Fisher-Hartwig Formula and XY Spin Chain

This award will support a study of entanglement in spin models. In XY spin chains, the entropy of several blocks of spins will be represented as a block Toeplitz determinant, and then the generalized Fisher - Hartwig formula will be used to evaluate asymptotic. The project will consider both von Neumann and Renyi entropies. The entanglement spectrum also will be evaluated. This question has attracted attention recently and requires rigorous mathematical treatment. Entanglement of two blocks of spins in ground states of spin models is important. After tracing the rest of the ground state, these two blocks are left in a mixed state. An appropriate measure of entanglement of mixed systems is negativity. The project will describe entanglement of 2 blocks of spins in the AKLT (Affleck, Lieb, Kennedy Takasi) spin chain by means of negativity. The ground state of the AKLT model is known as valence-bond-solid (VBS) state. It plays an important role in quantum information, especially in measurement based quantum computation. Entanglement is a special phenomenon of quantum systems, which cannot happen in the classical (macroscopic) case. It is an important resource for quantum control, which is central for quantum devices, including quantum computers. This has recently become important, because experimentalist who are working with optical lattices can now build models of interacting spins, which had only previously been studied from the mathematical point of view. This is a substantial contribution of mathematical physics towards the goal of quantum information processing, and progress in the laboratory now depends on further theoretical study. The award will support the mathematical analysis of such entanglement, in order to guide atom-molecular-optics experiments.