Flow Equations for Hamiltonians from Continuous Unitary Transformations

TL;DR

This thesis reviews the mathematical foundations and diverse applications of flow equations for Hamiltonians, including transformations, electron-phonon interactions, and models near phase transitions, highlighting recent methodological advances.

Contribution

It provides a comprehensive overview of flow equations, connecting mathematical frameworks with practical applications and recent developments like similarity renormalization group methods.

Findings

Flow equations can effectively diagonalize Hamiltonians.

Flow-dependent expectation values improve robustness near phase transitions.

Similarity renormalization group methods relate closely to Wegner's flow equations.

Abstract

This thesis presents an overview of the flow equations recently introduced by Wegner. The little known mathematical framework of the flow in the manifold of unitarily equivalent matrices, as discovered in the mathematical literature before Wegner's paper, is established in the initial chapter and used as a background for the entire presentation. The application of flow equations to the Foldy-Wouthuysen transformation and to the elimination of the electron-phonon coupling in a solid is reviewed. Recent flow equations approaches to the Lipkin model are examined thoroughly, paying special attention to their utility near the phase change boundary. We present more robust schemes by requiring that expectation values be flow dependent; either through a variational or self-consistent calculation. The similarity renormalization group equations recently developed by Glazek and Wilson are also reviewed. Their relationship to Wegner's flow equations is investigated through the aid of an instructive model.