Non-perturbative flow equations from continuous unitary transformations

TL;DR

This thesis develops a non-perturbative method for solving Wegner's flow equations by transforming them into a nonlinear PDE, enabling accurate analysis of models like Lipkin and Dicke across all coupling regimes.

Contribution

It introduces a scalar function parameterization of the Hamiltonian flow, allowing a non-perturbative PDE approach applicable to complex quantum models.

Findings

Accurate spectra and eigenstates for the Lipkin model across all couplings.

Identification of new phase structure features via non-perturbative analysis.

Effective Hamiltonian construction demonstrated with the Dicke model.

Abstract

The goal of this thesis is the development and implementation of a non-perturbative solution method for Wegner's flow equations. We show that a parameterization of the flowing Hamiltonian in terms of a scalar function allows the flow equation to be rewritten as a nonlinear partial differential equation. The implementation is non-perturbative in that the derivation of the PDE is based on an expansion controlled by the size of the system rather than the coupling constant. We apply this method to the Lipkin model and obtain very accurate results for the spectrum, expectation values and eigenstates for all values of the coupling and in the thermodynamic limit. New aspects of the phase structure, made apparent by this non-perturbative treatment, are also investigated. The Dicke model is treated using a two-step diagonalization procedure which illustrates how an effective Hamiltonian may be constructed and subsequently solved within this framework.