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-T)/
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.
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 .
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.
The randomly bond-dilute two-dimensional nearest-neighbor Ising model on the square lattice is studied by renormalization-group methods based on the Migdal-Kadanoff approximate recursion relations. 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.
Critical point scaling in a field
H applies for the limits $t\ensuremath{\rightarrow}0$ (where $t=T/{T}_{c}\ensuremath{-}1)$ and $H\ensuremath{\rightarrow}0$ but with the ratio $R=t/{H}^{2/\ensuremath{\Delta}}$ finite. $\ensuremath{\Delta}$ is a critical exponent of the zero-field transition. We study the replicon correlation length $\ensuremath{\xi}$ and from it the crossover scaling function
f(R) defined via $1/(\ensuremath{\xi}{H}^{4/(d+2\ensuremath{-}\ensuremath{\eta})})\ensuremath{\sim}f(R)$. We have calculated analytically
f(R) for the mean-field limit of the Sherrington-Kirkpatrick model. In dimension
d=3, we have determined the exponents and the critical scaling function
f(R) within two versions of the Migdal-Kadanoff (MK) renormalization group procedure. One of the MK versions gives results for
f(R) in
d=3 in reasonable agreement with those of the Monte Carlo simulations at the values of
R for which they can be compared. If there were a de Almeida-Thouless (AT) line for $d\ensuremath{\le}6$, it would appear as a zero of the function
f(R) at some negative value of
R, but there is no evidence for such behavior. This is consistent with the arguments that there should be no AT line for $d\ensuremath{\le}6$, which we review.