טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentBen Zickel
SubjectRandom Non-Hermitian Matrix Models and Their Applications
to Disordered Systems
DepartmentDepartment of Physics
Supervisor Dr. Feinberg Joshua


Abstract

There has been recent interest in systems described by non-hermitian hamiltonians. Such hamiltonians are applicable in the study of nuclear decay, dissipative systems, neural networks and QCD at a finite chemical potential.

D. Nelson and co-workers used a non-hermitian hamiltonian to describe vortex flux pinning in superconductors. Hatano and Nelson realized that the latter model is applicable to studying the Anderson localization-delocalization transition in disordered media. In this context their non-hermitian hamiltonian is known as the Hatano-Nelson model. This model is the focus of our interest in this thesis.

This thesis is divided into two parts. The first analytical part shows the properties of the non-hermitian tight binding hamiltonian. Using the Hatano-Nelson model we find an analytic expression for the localization length for the one dimensional Llloyd model, in agreement with a previous result obtained by Thouless. For two and three dimensional systems the Green’s function is not analytic inside the eigenvalue blob and therefore analytic methods cannot be used in order to study the spectrum of the hamiltonian inside the eigenvalue blob. The hermitization method is introduced which reduces the analysis to the familiar hermitian eigenvalue problem. The analytical results obtained for the single impurity hamiltonian via this method show the behavior of the DoS inside the eigenvalue blob.

The second part presents numerical simulations of non-hermitian many impurities hamiltonians. The participation ratio and the DoS are calculated for different disorder parameters. Also the effects of finite lattice sizes are studied numerically for two and three dimensional lattices. A criterion for finding the size of the eigenvalue blob is introduced and verified numerically.