The Division of Materials Research and the Division of Mathematical Sciences contribute funding to this award under the NSF-wide Mathematical Sciences Priority Area and contributes to cyberinfrastructure through fundamental research that provides the foundations of future cyberinfrastructure. This award supports research and education on the theory of strongly interacting condensed matter many-body systems. The objective is to gain insight into the origin and behavior of topological phases. These phases are states of matter that are not the product of spontaneous symmetry breaking, and therefore do not have an order-parameter, but possess instead "quantum order" in which the ground state degeneracy is determined by the topology of the space in which they live. The project addresses a range of important and difficult problems in currently fertile and therefore very active, areas of fundamental physics. It should lead to progress both in physics and in mathematics. The principal goals of the proposed work are two-fold: firstly, to understand how the local properties of the many-body wave-functions conspire to produce the global ground-state degeneracy that characterizes these systems, and, secondly, to explore a possible strategy for addressing the last part of the quantum computing problem that of reconnecting the isolated quantum CPU to the classical world and reading of the answer to the computation. The method will be to use the quantum field theory of many-body systems in combination with the representation theory of infinite dimensional Lie algebras. Broader Impact: The topological phases being studied in this project may have applications to quantum computing. The numerical simulation of even simple quantum systems requires exponentially large storage space and exponentially long computation times. The quantum state of a collection of N spin- 1/2 particles, or qubits, requires 2N classical variables for its description, and solving the Schrdinger equation to follow its time evolution requires a conventional computer to perform up to 2N operations per time step. A computer built out of quantum components would be able to exploit the massive parallelism inherent in quantum time evolution to provide fast solutions for problems that would require exponential time on conventional machines. The difficulties to be overcome in building a quantum computer are many. The quantum system at its core must be strongly isolated from the environment so as to avoid decoherence; the unitary transformations that constitute the elementary computational steps must be performed with sufficient precision that error-correcting codes remain effective; and, after all the quantum computation has been performed, the system must be capable of being reconnected to the outside world in such a way that the output can be read off via a measurement process that does not disturb the result. The internal Hilbert space of the topological-phase ground states has many of the features desired of a quantum computer CPU, with the exception of ease of read-out. This project addresses this last problem, and also the origin of the internal Hilbert space itself.
NON-TECHNICAL SUMMARY: The Division of Materials Research and the Division of Mathematical Sciences contribute funding to this award under the NSF-wide Mathematical Sciences Priority Area and contributes to cyberinfrastructure through fundamental research that provides the foundations of future cyberinfrastructure. This award supports research and education on the theory of strongly interacting condensed matter many-body systems. The research engages an important fundamental question: How are states of matter organized? This question is motivated in part by studies of quantum Hall systems and spin liquids that reveal states of matter that cannot be simply organized by symmetry and other principles that underlie the powerful standard theory that describes transformations among states of matter. These topological phases, as they are called, may underlie some of the most challenging problems at the frontiers of condensed matter physics, such as the nature of unusual states observed in high temperature superconductors and related materials, and how edge states that live at the boundaries of various complex quantum mechanical systems and control their properties at low temperature are organized. The PI will focus advanced quantum mechanical concepts of theoretical condensed matter physics and sophisticated mathematical methods that have not yet reached the mainstream of theoretical condensed matter physics on model systems to gain insight into these unusual states of matter and how they are organized. The manipulation of topological phases is a promising route to the realization of a robust implementation of a powerful new form of computation based on quantum mechanics. The PI will also engage a barrier to the realization of quantum computation, how to access in the macroscopic world information that is created by the operation of a quantum mechanical device. The research has potential impact on the field of mathematics and also on theoretical high energy physics.
