Non-equilibrium dynamics and large deviations in stochastic lattice models via tensor networks

Causer, Luke (2023) Non-equilibrium dynamics and large deviations in stochastic lattice models via tensor networks. PhD thesis, University of Nottingham.

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Over the last few decades, numerical tensor networks have revolutionized the study of quantum many-body systems. Despite this success, their application to classical stochastic problems has not yet been extensively explored. This thesis investigates how tensor network methods can be applied to studying the slow dynamics and the dynamical fluctuations of kinetically constrained models used in the modelling of structural glasses.

This thesis is divided into three parts. It first gives a brief introduction to stochastic dynamics, and explains how the statistics of dynamical observables can be understood through the framework of large deviations. Various approaches to calculating the dynamical large deviations are explained, including the estimation of leading eigenvectors of deformed Markov generators, and trajectory path sampling. It is then followed by an overview of tensor networks in one and two dimensions, which can be used to extract extremal eigenvectors from stochastic generators and simulate time evolution.

The second part then investigates two kinetically constrained models: the “XORFredrickson-Andersen” model, inspired by Rydberg atoms in their anti-blockade regime, and a stochastic Fredkin model, a direct stochastic generalization of the quantum spin

model. Their steady-state properties and non-equilibrium dynamics are studied through theoretical and numerical techniques, including tensor networks and Monte Carlo sampling. Both models display slow and glassy dynamics, motivating the study of their dynamical large deviations. This is done to a high precision via tensor network methods, uncovering and detailing first-order dynamical phase transitions for each model.

The final part of this thesis aims to further develop the application of tensor networks to dynamical fluctuations in classical stochastic dynamics. To this end, a novel method of directly sampling the rare trajectories associated with the large deviations in

one-dimensional stochastic dynamics is developed. This is then accompanied by a method which directly simulates the evolution of the master equation using time evolution methods with matrix product states, allowing for the study of biased dynamics at arbitrary times. The development of these new approaches allow for detailed characterizations of dynamical phase transitions of kinetically constrained models. This is demonstrated for the East, Fredrickson-Andersen and symmetric simple exclusion process, where the spatial and temporal finite-size scalings of their first-order phase transitions are determined. Finally, the methods are extended to two-dimensions via projected entangled-pair states.

Item Type: Thesis (University of Nottingham only) (PhD)
Supervisors: Garrahan, Juan P.
Powell, Stephen
Keywords: quantum dynamics, many-body problems, statistical mechanics, stochastic dynamics
Subjects: Q Science > QC Physics > QC170 Atomic physics. Constitution and properties of matter
Faculties/Schools: UK Campuses > Faculty of Science > School of Physics and Astronomy
Item ID: 73388
Depositing User: Causer, Luke
Date Deposited: 31 May 2023 13:49
Last Modified: 26 Jul 2023 04:40

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