Critical behavior of the dilute antiferromagnet in a magnetic field

Abstract

We study the critical behavior of the diluted antiferromagnet in a field with the tethered Monte Carlo formalism. We compute the critical exponents (including the elusive hyperscaling violations exponent θ). Our results provide a comprehensive description of the phase transition and clarify the inconsistencies between previous experimental and theoretical work. To do so, our method addresses the usual problems of numerical work (large tunneling barriers and self-averaging violations). Understanding collective behavior in the presence of quenched disorder has long been one of the most challenging and interesting problems in statistical mechanics. One of its simplest representatives is the random field Ising model (RFIM), which has been extensively studied both theoretically and experimentally. DOI1 The RFIM is physically realized by a diluted antiferromagnet in an applied magnetic field (DAFF).It is known that the D = 3 DAFF (or RFIM) undergoes a phase transition, but the details remain controversial, with severe inconsistencies between analytical, experimental, and numerical work. A scaling theory is generally accepted, where the dimension D of the system is replaced by D − θ in the hyperscaling relation. This third independent critical exponent, believed to be θ ≈ 1.5, is inaccessible both to a direct experimental measurement and to traditional Monte Carlo methods.The values of the remaining critical exponents, seemingly more straightforward, are also controversial. On the experimental front, different ansätze for the scattering line shape yield mutually incompatible estimates of the thermal critical exponent, namely, ν = 0.87 (7) On the other hand, the numerical determination of ν has steadily shifted, the most precise estimate being 1.37(9), 5 inconsistent with the experimental values and barely compatible with α ≈ 0. The value of α itself is very hard to measure in a numerical simulation. 6More fundamentally, the smallness of the magnetic exponent β, combined with the numerical observation of metastability, 7 has led some authors to suggest that the transition in the DAFF may be of first order.Ultimately, the physical reasons for this confusion betray the fact that the traditional tools of statistical mechanics are ill suited to systems with rugged free-energy landscapes. Both experimentally and numerically, the system gets trapped in local minima, with escape times that grow as log τ ∼ ξ θ (ξ is the correlation length). This not only makes it exceedingly hard to thermalize the system, but also generates a rare-events statistics, causing self-averaging violations. 8 In this Rapid Communication we study the DAFF with the tethered Monte Carlo (TMC) formalism.9 Our approach restores self-averaging and is able to negotiate the free-energy barriers of the DAFF to equilibrate large systems safely. It also provides direct access to the key parameter θ . We thus obtain a comprehensive picture of the phase transition, consistent both with analytical results for the RFIM and with experiments on the DAFF, and shed lig...

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

“…This method allows for a particularly transparent study of corrections to scaling, that up to now have been considered as the Achilles' heel in the study of the D 3 random-field problem. We should note that previous applications of the method include diluted antiferromagnets [48] and the spin-glass problem; see Ref. [108] and references therein.…”

“…Currently, despite the huge efforts recorded in the literature, a clear picture of the model's critical behavior is still lacking. Although the view that the phase transition of the RFIM is of second order is well established [48][49][50]64], the extremely small value of the exponent β continues to cast some doubts. Moreover, a rather strong debate exists with regards to the role of disorder: the available simulations are not able to settle the question of whether the critical exponents depend on the particular choice of the distribution for the random fields, analogous to the mean-field theory predictions [37].…”

“…Moreover, a rather strong debate exists with regards to the role of disorder: the available simulations are not able to settle the question of whether the critical exponents depend on the particular choice of the distribution for the random fields, analogous to the mean-field theory predictions [37]. Thus the whole issue of the model's critical behavior is under intense investigation [41,48,49,51,52,[74][75][76][77][78].…”

“…The small random magnetic fields make it possible to employ the full formalism that we derive in the following sections. Furthermore, the fluctuation-dissipation formulas elucidated below are also valid when working at finite (rather than zero) temperature, which is necessary for some algorithms [48,51].…”

“…However, typical Monte Carlo schemes get trapped into local minima with escape time exponential in ξ θ , where ξ denotes the correlation length. Although sophisticated improvements have appeared [48][49][50][51][52], these simulations produced low-accuracy data because they were limited to linear sizes of the order of L max 32. Larger sizes can be achieved at T = 0, through the wellknown mapping of the ground state to the maximum-flow optimization problem [53][54][55][56][57][58][59][60][61][62][63][64][65][66].…”