The random-field Ising model is studied within a Migdal-Kadanoff renormalization-group scheme. For dimension d=3 the recursion relations and the zero-temperature fixed point are studied numerically. There is a continuous phase transition with magnetization exponent \ensuremath{\beta}=0.02. The magnetization as a function of temperature displays an abrupt crossover from pure Ising to random-field behavior. Analytic results are obtained within a d=2+\ensuremath{\varepsilon} expansion.

Physical review. B, Condensed matterAug 1, 199352 citations

Physics Letters A

The surprising effectiveness of the Migdal-Kadanoff renormalization scheme

Robert H. Swendsen, Royce K. P. Zia

Published Jan 1, 1979• 11 citations

Publisher

DOI: 10.1016/0375-9601(79)90382-7

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The critical temperature, thermal and magnetic exponents, and the graphs of

\frac{d{T}_{\mathrm{eff}}}{\mathrm{dl}}

versus

{T}_{\mathrm{eff}}

(where

{T}_{\mathrm{eff}}

denotes "effective" temperature of the system and

l

the renormalization-group parameter) for the

d=3

n=2, 3

models are obtained using the Migdal-Kadanoff scheme. Results of finite-lattice approximations for the

d=2

n=2

model are also presented.