Markovian Approximations to the Time Evolution of Lead Coupled Spinless Hubbard Chains through Different Master Equation Approaches

Markovian Approximations to the Time Evolution of Lead Coupled Spinless Hubbard Chains through Different Master Equation Approaches

Hordijk, W.

Thijssen, J.M. (mentor)

Applied Sciences

Quantum Nanoscience

Theoretical Physics Research Group

2011-12-20

This thesis describes the process of deriving the Master Equation for the time evolution of the density operator for a lead coupled spinless Hubbard chain using a Markov approximation. The derivation is done in a direct, conventional way of expanding operators and switching pictures in order to separate the system Hamiltonian in a convenient way and through the Super Operator formalism, yielding a different form of the Master Equation. We will implement both results in software and we will iteratively find the stationary state of the chain from some initial state when the lead is kept at a constant chemical potential. We will show that for each initial state the chain density operator converges to this stationary state that is determined by the chemical potential of the lead, but at a rate that is independent of the chemical potential of the lead and that both the conventional and the Super Operator approach yield the exact same results. We will also show and discuss a small systematic error we find in the converged results yielding non-physical interpretations and how we can improve the numerics regarding the implementation of the model.

Markovian
Lead Coupled
Spinless Hubbard Chain
Master Equation

http://resolver.tudelft.nl/uuid:b4960dd4-788f-47df-98b0-c8ab804e304e

2011-12-20

Student theses

bachelor thesis

(c) 2011 Hordijk, W.