Tensor Network Methods for Nonequilibrium Statistical Mechanics

Citation

Helms, Phillip Laurence (2021) Tensor Network Methods for Nonequilibrium Statistical Mechanics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/kram-8g47. https://resolver.caltech.edu/CaltechTHESIS:06072021-223407413

Abstract

Large deviation theory has emerged as a powerful mathematical scaffolding for studying nonequilibrium statistical mechanics, particularly for characterizing the macroscopic effects of microscopic fluctuations. While the large deviation approach is firmly established, it relates the effects of fluctuations to the likelihood of exponentially rare events, which naively requires exponentially large simulation costs. This, in turn, necessitates the development of appropriate numerical simulation techniques. While the standard Monte Carlo toolkit has expanded to incorporate methods towards making rare events typical, in this thesis I propose and evaluate a powerful unorthodox approach adopted from quantum simulation, namely tensor network algorithms, which can work in concert with standard methods to deepen our understanding of nonequilibrium phenomena.

As a testbed for this novel approach, I consider the dynamical phase behavior of several versions of the simple exclusion process, a paradigmatic model of classical driven diffusion. Using a matrix product state, a one-dimensional tensor network ansatz, and the density matrix renormalization group algorithm, a corresponding optimization routine, I characterize the dynamical phase transition between a jammed and maximal current phase in both the one-dimensional and multi-lane simple exclusion processes. The matrix product state is found to be an efficient representation of the nonequilibrium steady-state biased to arbitrarily rare currents via large deviation theory. Because the one-dimensional ansatz is limited to finite-width systems, I extend this success to study the fully two-dimensional simple exclusion process. There, the projected entangled pair state, a two-dimensional tensor network ansatz, is used with the time evolution via block decimation algorithm to demonstrate that the phase transition observed in one-dimension persists in the fully two-dimensional system.

Towards the goal of making tensor network methods adaptable for a broad range of physically important systems, both classical and quantum, I also present progress towards studying systems in the continuum limit with interacting particles in two dimensions. This builds upon previous work proposing tensor network representations of quantum operators with long-range interactions in two dimensions by evaluating three operator representations in practice and finding two competitive and viable approaches.

Thesis Files