Order, disorder and phase transitions in quantum many body systems

TL;DR

This paper reviews key results in quantum many body theory, focusing on the universality of transport properties like quantum Hall conductivity and the interplay between mathematical and condensed matter physics.

Contribution

It highlights recent findings on the interaction-independence of quantum Hall conductivity and emphasizes the role of quantum conservation laws and correlation decay in this phenomenon.

Findings

Quantum Hall conductivity is independent of electron-electron interactions.

Quantum conservation laws influence transport properties.

Decay of correlation functions is crucial for universality.

Abstract

In this paper, I give an overview of some selected results in quantum many body theory, lying at the interface between mathematical quantum statistical mechanics and condensed matter theory. In particular, I discuss some recent results on the universality of transport coefficients in lattice models of interacting electrons, with specific focus on the independence of the quantum Hall conductivity from the electron-electron interaction. In this context, the exchange of ideas between mathematical and theoretical physics proved particularly fruitful, and helped in clarifying the role played by quantum conservation laws (Ward Identities), together with the decay properties of the Euclidean current-current correlation functions, on the interaction-independence of the conductivity.