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Starting from the seminal papers of Wegner [1] and Efetov [2], a field-theoretical description based on the nonlinear σ model (NLσM) has become one of the main analytical approaches to various problems in disordered electronic systems. The ensemble averaging over all configurations of disorder is performed either using bosonic [1, 3] or fermionic [4] n-replicated fields and taking the n→0 limit in the results, or using supersymmetric fields [2, 5]. The main advantage of this approach lies in formulating the theory in terms of low lying excitations (diffusion modes) which greatly simplifies perturbative and renormalization group calculations, and also allows a non-perturbative treatment.

Nonlinear Sigma Model for Disordered Media: Replica Trick for Non-Perturbative Results and Interactions