This research project concerns analytical investigation of exactly-solvable models of quantum spin systems, with particular emphasis on study of entropy and correlation functions for quantum spin chains using the Fisher-Hartwig formula for the determinant of a Toeplitz matrix. The work has implications for measures of entanglement in quantum information science. The project will: (1) study entropy and entanglement in the XXZ spin chain model of statistical physics; (2) evaluate space-, time- and temperature-dependent correlation functions in the XXZ model in a critical regime; (3) calculate asymptotics for the spectrum of the density matrix in the XXZ model in the large-number limit; (4) study the density matrices of blocks of spins in a model of interacting spins due to Affleck, Kennedy, Lieb and Tasaki; and (5) calculate the entanglement entropy in models generalized to other Lie groups.
This project is an investigation of fundamentals in the physics of information. The project studies entanglement, a central concept in quantum information science. Because of its applications to cryptography and secure transmission of sensitive information, quantum information science is important for national security and is an area of federal strategic interest. This project is related to an approach to quantum computation based on measurements in quantum spin systems.
Quantum mechanics provides an oportunity for technological breakthrough. It helps to accelerate computation, provides new security for information transfer and helps developement of new materials. One of the ways of building quantum systems is to use optical lattices and photonic crystals [it operates with particles of light]. The project study entanglement: it is a property which is unique for quantum mechanics [it separates classical world from quantum one]. Over the years experts developed different measures of entanglement: von Neumann entropy, Renyi entropy and negativity. PI studied these measures in the models, which can be implemented in photonic crystals and optical lattices: Affleck-Lieb-Kennedy-Tasaki model, XY spin chain, Lieb-Liniger model of interacing Bose gas and Hubbard model of strongly correlated electrons. PI used advanced mathematics [Fisher- Hrtwig formula for calculation of determinant of Toepliz matrix] for analytical study of the entropies. Some of the outcomes of the project are: PI suggested an architecture for building quantum internet. Another outcome is quantum simulation: PI suggested to use photonic crystals to construct a cetral model of quantum field theory: massive Thirring model [it is equivalent to quantum Sine Gordon model ]. Currently PI works with experimentlists in Stony Brook. Sucessful implementation of this model would mean creation of a controlable model of many body quantum mechanics.
