnetic interactionJ(d)=1/4d. We show that asd→∞ the+phase (and the — phase) approaches a product measure with density given by the mean field approximation. In particular the spontaneous magnetization converges to its mean field value. A similar result holds for the unique Gibbs measure of the system subject to an external fieldh≠0.
Relaxation to equilibrium for two dimensional disordered Ising systems in the Griffiths phase
Communications in Mathematical Physics, 1997
We consider Glauber-type dynamics for two dimensional disordered magnets of Ising type. We prove that, if the disorder-averaged influence of the boundary condition is sufficiently small in the equilibrium system, then the corresponding Glauber dynamics is ergodic with probability one and the disorder-average C(t) of time-autocorrelation function satisfies C(t) e −m(log t) 2 (for large t). For the standard two dimensional dilute Ising ferromagnet with i.i.d. random nearest neighbor couplings taking the values 0 or J 0 > 0, our results apply even if the active bonds percolate and J 0 is larger than the critical value J c of the corresponding pure Ising model. For the same model we also prove that in the whole Griffiths' phase the previous upper bound is optimal. This implies the existence of a dynamical phase transition which occurs when J crosses J c .
Physica A: Statistical Mechanics and its Applications, 1999
We derive high-temperature series expansions for the free energy and susceptibility of the two-dimensional random-bond Ising model with a symmetric bimodal distribution of two positive coupling strengths J1 and J2 and study the in uence of the quenched, random bond-disorder on the critical behavior of the model. By analysing the series expansions over a wide range of coupling ratios J2=J1, covering the crossover from weak to strong disorder, we obtain for the susceptibility with two di erent methods compelling evidence for a singularity of the form ∼ t −7=4 |ln t| 7=8 , as predicted theoretically by Shalaev, Shankar, and Ludwig. For the speciÿc heat our results are less convincing, but still compatible with the theoretically predicted log-log singularity.
The quenched-disordered Ising model in two and four dimensions
2009
We briefly review the Ising model with uncorrelated, quenched random-site or random-bond disorder, which has been controversial in both two and four dimensions. In these dimensions, the leading exponent α, which characterizes the specific-heat critical behaviour, vanishes and no Harris prediction for the consequences of quenched disorder can be made. In the two-dimensional case, the controversy is between the strong universality hypothesis which maintains that the leading critical exponents are the same as in the pure case and the weak universality hypothesis, which favours dilution-dependent leading critical exponents. Here the random-site version of the model is subject to a finite-size scaling analysis, paying special attention to the implications for multiplicative logarithmic corrections. The analysis is fully supportive of the scaling relations for logarithmic corrections and of the strong scaling hypothesis in the 2D case. In the four-dimensional case unusual corrections to scaling characterize the model, and the precise nature of these corrections has been debated. Progress made in determining the correct 4D scenario is outlined.
Dynamical properties of random-field Ising model
Physical Review E, 2013
Extensive Monte Carlo simulations are performed on a two-dimensional random field Ising model. The purpose of the present work is to study the disorder-induced changes in the properties of disordered spin systems. The time evolution of the domain growth, the order parameter and the spin-spin correlation functions are studied in the non equilibrium regime. The dynamical evolution of the order parameter and the domain growth shows a power law scaling with disorder-dependent exponents. It is observed that for weak random fields, the two dimensional random field Ising model possesses long range order. Except for weak disorder, exchange interaction never wins over pinning interaction to establish long range order in the system.
Critical aspects of the random-field Ising model
The European Physical Journal B, 2013
We investigate the critical behavior of the three-dimensional random-field Ising model (RFIM) with a Gaussian field distribution at zero temperature. By implementing a computational approach that maps the ground-state of the RFIM to the maximum-flow optimization problem of a network, we simulate large ensembles of disorder realizations of the model for a broad range of values of the disorder strength h and system sizes V = L 3 , with L ≤ 156. Our averaging procedure outcomes previous studies of the model, increasing the sampling of ground states by a factor of 10 3 . Using well-established finitesize scaling schemes, the fourth-order's Binder cumulant, and the sample-to-sample fluctuations of various thermodynamic quantities, we provide high-accuracy estimates for the critical field hc, as well as the critical exponents ν, β/ν, andγ/ν of the correlation length, order parameter, and disconnected susceptibility, respectively. Moreover, using properly defined noise to signal ratios, we depict the variation of the self-averaging property of the model, by crossing the phase boundary into the ordered phase. Finally, we discuss the controversial issue of the specific heat based on a scaling analysis of the bond energy, providing evidence that its critical exponent α ≈ 0 − .