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arXiv:1603.04444 · Journal of Physics A Mathematical and Theoretical

The random-bond Ising model in 2.01 and 3 dimensions

Zohar Komargodski, David Simmons-Duffin

Published Feb 15, 2017• 112 citations
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Abstract

Abstract We consider the Ising model between 2 and 4 dimensions perturbed by quenched disorder in the strength of the interaction between nearby spins. In the interval 2 < d < 4 this disorder is a relevant perturbation that drives the system to a new fixed point of the renormalization group. At d = 2 such disorder is marginally irrelevant and can be studied using conformal perturbation theory. Combining conformal perturbation theory with recent results from the conformal bootstrap we compute some scaling exponents in an expansion around d = 2. If one trusts these computations also in d = 3, one finds results consistent with experimental data and Monte Carlo simulations. In addition, we perform a direct uncontrolled computation in d = 3 using new results for low-lying operator dimensions and OPE coefficients in the 3d Ising model. We compare these new methods with previous studies. Finally, we comment about the O (2) model in d = 3, where we predict a large logarithmic correction to the infrared scaling of disorder.

DOI
10.1088/1751-8121/aa6087

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Abstract We consider the Ising model between 2 and 4 dimensions perturbed by quenched disorder in the strength of the interaction between nearby spins. In the interval 2 < d < 4 this disorder is a relevant perturbation that drives the system to a new fixed point of the renormalization group. At d = 2 such disorder is marginally irrelevant and can be studied using conformal perturbation theory. Combining conformal perturbation theory with recent results from the conformal bootstrap we compute some scaling exponents in an expansion around d = 2. If one trusts these computations also in d = 3, one finds results consistent with experimental data and Monte Carlo simulations. In addition, we perform a direct uncontrolled computation in d = 3 using new results for low-lying operator dimensions and OPE coefficients in the 3d Ising model. We compare these new methods with previous studies. Finally, we comment about the O (2) model in d = 3, where we predict a large logarithmic correction to the infrared scaling of disorder.
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In this paper we provide new analytic results on two-dimensional q q -Potts models ( q ≥ 2 q ≥ 2 ) in the presence of bond disorder correlations which decay algebraically with distance with exponent a a . In particular, our results are valid for the long-range bond disordered Ising model ( q=2 q = 2 ). We implement a renormalization group perturbative approach based on conformal perturbation theory. We extend to the long-range case the RG scheme used in [V. Dotsenko et al., Nucl. Phys. B 455 701-23] for the short-range disorder. Our approach is based on a 2 2 -loop order double expansion in the positive parameters (2-a) ( 2 − a ) and (q-2) ( q − 2 ) . We will show that the Weinrib-Halperin conjecture for the long-range thermal exponent can be violated for a non-Gaussian disorder. We compute the central charges of the long-range fixed points finding a very good agreement with numerical measurements.
Thanks to the impressive progress of conformal bootstrap methods we have now very precise estimates of both scaling dimensions and operator product expansion coefficients for several 3D universality classes. We show how to use this information to obtain similarly precise estimates for off-critical correlators using conformal perturbation. We discuss in particular the $⟨\ensuremath{\sigma}(r)\ensuremath{\sigma}(0)⟩$, $⟨\ensuremath{\epsilon}(r)\ensuremath{\epsilon}(0)⟩$ and $⟨\ensuremath{\sigma}(r)\ensuremath{\epsilon}(0)⟩$ two-point functions in the high and low temperature regimes of the 3D Ising model and evaluate the leading and next to leading terms in the $s={\mathrm{tr}}^{{\mathrm{\ensuremath{\Delta}}}_{t}}$ expansion, where ttt is the reduced temperature. Our results for $⟨\ensuremath{\sigma}(r)\ensuremath{\sigma}(0)⟩$ agree both with Monte Carlo simulations and with a set of experimental estimates of the critical scattering function.
We have investigated the critical behavior of a three-dimensional random-bond Ising model for a series of the disorder strength by a finite-time scaling combining with Monte Carlo renormalization-group method in the presence of a linearly varying temperature. The method enables us to estimate a lot of critical exponents of both static and dynamic nature independently as well as the critical temperatures. The static exponents obtained agree well with most existing results, verify both the hyperscaling and the Rushbrooke scaling laws and their combined scaling law, which in turn validate their asymptotic nature, and corroborate the universality of the relevant random fixed point with respect to the forms of disorder. The dynamic critical exponent z is estimated to 2.114(51), which is compatible with those obtained from experiments and renormalization-group analyses. The exponents at low and high disorder strengths do not satisfy all scaling laws and are argued to be crossover exponents that reflect crossover from the random fixed point to the pure and the percolation fixed point. They also indicate that the exponents that were previously suggested to be a distinct universality class for strong disorder strength in the literature may be just crossover. Our results demonstrate the effectiveness of the finite-time scaling method.
We consider the critical behavior of two-dimensional Potts models in the presence of a bond disorder in which the correlation decays as a power law. In some recent work the thermal sector of this theory was investigated by a renormalization group computation based on perturbed conformal field theory. Here we apply the same approach to study instead the magnetic sector. In particular, we compute the leading corrections to the Potts spin scaling dimension. Our results include as a special case the long-range disorder Ising model. We compare our prediction to Monte Carlo simulations. Finally, by studying the magnetization scaling function, we show a clear numerical evidence of a cross-over between the long-range and the short-range class of universality.
The random-field Ising model is one of the few disordered systems where the perturbative renormalization group can be carried out to all orders of perturbation theory. This analysis predicts dimensional reduction, i.e., that the critical properties of the random-field Ising model in D dimensions are identical to those of the pure Ising ferromagnet in D-2 dimensions. It is well known that dimensional reduction is not true in three dimensions, thus invalidating the perturbative renormalization group prediction. Here, we report high-precision numerical simulations of the 5D random-field Ising model at zero temperature. We illustrate universality by comparing different probability distributions for the random fields. We compute all the relevant critical exponents (including the critical slowing down exponent for the ground-state finding algorithm), as well as several other renormalization-group invariants. The estimated values of the critical exponents of the 5D random-field Ising model are statistically compatible to those of the pure 3D Ising ferromagnet. These results support the restoration of dimensional reduction at D=5. We thus conclude that the failure of the perturbative renormalization group is a low-dimensional phenomenon. We close our contribution by comparing universal quantities for the random-field problem at dimensions 3≤D<6 to their values in the pure Ising model at D-2 dimensions, and we provide a clear verification of the Rushbrooke equality at all studied dimensions.