Transport around criticalities: Localization-delocalization and paramagnetic-ferromagnetic transitions

Abstract

We study two examples of transport properties of electron systems in the vicinity of a critical point. The ®rst example, a one-body problem, is the disorder-driven metal±insulator transition in the integer quantum Hall system. We have numerically examined the multifractality of the critical wavefunction, which is relevant to an anomalous diOE usion of electrons. The second example, a many-body problem, is the paramagnetic±ferromagnetic transition in a strongly correlated clean electron system. We show that the proximity to ferromagnetism can give rise to a negative magnetoresistance. These exemplify how a criticality manifests itself in transport properties.} 1. Introduction Central to the understanding of electron systems are critical phenomena which occur in various manners, in both one-body problems and many-body problems. It is then an intriguing question to ask how electronic transport properties can re¯ect the criticality when the system is at or around a critical point.In the one-body picture an example is the localization±delocalization transition in disordered systems. We can start by asking ourselves whether this transition may be regarded as a phase transition. The present author (Aoki 1983(Aoki , 1986 suggested that wavefunctions at the Anderson transition should be self-similar (fractal) just as the critical point in a phase transition is characterized by a diverging correlation length with scale-invariant¯uctuations. To be more precise, the self-similarity in critical wavefunctions extends beyond a single scale transformation , so the idea has been subsequently developed into the multifractal analysis (Castellani and Peliti 1986, Schreiber and Grassback 1991, Avishai et al. 1995.The criticality of the wavefunction proposed in general has been analysed for the quantum Hall system in particular (Aoki 1983(Aoki , 1986. In discussing the integer quantum Hall eOE ect, Aoki and Ando (1981) have pointed out, ®rstly, that localization±delocalization transition has to exist to have the integer quantum Hall eOE ect and, secondly, that we can prove that the transition does indeed exist in the limit of strong magnetic ®eld. In the language of the scaling theory of localization, the delocalized states are marginally allowed to appear in two dimensions …dˆ2 † when a magnetic ®eld exists. Later it was revealed that the marginal situation is realized as the delocalized region coalescing into a single point (the centre E 0 ) of each Landau level on the energy axis, where the localization length (proportional to