The problem of the lower critical dimension (LCD) of spin systems in random fields is discussed. The different theoretical predictions for the Ising case-i.e. LCD=2 from considerations of domain wall roughening but LCD=3 from diagrammatic renormalisation-are related to different results for the spectral function Delta (q) of the dressed static-field fluctuations-i.e. LCD=3 is obtained if Delta (q) diverges as xi eta with increasing thermal correlation length xi , while LCD=2 corresponds to a non-divergent Delta (q). This difference is analogous to the difference between the true value of the dynamical critical exponent z of a pure ferromagnetic spin system and the 'conventional approximation' z=2- eta . In this way, the weak point of the arguments leading to LCD=2 is discovered, namely the neglect of certain 'mode couplings' between fluctuations with different q or, more pictorially, the neglect of 'domains within domains', and thus the result LCD=2 is seen to be no longer credible. The domain argument leading to LCD=2 are modified by replacing the bare fields by 'dressed' ones; in this way, they are reconciled with the field theoretic result LCD=3.
Spin-models in random fields (RFs) are good representations of many impure materials. Their macroscopic collective behaviour is dominated by the fluctuations in the random fields which accumulate on large scales even if the local field is arbitrarily small. This feature is shared by other weakly disordered models, like flux lines or domain walls in random media. We review some of the main theoretical attempts to describe such systems. A modification of Harris’ argument demonstrates that at the critical point the RF disorder is relevant and that (hyper)scaling must be changed. A domain argument invented by Imry and Ma shows that long-range order is not destroyed by weak RFs in more than d=2 dimensions. This result is supported both by a more refined treatment of the domain argument and by considering the roughness of an isolated domain wall due to the randomness. The wall (or flux line) becomes rough due to disorder but if d>2 the wall remains a well-defined object in RF systems. Different approaches are used to calculate the roughness exponent ζ for walls and lines. Some applications of ζ for the description of type-II superconductors and incommensurate systems are given. More detailed calculations are possible for one-dimensional, Bethe-lattice or the hierarchical Dyson model systems, which confirm as a rule the more approximate treatment of the other sections. In one dimension there is an interesting relation between the statistical mechanics of these models and nonlinear dynamics. Non-classical critical behaviour occurs in RF systems for d<6 and is determined in general by three independent exponents which fulfil certain inequalities. The new exponent θ≡yJ>0 is related to the violation of conventional hyperscaling and is determined by the energy ~H0ξ0 of a correlated region of size ξ. In a renormalization group treatment, the temperature T turns out to be a (dangerous) irrelevant variable which is the most prominent property of the systems considered in this review. The irrelevance of thermal fluctuations on large scales produces metastability and hysteresis effects both in the transition region and in the ordered phase, only briefly considered here. These features occur also in other systems with a disordered T=0 fixed point like in the ordered phase of a spin-glass.
A review is given on some recent developments in the theory of the Ising model in a random field. This model is a good representation of a large number of impure materials. After a short repetition of earlier arguments, which prove the absence of ferromagnetic order in
d≤2 space dimensions for uncorrelated random fields, we consider different random field correlations and in particular the generation of uncorrelated from anti-correlated random fields by thermal fluctuations. In discussing the phase transition, we consider the transition to be characterized by a divergent correlation length and compare the critical exponents obtained from various methods (real space RNG, Monte Carlo calculations, weighted mean field theory etc.). The ferromagnetic transition is believed to be preceded by a spin glass transition which manifests itself by replica symmetry breaking. In the discussion of dynamical properties, we concentrate mainly on the zero temperature depinning transition of a domain wall, which represents a critical point far from equilibrium with new scaling relations and critical exponents.
A variety of phenomena is observed experimentally in random-field (RF) systems realized by the application of an external field to diluted antiferromagnets. At low temperatures, infinitely long hysteretic effects are manifested by the history dependence of the final states: long-range order is observed if the field is applied after cooling, while domain states are reached when field cooled. While no indications for critical fluctuations are detected in 2-D systems, scaling behavior, for both the correlation length and the specific heat, is observed in 3-D systems over an intermediate range of temperatures. The related critical properties seem to be well described by the corresponding ones in the 2-D pure Ising model. The renormalization-group approach, which yields for the equilibrium critical exponents their values of the pure model in d−2 dimensions, is reviewed. A generalization of the dimensional-reduction approach, which accounts self-consistently for renormalized responses of the RF system, is presented. The dynamical effects are implicitly incorporated through the variation in the critical response between the local and the global regimes, associated with short- and long-time scales, respectively. In both regimes the lower critical dimension is found to be d=2 in accordance with stability arguments. The short-time critical behavior indicates a dimensional reduction by one for the 3-D thermal exponents, in agreement with the experimental results.
Analytic phenomenological scaling is carried out for the random field Ising model in general dimensions d using a bar geometry. Domain wall configurations and their decorated profiles and associated wandering and other exponents (ζ,γ,δ,μ) are obtained by free-energy minimization. Scaling between different bar widths provides the renormalization group (RG) transformation. Its consequences are i) criticality at h = T = 0 in d ⩽ 2 with correlation length ξ( h , T ) diverging like ξ( h ,0) ∝ h −2/(2 − d ) for d < 2 and ξ( h ,0) ∝ exp [1/( c 1 γ h γ )] for d = 2, where c 1 is a decoration constant; ii) criticality in d = 2 + ε dimensions at T = 0, h * = (ε/2 c 1 ) 1/γ , where ξ ∝ [( s − s *)/ s ] −2ε/γ , s ≡ h γ . Finite-temperature generalizations are outlined. Numerical transfer matrix calculations and results from a ground-state algorithm adapted for strips in d = 2 confirm the ingredients which provide the RG description.
The dynamic critical behavior of one-dimensional random spin systems in a field at zero temperature is investigated by a generalized transfer-matrix scaling technique. Three spin models are considered: Heisenberg in a longitudinal field, XY in a transverse field, and Ising in a transverse field, with either spin-glass or random-field disorder. Near the transition, i.e., for small values of the reduced field \ensuremath{\Delta}=(H-
Hc), where
Hc is the critical field, one finds that the dispersion relation for the low-frequency \ensuremath{\omega}, small-wave-vector k spin-wave excitations takes the scaling form \ensuremath{\omega}=
kzf(\ensuremath{\Delta}/
kcphi), and exact results are given for the dynamic exponent z and crossover exponent cphi. This form contains a crossover of the frequency \ensuremath{\omega} between two different asymptotic behaviors:
kz for \ensuremath{\Delta}\ensuremath{\ll}
kcphi and ${h}^{\ensuremath{\delta}}$ for \ensuremath{\Delta}\ensuremath{\gg}
kcphi, where the field exponent is \ensuremath{\delta}=z/cphi. In the case of the Heisenberg systems the spin-glass disorder gives rise to the nontrivial dynamic exponent z=(3/2, whereas in the random-field case it is the exponent associated to the field that becomes nontrivial, taking the value (4/3. For the transverse XY systems the random-field disorder implies nontrivial values for both the dynamic and field exponents, (3/2 and (3/4, respectively, whereas the spin-glass disorder does not affect the dynamics of the system, which behaves like a pure transverse XY ferromagnet. Finally, for the transverse Ising systems neither the random field nor the spin-glass disorder affects the dynamics, which is the same as for a pure transverse Ising ferromagnet.