Two Spatial Correlation Estimates in Quantum Statistical Mechanics

Abstract

This thesis presents the results of two studies done by the author and his collaborators with the common thread of obtaining spatial correlation estimates for quantum states of interest in quantum statistical mechanics. The main body (Chapters 1, 2, and 3) is concerned with the decay properties of the spatial correlations of finite-volume steady states in dissipative quantum lattice systems. In it we show that the steady states with respect to a power-law decaying interaction have correlations that decay polynomially in space. To do so, we develop a general Lieb-Robinson bound which grows linearly in time for certain small time-scales, and employ it to give estimates on various finite-volume approximations to the associated dynamics. Some of the results contained in these chapters appear in the preprint [75]. Chapter 4 is a summary of the results of a co-authored study [65]. This paper studies a stochastic quantum model where states are generated by a stochastic process in a von Neumann algebra. In particular, Nelson and Roon construct an example of a random translation co-variant state on a quantum spin chain with infinite-dimensional on-site observable algebras. An estimate for the spatial correlations of a random co-variant state is derived from a multiplicative ergodic theorem for random quantum channels. The preprint [65] is included in Appendix C. While the models considered in each study are quite different, the underlying principle is to reduce the estimation of spatial correlations to a kind of completely positive dynamics: in the first case, these arise as quantum dynamical semigroups of the local interaction; in the second case, these dynamics arise as a kind of ``factorization'' of the random state we construct. In each case, the properties of the dynamics give rise to an upper estimate on the correlations.