# Scale-dependent theory of the disordered electron liquid

## TL;DR

This paper reviews the scaling theory of disordered interacting electrons, focusing on the metal-insulator transition and thermal transport, using the non-linear sigma model framework to connect microscopic theories with observable phenomena.

## Contribution

It introduces a comprehensive NLSM-based approach to analyze disordered electron liquids, including the effects of interactions on metal-insulator transitions and thermal conductivity.

## Key findings

- Accurate description of the metal-insulator transition in 2D and 3D.
- Explanation of non-monotonic resistance behavior in Si-MOSFETs.
- Thermal conductivity corrections due to disorder and interactions.

## Abstract

We review the scaling theory of disordered itinerant electrons with e-e interactions. We first show how to adjust the microscopic Fermi-liquid theory to the presence of disorder. Then we describe the non-linear sigma model (NLSM) with interactions (Finkel'stein's model). This model is closely connected to the Fermi liquid, but is more generally applicable, since it can also be viewed as a minimal effective functional describing disordered interacting electrons. Our discussion emphasizes the general structure of the theory, and the connection of the scaling parameters to conservation laws. We then move on to discuss the metal-insulator transition (MIT) in the disordered electron liquid in two and three dimensions. This MIT is a non-trivial example of a quantum phase transition. The NLSM approach allows to identify the dynamical exponent connecting the spatial and energy scales, which is central for the description of the kinetic and thermodynamic behavior in the critical region of the MIT in three dimensions. In two dimensions, the system can be discussed in terms of a flow in the disorder-interaction phase plane, which is controlled by a fixed point. We demonstrate that the two-parameter RG-equations accurately describe electrons in Si-MOSFETs including the observed non-monotonic behavior of the resistance and its strong drop at low temperatures. The theory can also be applied to systems with an attractive interaction in the Cooper channel, where it describes the suppression of superconductivity in disordered amorphous films. We extend our discussion to heat transport in the two-dimensional electron liquid. Similar to the electric conductivity, the thermal conductivity also acquires logarithmic corrections induced by electron interaction and disorder. The resulting thermal conductivity can be calculated in the NLSM formalism after introducing so-called gravitational potentials.