Disorder and interaction in low-dimensional electronic systems

Abstract

All electrons are identical and subject to the same forces, yet the materials they make up possess a dazzling diversity of physical properties. The interplay of electron-electron interaction and random disorder is largely responsible for this emergent complexity. This thesis presents some examples of non-trivial behavior arising from simple theoretical models of electronic systems in low dimensions.

The first part of this thesis describes modifications to two well-studied models of disorder in one-dimensional non-interacting systems. In the “Dyson’’ model, electrons on a chain hop randomly between nearest neighbors in the absence of an on-site potential. It is shown that a choice of singular hopping distributions induces continuously tunable non-universal behavior in the localization length and density of states at the band center. The next topic is the Anderson model, in which a random potential causes complete breakdown of transport. Adding an energy penalty for the propagation of electrons along one direction truncates the Hilbert space; this leads to the possibility of multifractal behavior, and a mobility edge between localized and extended states.

The second part of this thesis focuses on aspects of the quantum Hall effect of a two-dimensional interacting electron gas in a high magnetic field, observed in semiconductor heterostructures, graphene and layered compounds. Here the single particle spectrum resolves into dispersionless and highly degenerate Landau levels (LLs). We study the response of the many-body ground state to band mass anisotropy in the lowest LL and find it to be a nontrivial (and non-analytic) function of interaction type, filling fraction and particle statistics. Finally, this thesis reports on the influence of disorder on nearly flat subbands of the lowest LL, in both the interacting and non-interacting settings. These subbands are obtained by adding a carefully tuned periodic potential. Topologically trivial subbands, with Chern number zero, may host a many-body localized (MBL) phase at large disorder. However, MBL is absent in topologically nontrivial subbands, with nonzero Chern number. This approach enables one to disentangle the roles of topology and dimensionality in destabilizing MBL in the quantum Hall system.