Supersymmetry for disordered systems with interaction

Abstract

Considering disordered electron systems we suggest a scheme that allows us to include an electronelectron interaction into a supermatrix σ-model. The method is based on replacing the initial model of interacting electrons by a fully supersymmetric model. Although this replacement is not exact, it is a good approximation for a weak short range interaction and arbitrary disorder. The replacement makes the averaging over disorder and further manipulations straightforward and we come to a supermatrix σ-model containing an interaction term. The structure of the model is rather similar to the replica one, although the interaction term has a different form. We study the model making perturbation theory and renormalization group calculations. We check the renormalizability of the model in the first loop approximation and in the first order in the interaction. In this limit we reproduce the renormalization group equations known from earlier works. We hope that the new supermatrix σ-model may become a new tool for nonperturbative calculations for disordered systems with interaction.

Nonanalytic magnetic response of Fermi and non-Fermi liquids

“…(1) by its exact value. The same result was obtained within the supersymmetric theory of the Fermi liquid [10], and in the analysis of the scattering amplitude in the Cooper chanel, Γ C s (θ) [11]. Since an extension of the second-order result for δχ s hints at far-…”

“…Since Γ s (π) is equivalent to the spin component of the particle-particle scattering amplitude Γ C s (π) (Refs. [3,11]), the repulsive interaction in the Cooper channel leads to a logarithmic reduction of Γ s (π) [10,11,13]. For T = 0, H → 0, Γ s (π) ∝ 1/ log |H| and therefore δχ BS s ∝ |H| / log 2 |H| [14].…”

Keldysh approach to the renormalization group analysis of the disordered electron liquid

“…Its application to interacting systems, however, is a formidable challenge, and progress in this direction is so far limited. 46 ] In order to lighten notations, starting from Sec. IV C we will leave out the hats symbolizing matrices in Keldysh space.…”

Are Bosonic Replicas Faulty?

“…In particular, the exact Keldysh [1] and approximate supersymmetry [2] techniques have been conceived to offer a nonperturbative alternative to the notoriously known replica field theories [3,4,5] whose legitimacy has been questioned [6] for more than two decades. Sadly, the newly proposed field theoretic approaches [1,2] have not yet evolved into efficient calculational tools, and their success [7] has been very limited.…”

“…Upon assigning Z the meaning of a quantum partition function Z(ς) = p α=1 det(ς α − H) of a system characterised by a stochastic Hamiltonian H, the identity (1) can be utilised to represent the average p-point Green function G(ς) = p α=1 Tr (ς α − H) −1 in terms of the average characteristic polynomials (2) to be referred to as the replica partition function. Notice that Eq.…”