The non-equilibrium random-field Ising model is well studied, yet there are outstanding questions. In two dimensions, power law scaling approaches fail and the critical disorder is difficult to pin down. Additionally, the presence of faceting on the square lattice creates avalanches that are lattice dependent at small scales. We propose two methods which we find solve these issues. First, we perform large scale simulations on a Voronoi lattice to mitigate the effects of faceting. Secondly, the invariant arguments of the universal scaling functions necessary to perform scaling collapses can be directly determined using our recent normal form theory of the Renormalization Group. This method has proven useful in cleanly capturing the complex behavior which occurs in both the lower and upper critical dimensions of systems and here captures the 2D NE-RFIM behavior well. The obtained scaling collapses span over a range of a factor of ten in the disorder and a factor of
104 in avalanche cutoff. They are consistent with a critical disorder at zero and with a lower critical dimension for the model equal to two.
The random transverse-field Ising ferromagnet (RTFIF) is a highly disordered quantum system which contains randomness in the coupling strengths as well as in the transverse-field strengths. In one dimension, the critical properties are governed by an infinite-randomness fixed point (IRFP), and renormalization-group studies argue that the two-dimensional (2D) model is also governed by an IRFP. However, even the location of the critical point remains unsettled among quantum Monte Carlo (QMC) studies. In this work, we perform extensive QMC simulations to locate the quantum critical point and attempt a finite-size scaling analysis to observe the critical behavior. We estimate the critical field strength of the 2D RTFIF as ${\mathrm{\ensuremath{\Gamma}}}_{c}=7.52(2)$, together with critical exponents such as $\ensuremath{\beta}=1.5(3), \ensuremath{\nu}=1.6(3)$, and
z=3.3(3) or $\ensuremath{\psi}=0.50(3)$. We have also considered the McCoy-Wu model, which has randomness in the ferromagnetic coupling strengths but not in the transverse-field strength. Our QMC calculation shows that the critical behavior of the 2D McCoy-Wu model is closer to that of the 2D transverse-field Ising spin glass than to that of the 2D RTFIF. These numerical findings enhance our understanding of disordered 2D quantum systems.
A formalism is given whereby high-temperature series for the random-field Ising model are obtained by partitioning the vertices of the pure Ising series' diagrams; for any given order in the high-temperature variable, the random fields are thereby treated exactly. Series for the susceptibility up to seventh order in the high-temperature variable have been obtained for an arbitrary distribution of the random fields. Series for a bimodal and a Gaussian distribution were analyzed by Dlog-Pad\'e methods. The analysis predicts that, for a bimodal distribution of random fields, the tricritical point predicted by mean-field theory persists in all dimensions above the lower critical dimension, but moves to weaker fields as the dimension is reduced. For the Gaussian distribution, the transition is predicted to be second order over the whole phase boundary for dimension d\ensuremath{\ge}4; however, for d<4 the analysis suggests that fluctuations drive the transition first order for sufficiently strong fields implying that d=4 is a ``critical'' dimension for this model. Above four dimensions the value of the susceptibility exponent determined from the series is consistent with the predictions of the d\ensuremath{\rightarrow}d-2 rule for both distributions.