Princeton University

School of Engineering & Applied Science

Computational Studies of Model Disordered and Strongly Correlated Electronic System

Sonika Johri
Engineering Quadrangle B327
Friday, November 14, 2014 - 3:30pm to 5:00pm

In this thesis, I address some salient problems in disordered and correlated electronic systems using modern numerical techniques like sparse matrix diagonalization, density matrix renormalization group (DMRG), and large disorder renormalization group (LDRG) methods.
The pioneering work of P. W. Anderson, in 1958, led to an understanding of how an electron can stop diffusing and become localized in a region of space when a crystal is sufficiently disordered. Our work has uncovered a new singularity in the disorder-averaged inverse participation ratio of wavefunctions within the localized phase, arising from resonant states. The effects of system size, dimension and disorder distribution on the singularity have been studied. A novel wavefunction-based LDRG technique has been designed for the Anderson model which captures the singular behaviour.
While localization is well established for a single electron in a disordered potential, the situation is less clear in the case of many interacting particles. Most studies of a many-body localized phase are restricted to a system which is isolated from its environment. Such a condition cannot be achieved perfectly in experiments. A chapter of this thesis is devoted to studying signatures of incomplete localization in a disordered system with interacting particles which is coupled to a bath.
Strongly interacting particles can also give rise to topological phases of matter that have exotic emergent properties, such as quasiparticles with fractional charges and anyonic, or perhaps even non-Abelian statistics. These particles (e.g. Majorana fermions) may be the building blocks of future quantum computers. The third part of my thesis focuses on the best experimentally known realizations of such systems - the fractional quantum Hall effect (FQHE) which occurs in two-dimensional electron gases in a strong perpendicular magnetic field. I have developed software for exact diagonalization of the many-body FQHE problem on the surface of a cylinder, a geometry which turns out to be optimal for the DMRG algorithm. Using this new geometry, I have studied properties of various fractionally-filled states such as their edge excitations, entanglement spectra and quasiparticle sizes. I have also designed numerical probes of the recently discovered geometric degree of freedom of FQHE states.