Non-perturbative flow equations from continuous unitary transformations

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.

81 Pages. 43 Figures. Masters thesis. Supervisors : Prof F.G. Scholtz, Prof H.B. Geyer; added and updated references

References in corpus (10)

• Large N Scaling Behavior of the Lipkin-Meshkov-Glick Model
• Excitations in one-dimensional S=1/2 quantum antiferromagnets
• Flow equations for band--matrices
• Asymptotic Freedom and Bound States in Hamiltonian Dynamics
• Flow equation approach to the sine-Gordon model
• Hamiltonian Flow Equations for a Dirac Particle in an External Potential
• Hamiltonian Flow Equations for the Lipkin Model
• Flow equation renormalization of a spin-boson model with a structured bath
• Flow equations for Hamiltonians: Contrasting different approaches by using a numerically solvable model
• Adiabatic approximation from a renormalization group inspired method

Cited by in corpus (8)

• Continuous unitary transformations and finite-size scaling exponents in the Lipkin-Meshkov-Glick model
• Large N Scaling Behavior of the Lipkin-Meshkov-Glick Model
• Flow Equations and Normal Ordering. A Survey
• IMSRG with flowing 3 body operators, and approximations thereof
• On the Thermodynamic Limit of the Lipkin Model
• Flow equation approach to the linear response theory of superconductors
• Strong-coupling expansion and effective hamiltonians
• Eigenvalue distributions from a star product approach