Project: Tensor Renormalization Group (Level 4)

Phase transitions occur when the properties of a material change suddenly at a specific point. For example, water undergoes a phase transition at one-hundred degrees, from a liquid phase to a gaseous phase. Reflecting on the fact that water is just a huge collection of identical molecules, it is natural to wonder how to predict the occurrence of phase transitions starting from a molecular/microscopic description of materials. How does the microscopic description translate to the emergent (macroscopic) behaviour? What happens exactly at points of phase transition? The latter question is particularly challenging when so-called continuous phase transitions occur. In this case there is no characteristic length scale, meaning the macroscopic behaviour is not determined by a single (or even finitely many) dominant contributions. You can see a visual consequence in critical opalescence — a phenomenon in which a clear substance becomes cloudy when at a critical point.

The renormalization group refers to a set of methods developed in theoretical physics, used to understand critical phenomena (continuous phase transitions), quantum field theories, and more. The use of renormalization group ideas is ubiquitous in theoretical physics, and Kenneth Wilson received the Nobel prize for his work in this area.

There have been notable (and celebrated) successes in making renormalization group methods mathematically rigorous, but most of them are rather complicated. This project concerns a recent, and relatively simpler, approach to implementing renormalization group ideas. This is the so-called tensor renormalization group (tensor RG). Tensor RG applies to mathematical models which can be reformulated in terms of tensor network contractions. This includes many interesting problems in statistical mechanics (e.g., the Ising model) and computer science (Holant problems). Recent work of Kennedy and Rychkov has shown how the tensor RG can be made mathematically rigorous, at least away from critical points. The mathematics involved is a combination of linear algebra and analysis, and enables one to prove interesting results concerning probabilistic models of statistical mechanics.

The goal of this project is to understand these recent developments, to present the theory in a self-contained way, and possibly to carry out tensor RG investigations of interesting problems via simulation or theory. A mix of rigorous and non-rigorous work is possible.

Prerequisites: Linear Algebra, AMV, Complex Analysis. Some exposure to probability (e.g., Probability II) may be beneficial. Taking Statistical Mechanics IV simultaneously might be advantageous, but by no means necessary. Please contact me in advance with questions about prerequisites. This project concerns very recent research developments, and we will be learning together.

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