DMRG for the Quantum Hall Effect and Application in an Exact Model

Abstract

The fractional quantum Hall effect (FQHE) forms an extreme case of interacting quantum systems, wherein the kinetic energy of the system is “frozen”. In a chiral constrained space of constant kinetic energy, the states are highly degenerate, and the inter-particle interactions become infinitely stronger than the kinetic energy. An important numerical tool in the study of such strongly interacting systems, particularly 1D lattice problems, is the density matrix renormalization group (DMRG). The FQH problem—particularly in the cylindrical or torus geometries—can be mapped to a 1D lattice problem, making the system amenable to DMRG calculations. However, this quasi-one-dimensional problem has an effective interaction that has a longer range than that typical of lattice problems studied using DMRG, making this a challenging problem. The principal achievement of this work is the realization of a working DMRG algorithm tailored specifically for FQH calculations. In this thesis, we present a thorough discussion of the algorithm and its implementation, before presenting the results of our calculations for the V_1 interaction and the Coulomb interaction at ν = 1/3 and for a model interaction at ν = 2/5. The current algorithm and its implementation can produce reliable and reproducible results in systems where gaps in the spectra are robust. We identify crucial directions for improvement on the current setup that will be key to studying more challenging physical systems such as the gapless boundaries occurring at the interfaces of different topological orders.