Random‐field ising systems: A survey of current theoretical views

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Abstract
Ising or Ising-like models in random fields are good representations of a large number of impure materials. The main attempts of theoretical treatments of these models--as far as they are relevant from an experimental point of view--are reviewed. A domain argument invented by Imry and Ma shows that the long-range order is not destroyed by weak random-fields in more than D = 2 dimensions. This result is supported by considerations of the roughening of an isolated domain wall in such systems: domain walls turn out to be well defined objects for D > 2, but arbitrarily convoluted for D < 2. Different approaches for calculating the roughness exponent ζ yield ζ= (5 - D)/3 in random-field systems. The application of ζ in incommensurate-commensurate critical behaviour is discussed. Non-classical critical behaviour occurs in random-field systems below D = 6 dimensions which is determined in general by three independent exponents. The new exponent yJ = θ= D/2 - σ corresponds to random-field renormalization or, to say it differently, to the irrelevance of the temperature at the zero-temperature fixed point, which is believed to rule the critical behaviour. The inequalities satisfied by these exponents are investigated, as well as the relations between the eigenvalue and the critical exponents and their numerical values found in the literature. Domain wail roughening due to random fields produces also metastability and hysteresis. It is shown that when cooling a 3D system into the low-temperature phase in an applied random field, the system runs into a metastable domain state, in agreement with the experimental observation. The further approach of the system to the ordered equilibrium state is hindered by pinning which leads to domain size increasing only logarithmically in time. Domain wall roughness and pinning in random bond systems is also considered.

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