On the lower critical dimension of spin systems in random fields

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.