Novel Critical Phenomena

Abstract

The renormalization group is nearly fifty years old, and has proven to be a comprehensive description of equilibrium critical phenomena. Fifty years may seem a long time, but after fifty years of quantum mechanics physicists were only just coming to terms with the significance of entanglement. New frontiers abound, but dark corners in our understanding remain. This thesis, in various ways, pokes around some of those dark corners.

One burgeoning frontier is in the critical phenomena of nonequilibrium systems. Chapter 1 describes the skeleton of a new scaling theory for quasibrittle fracture as a departure from the percolation fixed point along an invariant surface that becomes trivial in the thermodynamic limit. After reviewing the set of numeric tools needed to simulate a popular model for quasibrittle fracture across a wide scale of disorders, we compare simulation results to several versions of the scaling theory. Qualities of the numeric model at critical stress and after rupture that are singular at the percolation fixed point seem well-described by the simplest versions of that theory, while dynamic qualities that are nonsingular at the percolation fixed point appear to be governed by different scaling. At the end we reflect on possible explanations for this discrepancy, and paint a path towards distinguishing them.

If the first chapter demonstrates the importance of numeric experiments for assessing the predictions of scaling theories, the next two offer some new tools for performing certain numeric experiments. Chapter 2 describes an extension to cluster Monte Carlo algorithms that permit their efficient use in the presence of arbitrary on-site potentials. These algorithms were vital for making precision numeric measurements near the critical points of lattice models, and our extension offers new opportunities for study. We give two examples: determining the relevance of symmetry-breaking fields on the 2D XY model, and constraining the form of universal scaling functions in the metastable state. Chapter 3 describes how our extension works equally well for the application of inhomogeneous potentials in geometric cluster flip algorithms for the simulation of colloidal or atomistic systems. We detail the form of the extension and speculate on possible uses, from the simulation of novel phase transition boundaries in gravitational fields to improving the equilibration of computer glasses.

After building some useful tools, we return to a dark corner. Chapter 4 examines the influence that an obscure and subtle singularity near lines of abrupt phase transitions has on the universal scaling at the critical points that terminate them. We show that a simple ansatz for the scaling functions in the metastable state, across the abrupt transition, gives a close prediction for the scaling functions and their derivatives in the stable state, at least for the 2D Ising model. We discuss techniques for improving the simplest prediction perturbatively and speculate on why the description is so much worse in 3D.

Chapter 5 addresses another subtle question: what makes two critical points the same? Specifically, it examines the breadth of phenomena, equilibrium or not, whose critical behavior is infinite order and whose critical singularities are not power laws, but stretched exponentials. We question whether every transition claimed to be in the Berezinskiĭ–Kosterlitz–Thouless universality class actually is, and go hunting for corrections to scaling in a model of growing networks that would prove otherwise.

Finally, we return to more familiar waters. Chapter 6 uses functional mean field theory to explain anomalous singular behavior near the critical point of URu₂Si₂. Data from experiments that measure components of the modulus tensor across an array of temperatures reveal behavior in one mode of the modulus that appears to decay like a power law above the critical point but is not singular at it. We show that this can be explained by the critical point of an order parameter with the same symmetry of that mode, but which becomes modulated, not uniform, in the low-temperature phase. We reflect on other evidence for this idea and on future experiments that could support or falsify it.