Renormalization, Conservation Laws and Transport in Correlated Electron Systems

TL;DR

This thesis explores the functional renormalization-group method to analyze symmetries, conservation laws, Friedel oscillations, and transport phenomena in correlated electron systems, especially Luttinger liquids with impurities.

Contribution

It demonstrates how the fRG framework respects Ward identities under gauge invariance and applies it to study impurity effects and conductance in Luttinger liquids.

Findings

Ward identities hold for gauge-invariant cutoff actions

Friedel oscillations follow expected power laws in large systems

Conductance exhibits universal power-law regimes and crossovers

Abstract

This thesis comprises two parts centered around the functional renormalization-group framework: in the first part, I study the role of symmetries and conservation laws in approximate solutions, while in the second part I analyze Friedel oscillations and transport in Luttinger liquids with impurities. The functional renormalization group (fRG) has been developed as a new computational tool in the theory of interacting Fermi systems. The effective behavior of a given microscopic model is calculated by solving coupled differential flow equations for the Green functions with an energy scale as the flow parameter. The symmetries of the microscopic model imply Ward identities between Green and response functions. It is shown that solutions of truncated flow-equation hierarchies satisfy Ward identities if the cutoff bare action is gauge invariant. However, truncations are generally not self-consistent approximations in the sense of Baym and Kadanoff. The fRG is then applied to study Luttinger liquids. By computing the full spatial effective potential of a single impurity, long-range Friedel oscillations are observed in the density profile with the expected power laws for systems with up to 10^7 lattice sites. For a double barrier enclosing a dot region we find temperature regimes in which the conductance follows power laws with universal exponents, as well as non-universal crossover regimes in intermediate parameter regions.