Quantum States of Matter, Topological Invariants, and Spectral Gaps

Quantum states of matter have unique properties due to the presence of quantum entanglement. They appear at low temperatures in a variety of materials. Some of these materials are minerals found in nature or are derived from them, and others are synthesized in the cutting-edge laboratories of today using trapped cold atoms or optical lattices. Mathematically, these systems are studied in the ground state, which is the state of smallest possible energy of the systems. In experiments, this corresponds to temperatures near absolute zero. These special states of matter have attracted a great deal of attention in the past few decades especially due to their relevance for quantum information science and technology. Any advantage of quantum information processing (including computation) derives from the special properties of quantum entanglement. It is therefore important to characterize the structure of such states in a rigorous manner. Topological phases represent types of specially entangled quantum states. They can be classified by so-called topological invariants which explain why these phases are particularly robust against noise and other perturbations. This robustness is an essential requirement in applications of topological insulators to quantum memory. The term `insulator’ refers to the appearance of a gap in the energy spectrum above the ground state. A key property is the stability of the ground state gap under moderate perturbations of the Hamiltonian. The principal investigator studies this spectral gap in a series of paradigmatic model systems. The topological characterization translates into a succinct description of the elementary excitations of the systems. In two-dimensional systems (thin layers) this description is in terms of a novel type particles called anyons. This study is motivated by the possibility of anyon based quantum computation in solid state devices. Attracting and educating talented junior researchers and preparing them for the future quantum science and technology workforce is an integral par of the project.

Mathematics is crucial to the precise and quantitative description of topological phases. At the highest level, different phases in two dimensions are often characterized by a modular tensor category. The principal investigator works with junior collaborators to prove results about the existence and stability of the spectral gap for types of many-body quantum systems for which such mathematical results are currently lacking. For example, in the case of the pseudo-potential models introduced by Haldane and others, the detailed analysis of the operator product structure of the ground states undertaken in this project, may open a new avenue for progress on the spectral gap problem. A related topic is the study of ground state phase diagrams. In addition to the study of topological indices, the principal investigator and his graduate students work on obtaining more detailed descriptions of the entangled ground state phases using string order parameters and new dualities, which have been conjectured in the recent literature. At the technical level, this involves applying the techniques of operator algebras and functional analysis, and also requires further development of those techniques. Spectral analysis for many-body fermions subject to a magnetic field in two dimensions, is an example where new technical advances are pursued. National and international collaborations will be a key part of this activity, both by using online collaboration tools and in person conferences, schools, and research visits.

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.

Award Number: 2510824
Principal Investigator: Bruno Nachtergaele
Funds Obligated: $275,000
State: CA