Gapped ground state phases of quantum lattice systems

The next generation of quantum technologies is poised to exploit new quantum phases of matter with properties quite unlike the well-known quantum materials exhibiting superfluidity and superconductivity. Good mathematical models for most aspects of superconductivity already exist, and superconducting materials have been successfully applied in many technological applications for decades.

The new phases that will be studied in this project, however, are described in terms of particle-like entities called anyons, which have properties quite distinct from the traditional bosons and fermions that describe traditional materials (including superfluids and superconductors). Anyons are potential candidates to build quantum memory and other quantum information devices.

Anyons have already been experimentally realized in fractional quantum Hall materials, and surely still hold surprises to be revealed in further studies. In this project, the principal investigator (PI) and collaborators will develop fundamental mathematical techniques required for the effective description of these new materials. This research falls squarely within NSF's Big Idea 'Quantum Leap'.

Graduate and undergraduate students will be actively involved with the project and will be offered opportunities to interact with a range of researchers in pure and applied science and encouraged to pursue internships in technology companies as part of their education. Thus, the project will help prepare the workforce for the emerging quantum technology industry.

Specific research problems to be pursued in this project fall in two main directions. 1) Stability of the gap above the ground state under small modifications of the interactions in the Hamiltonian. In particular, the PI will prove results for the bulk gap (away from boundaries) for systems with multiple ground states, for lattice fermion models, and for important classes of models that do not satisfy the usual frustration-freeness condition.

To drop the frustration-freeness assumption will prompt us to develop new techniques. 2) Spectral properties of fractional quantum Hall (FQH) systems. Building on their recent work for the 1/3 filled case, the principal investigator and collaborators will analyze the situation of other filling fractions. The project will also study the stability of FQH phases and investigate the derivation of Haldane's pseudo-potential models beyond the existing 'finite-N' treatments.

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.