The Random Field Ising Model (RFIM) represents one of the simplest models of cooperative behaviour with quenched disorder, and it is, in a way, complementary to the Ising Spin Glass which will be extensively treated later in this book. It accounts for the presence of a random external magnetic field which antagonizes the ordering induced by the ferromagnetic spin–spin interactions. From an experimental point of view, on the other hand, as shown by Fishman and Aharony (1979) and Cardy (1984), it is equivalent to a dilute anti-ferromagnet in a uniform field (see Belanger, 1998 for a recent review on experimental results). Despite twenty-five years of active and continuous research the RFIM is not yet completely understood. The problem seems related to the presence of bound states in the ferromagnetic phase, which make the standard theoretical approaches not adequate to analyze the critical behaviour. Here we discuss the RFIM in the context of perturbative field theory. The chapter is organized as follows: in Section 2.1 we define the model and outline the main expectations for its qualitative behaviour. In Section 2.2 we introduce an effective replicated ϕ 4 field model where the disorder has been integrated out. Then we perform a perturbative analysis on this model (Section 2.3) and illustrate how the so-called dimensional reduction arises (Section 2.4). Finally, in Section 2.5 we introduce some generalized couplings which need to be taken into account to properly describe the system; we perform a perturbative Renormalization Group (RG) close to the upper critical dimension (Sections 2.6 and 2.7) and discuss the occurrence of a vitrous transition (Section 2.8).
The Random Field Ising Model (RFIM) represents one of the simplest models of cooperative behaviour with quenched disorder, and it is, in a way, complementary to the Ising Spin Glass which will be extensively treated later in this book. It accounts for the presence of a random external magnetic field which antagonizes the ordering induced by the ferromagnetic spin–spin interactions. From an experimental point of view, on the other hand, as shown by Fishman and Aharony (1979) and Cardy (1984), it is equivalent to a dilute anti-ferromagnet in a uniform field (see Belanger, 1998 for a recent review on experimental results). Despite twenty-five years of active and continuous research the RFIM is not yet completely understood. The problem seems related to the presence of bound states in the ferromagnetic phase, which make the standard theoretical approaches not adequate to analyze the critical behaviour. Here we discuss the RFIM in the context of perturbative field theory. The chapter is organized as follows: in Section 2.1 we define the model and outline the main expectations for its qualitative behaviour. In Section 2.2 we introduce an effective replicated ϕ 4 field model where the disorder has been integrated out. Then we perform a perturbative analysis on this model (Section 2.3) and illustrate how the so-called dimensional reduction arises (Section 2.4). Finally, in Section 2.5 we introduce some generalized couplings which need to be taken into account to properly describe the system; we perform a perturbative Renormalization Group (RG) close to the upper critical dimension (Sections 2.6 and 2.7) and discuss the occurrence of a vitrous transition (Section 2.8).
The random field Ising model (RFIM) has been investigated extensively both theoretically and experimentally in recent years [1–4]. Some questions, such as the lower critical dimension and the existence of a static phase transition, have already been solved from the theoretical point of view, but other questions, such as the existence of the tricritical point, are still open. However, it is now understood that the RFIM describes the essential physics of a rich class of experimentally accessible disordered systems. These include structural phase transitions in random alloys, commensurate charge-density-wave systems with impurity pinning, binary fluid mixtures in random porous media, and the melting of intercalates in layered compounds such as TiS2. The behavior of the RFIM is governed by the competing tendencies of the spins to align ferromagnetically under impetus of the exchange or to follow the local fields and thus be uncorrelated. The RFIM has been studied using different methods. One of the authors [5] of the present paper has developed a new version of effective field theory, which extends the approximation methods [6, 7] to random Ising systems, in which randomness can be described by discrete random variables and the geometry of the lattice can be easily taken into account. This method is based on the determination of the probability distribution of the discrete variables, which takes into account exactly the self-site correlations. This approach has been used to study cooperative phenomena and phase transitions in various systems, and gives useful qualitative and quantitative insights for the critical behavior of a wide variety of classical and quantum lattice spin systems [8–10]. In recent decades, the renormalization group theory of phase transitions and critical phenomena represents one of the most powerful tools of theoretical physics. Therefore, employing this theory (which can be applied for weak fields), the proof
The random-field Ising model (RFIM) is one of the simplest statistical-mechanical models that captures the anomalous irreversible collective response seen in a wide range of physical, biological, or socioeconomic situations in the presence of interactions and intrinsic heterogeneity or disorder. When slowly driven at zero temperature, it can display an out-of-equilibrium phase transition associated with critical scaling (``crackling noise''), while it undergoes at equilibrium, under either temperature or disorder-strength changes, a thermodynamic phase transition. We show that the out-of-equilibrium and equilibrium critical behaviors are in the same universality class: they are controlled, in the renormalization-group (RG) sense, by the same zero-temperature fixed point. We do so by combining a field-theoretical formalism that accounts for the multiple metastable states and the exact (functional) RG. As a spin-off, we also demonstrate that critical fluids in disordered porous media are in the same universality class as the RFIM, thereby unifying a broad spectrum of equilibrium and out-of-equilibrium phenomena.
We have studied the equilibrium properties of the random-field (H) ferromagnetic Ising model (RFIM) and the dilute antiferromagnetic Ising model (DAF) in a uniform field. For RFIM we have used the transfer-matrix technique, thus bypassing serious equilibration problems which would arise in a Monte Carlo calculation for random systems. We have calculated the structure factor S versus H. Our results are consistent with S\ensuremath{\sim}exp(c/H2) as predicted by theories which yield a lower critical dimension dc=2, whereas the expected behavior for dc=3, S\ensuremath{\sim}${H}^{\mathrm{\ensuremath{-}}4}$ is inconsistent with our results. We have carried out analogous calculations for DAF. Our results for DAF suffer seriously from finite-size effects. To circumvent this difficulty, we have also performed calculations on interface models. Together, these results indicate that DAF crosses over to RFIM behavior at H\ensuremath{\simeq}J.
Abstract Perturbation theory for the random-field Ising model (RFIM) has the infamous attribute that it predicts at all orders a dimensional-reduction property for the critical behavior that turns out to be wrong in low dimensions. Guided by our previous work based on the nonperturbative functional renormalization group (NP-FRG), we show that one can still make some use of the perturbation theory for a finite range of dimensions below the upper critical dimension, d = 6. The new twist is to account for the influence of large-scale zero-temperature events known as avalanches. These avalanches induce nonanalyticities in the field dependence of the correlation functions and renormalized vertices, and we compute in a loop expansion the eigenvalue associated with the corresponding anomalous operator. The outcome confirms the NP-FRG prediction that the dimensional-reduction fixed point correctly describes the dominant critical scaling of the RFIM above some dimension close to 5 but not below.
"Renorma1ization-group methods have been applied in the study of quenched random magnetic systems in recent years. We begin with a brief review of second-order phase transitions in pure, homogeneous systems and also of the .. renorma1ization group framework. Then we provide an introduction to quenched random magnetic systems. Next, momentum-space methods and position-space techniques as applied to quenched random magnets are outlined and compared. Grinstein and Luther applied the Wilson-Fisher E-expansion to random n-vector models; Khme1'nitsky discovered that the random Ising model (n = 1) possessed a ""random"" fixed point of 0(squareroot(e)). This fixed point was found to have one marginal and one irrelevant operator. We have investigated the stability of this fixed point using Ca11an-Symanzik equations and renorma1ized perturbation theory. We find the fixed point stable in the next order; we have also obtained critical exponents to one higher order. Next, position-space techniques are used to study some simple model systems. In addition to critical exponents, global thermodynamic properties are determined. These calculations are based on the Migda1-Kadanoff approximate recursion relations suitably generalized to the inhomogeneous case. Firstly we study the randomly bond-dilute two-dimensional nearest- neighbor Ising model on a square lattice. Calculations give both thermal and magnetic exponents associated with the percolative fixed point. Differential recursion relations yield a phase diagram which is in quantitative agreement with all known results. Curves for the specific heat, percolation probability, and magnetization are displayed. The critical region of the specific heat becomes unobservably narrow well above the percolation threshold Pc. This provides a possible explanation for the apparent specific-heat rounding in certain experiments. We then study the Edwards-Anderson model of a spin glass. The current theoretical situation, which is far from satisfactory at present, is briefly reviewed. We treat the spin-l/2 Ising model with independently random nearestneighbor interactions in dimensionalities d = 2, 3, and 4. The phase diagram, which is in qualitative agreement with mean-field results, exhibits paramagnetic, ferromagnetic, antiferromagnetic, and spin-glass phases. The spinglass and paramagnetic phases meet along an extended second-order phase boundary, which terminates in two tricritical points. Critical and tricritical exponents are calculated. The spin-glass specific-heat exponent turns out to be large and negative, compatibly with recent experiments which show a rounded specific heat anomaly. Global specific-heat curves are also displayed for d = 2."
PACS. 05.10.Cc – Renormalization group methods. PACS. 64.60.-i – General studies of phase transitions. PACS. 75.10.Hk – Classical spin models. Abstract. – A study of the influence of different types of disorder on systems in the Ising universality class by employing both a dynamical field theory approach and extensive Monte Carlo simulations is presented. We reproduce some well-known results for the case of quenched disorder (random temperature and random field), and analyze the effect of four different types of time-dependent disorder scarcely studied so far in the literature. Some of them are of obvious experimental and theoretical relevance (as, for example, globally fluctuating temperatures or random fields). All the predictions coming from our field-theoretical analysis are fully confirmed by extensive simulations in two and three dimensions, and novel qualitatively different, non-Ising transitions are reported. Possible experimental setups designed to explore the described phenomenologies are also briefly discussed. The Ising model has played a central role in the development of modern statistical mechanics. Its universality class includes a broad variety of models and real materials, as magnetic