Exact model for extremely correlated electrons in a magnetic field

NONTECHNICAL SUMMARY

The study of interacting electrons is crucial to understanding of the properties of metals, semiconductors, and superconductors, which form the foundation for today's technology. This award supports theoretical and computational research and education focused on interacting electrons confined to two-dimensions, 'flatland,' and with phenomena that occur when they are cooled to very low temperatures and exposed to a high magnetic field. This system of elections which can be realized at the interface of two semiconductors is known to produce very unusual quantum mechanical phases of matter, most notably those for which a component of resistance, known as the Hall resistance, to the flow of electrical current is a multiple of a quantum of resistance that is independent of the material and depends only on fundamental constants like the charge of an electron. That this macroscopic resistance becomes quantized was unexpected; this phenomenon is called the fractional quantum Hall effect. It has served as the springboard for the advent of new phenomena that occur in two-dimensional quantum materials, such as semiconductor structures known as quantum wells, single and multi-layer graphene, and a class of materials known as transition metal dichalcogenides that may be an alternative to graphene for some technical applications. An additional aspect of these systems is the emergence of new and exotic types of particles, which are not found elsewhere in nature, and which have generated innovative ideas for future quantum device technologies.

To gain further insight into the fractional Hall effect, the PI will investigate the consequences of a model interaction that is exactly solvable and exhibits several aspects of the observed phenomenology of the fractional quantum Hall effect. A crucial aspect in which this model deviates from the previous models is that the interaction is taken to be infinitely strong compared to the kinetic energy. This project involves the investigation of exactly solvable models to provide useful insights into existing concepts and to reveal fruitful new directions of inquiry to advance understanding of the fascinating physic of electrons confined to two dimensions. Aspects of the fractional quantum Hall effects that will be investigated include proposed new particles that are fundamentally different from the electron or photon - the quantum of light, and proposed particles that behave as if they were a fraction of an electron. The PI also plans to explore generalizations to other physical systems beyond electrons in 'flatland.'

The students supported by this project will be trained in advanced numerical techniques, analytical field theoretical methods, topological concepts, and high-performance computing. They will also be exposed to international collaboration. Participation in various education and outreach activities will be an integral part of their training because it not only promotes public understanding of the STEM fields but also is of utmost importance for the professional growth of the graduate students and will help build essential skills that will serve them well in their future careers.

TECHNICAL SUMMARY

This award supports theoretical and computational research and education to investigate new directions in the study of the fractional quantum Hall effect. This effect has given rise to new concepts, such as particles with fractional charge and fractional statistics, composite fermions, topological superconductivity, Majorana fermions, and chiral Luttinger liquids. Some of these have found homes in other platforms and have driven ideas related to topological quantum computation. Exact parent Hamiltonians have been constructed for certain fractional quantum Hall wave functions, the earliest example being Haldane's model that obtained the Laughlin wave function as the exact ground state. These models assume that the cyclotron energy is infinitely large compared to the interaction energy, so that electrons are confined to the lowest Landau level.

The primary objective of this project will be to investigate several directions opened by a recently developed model that considers an interaction that is infinitely strong compared to the cyclotron energy and is solvable for all eigenstates and eigen-energies. Various topological features will be explicitly evaluated from the exact wave functions, which have a rather complex form. In particular, the entanglement spectrum and braid statistics of the quasiparticles will be evaluated. The wave functions will be extended to the spherical and torus geometries, which should bring out their topological character. The physics of pairing instability and of non-Abelian statistics will be addressed within this model, which will also be generalized to fractional quantum Hall effects of bosons, spinful electrons and bilayer systems, and to include structures inspired by the parton construction. An important goal will be to ascertain what aspects of the solution relate to the laboratory experiments and whether hitherto unknown structures emerge.

The students supported by this project will employ advanced analytical as well as computational techniques, such as Chern-Simons field theory, exact diagonalization studies, and quantum Monte Carlo method. They will also organize education and outreach activities for pre-college students at appropriate summer camps at Penn State as well as at the annual Arts Festival at State College.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.