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Journal of Applied Physics

Equimagnetization lines in the hybrid-order phase diagram of the d=3 random-field Ising model (invited)

Susan R. McKay, A. Nihat Berker

Published Nov 15, 1988• 31 citations
Publisher

Abstract

The three-dimensional random-field Ising model has been approximately solved by pursuing the global renormalization-group trajectories of the full, coupled probability distributions of local fields and bonds. This study, which in effect is the global analysis of the renormalization-group flows of 220 physical quantities (based on a generalization of the Migdal–Kadanoff approximation), yields several results that were previously unsuspected. The boundary between the ferromagnetic and paramagnetic phases is hybrid order, in the sense that the magnetization is discontinuous (as in a first-order transition) and the specific heat has a power-law singularity (as in a second-order transition). The magnetic properties of the d=3 random-field Ising model are presented in the form of equimagnetization lines. The study has been repeated for a variety of dimensions d. The lower critical dimension dl=2 is correctly obtained. Moreover, this study indicates the existence of an intermediate-critical dimension dt (between the lower- and upper-critical dimensions) where a singularity occurs in the critical properties as a function of dimension.

DOI
10.1063/1.342230

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The three-dimensional random-field Ising model has been approximately solved by pursuing the global renormalization-group trajectories of the full, coupled probability distributions of local fields and bonds. This study, which in effect is the global analysis of the renormalization-group flows of 220 physical quantities (based on a generalization of the Migdal–Kadanoff approximation), yields several results that were previously unsuspected. The boundary between the ferromagnetic and paramagnetic phases is hybrid order, in the sense that the magnetization is discontinuous (as in a first-order transition) and the specific heat has a power-law singularity (as in a second-order transition). The magnetic properties of the d=3 random-field Ising model are presented in the form of equimagnetization lines. The study has been repeated for a variety of dimensions d. The lower critical dimension dl=2 is correctly obtained. Moreover, this study indicates the existence of an intermediate-critical dimension dt (between the lower- and upper-critical dimensions) where a singularity occurs in the critical properties as a function of dimension.
A highly detailed renormalization-group study is applied to the equilibrium statistical mechanics of the random-field Ising model in three dimensions, with analysis of the renormalization-group flows of over 8 500 000 quantities that determine the quenched probability distribution. We determine the strong-coupling fixed distribution that controls the finite-temperature phase boundary. This boundary is second order but has a magnetization behavior that is phenomenally close to a discontinuity, with the exponent \ensuremath{\beta}=0.0200\ifmmode\pm\else\textpm\fi{}0.0005. Magnetization curves are obtained, demonstrating that, for example, the calculated M changes from 0 to 0.407, almost half saturation, as (Tc{\mathit{T}}_{\mathit{c}}Tc​-T)/Tc{\mathit{T}}_{\mathit{c}}Tc​ is varied from 0 to 0.2947\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}16}$. However, a rigorous argument is given showing that the magnetization M cannot be truly discontinuous at phase transitions controlled by a strong-coupling random-field fixed distribution, in renormalization-group studies that employ the Migdal-Kadanoff local recursion relations.
An Ising model including field randomness and $\ifmmode\pm\else\textpm\fi{}J$ bond randomness is studied by renormalization-group theory in spatial dimensions d=2d=2d=2 and 3. In d=3,d=3,d=3, with field randomness but no $\ifmmode\pm\else\textpm\fi{}J$ randomness, ferromagnetic and paramagnetic phases occur with no intervening spin-glass phase. Also in d=3,d=3,d=3, at sufficient $\ifmmode\pm\else\textpm\fi{}J$ randomness, a spin-glass phase occurs, but is replaced by the paramagnetic phase for any nonzero random field, also implying its disappearance for any nonzero uniform field. In d=2,d=2,d=2, no ferromagnetic phase under random fields and no spin-glass phase occur. Global phase diagrams uniting the random-field and spin-glass problems are evaluated. The strong violation of critical phenomena universality, previously found adjoining random-bond tricriticality in d=3,d=3,d=3, is seen here in d=2.d=2.d=2.
We present an analytic real-space renormalization group calculation for the random-field Ising model. We apply the Migdal-Kadanoff approximation for the renormalization of a cubic cell in dimensions d , introducing a new field partitioning scheme which allows us to treat the random-field fluctuations in a coherent manner. Our scheme leads naturally to a lower critical dimensionality and allows us to calculate a complete set of three independent exponents in arbitrary dimension. In three dimensions the magnetization exponent and the Schwartz-Soffer inequality is almost satisfied as an equality. We expand analytically in . Further, we show that and the magnitude of the inequality go to zero exponentially with . We calculate the crossover exponent, from pure to the random-field system and find surprisingly good agreement with experimental values. We find that satisfies the Schwartz-Soffer inequality: , the susceptibility exponent of the pure system. We expand in and find that the magnitude of the inequality varies exponentially in . Finally we find that dimensional reduction is satisfied to first order in , with the reduced dimension .
"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."
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