On the lower critical dimension of spin systems in random fields

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Abstract
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.