Ph.D Thesis
Ph.D Student Levit Anna Stochastic Geometric Methods in Statistical Mechanics Department of Industrial Engineering and Management Professor Dmitry Ioffe

Abstract

Stochastic geometric methods are known to be a very useful tool for solving a variety of problems in statistical mechanics. Furthermore, problems arising due to this approach are interesting in their own right. In this dissertation, we develop path integral representations for Quantum Ising models. To familiarize the reader with our methods and to connect with classical Fortuin-Kasteleyn (FK) representation, we begin by presenting the FK representation for Classical Ising model via the language of Poisson Point Processes. We then pass to Quantum Ising models using Poisson Point Processes and the Lie-Trotter product formula to interpret exponential sums of operators as random products of operators. Contracting these products in different orthonormal bases gives different Stochastic Geometric reprsentations. In particular, we show how to derive a general FK representation for Quantum Ising model. This representation was originally derived by M. Campanino, A. Klein, J.F. Perez (1991) and M. Aizenman, A. Klein, C.M. Newman (1993).

We apply the above Stochastic Geometric reprsentations to the Quantum Curie-Weiss model in transversal field (the Quantum Ising model on complete graph). First, we present the full FK representation of the model. Examining the form of the resulting measure and dropping the weight component from it leads to the natural extension of the Erdős - Rényi random graphs.  We are able to compute explicitly the critical curve for this Quantum Random Graphs. Thereafter, we return to the Quantum Curie-Weiss model and use a "partial" Lie-Trotter decomposition to obtain modified FK representation. The critical curve for this model is computed using a large deviation approach.

Finally, we consider the ground state of the quantum Curie-Weiss model. Contrary to the classical models this question is not trivial for the quantum ones. Again, we use Lie-Trotter formula to arrive at a rigorous path integral representation, albeit looking very different from the FK representation used above. The description which arises is a Markov chain in a killing potential. We prove the existence of a phase transition in the ground state when the strength of the transversal field equals one.