Efficient numerical methods for the random-field Ising model: Finite-size scaling, reweighting extrapolation, and computation of response functions

Abstract

It was recently shown [Phys. Rev. Lett. 110, 227201 (2013)] that the critical behavior of the random-field Ising model in three dimensions is ruled by a single universality class. This conclusion was reached only after a proper taming of the large scaling corrections of the model by applying a combined approach of various techniques, coming from the zero-and positive-temperature toolboxes of statistical physics. In the present contribution we provide a detailed description of this combined scheme, explaining in detail the zero-temperature numerical scheme and developing the generalized fluctuation-dissipation formula that allowed us to compute connected and disconnected correlation functions of the model. We discuss the error evolution of our method and we illustrate the infinite limit-size extrapolation of several observables within phenomenological renormalization. We present an extension of the quotients method that allows us to obtain estimates of the critical exponent α of the specific heat of the model via the scaling of the bond energy and we discuss the self-averaging properties of the system and the algorithmic aspects of the maximum-flow algorithm used.

Restoration of dimensional reduction in the random-field Ising model at five dimensions

“…3D RFIM [22,23] 4D RFIM [6,38] 5D RFIM (current work) 2D IM [39] not only for the RFIM. Still, the RFIM is unique among other models due to the existence of very fast algorithms that make the study of these questions numerically feasible.…”

“…Here we report large-scale zero-temperature numerical simulations of the RFIM at five spatial dimensions. Our analysis benefits from recent advances in finitesize scaling and reweighting methods for disordered systems [22,23]. By using two different random-field distribution we are able to show the universality of the critical exponents characterizing the transition.…”

Evidence for Supersymmetry in the Random-Field Ising Model at D=5

“…Our simulations and analysis closely follows the methodology outined in our previous works at D = 3 and 4 [23,25] (for full technical details see Ref. [24]).…”

Phase Transitions in Disordered Systems: The Example of the Random-Field Ising Model in Four Dimensions

“…Applications in hard and soft condensed matter Physics are many (see e.g. [3][4][5]), and their numbers increase [6][7][8]. The RFIM Hamiltonian is…”

“…Noteworthy, claims of universality violations for the RFIM at D ≥ 3 have been quite frequent when comparing different distributions of random fields [34][35][36][37]. Fortunately, using new techniques of statistical analysis [5], it has been possible to show that, at least in D = 3, these apparent universality violations are merely finite-size corrections to the leading scaling behavior [44,49]. We also note the numerical bound 2η −η ≤ 0.0026(10) [44] which is valid in D = 3 [50].…”