Condensed Matter Theory

Abstract:

`Active matter' is a term used to describe non-equilibrium systems, such as living organisms and tissues, that do work using an external source of energy. Epithelial cells are a subclass of living systems that derive their energy from ATP hydrolysis and are driven by active processes in the cytoskeleton. Epithelial cells can migrate as individuals, but also organise into two-dimensional monolayers and give rise to rich emergent behaviours such as collective migration and nematic liquid-crystalline order, with implications for morphogenesis, growth, and cancer metastasis. In this Thesis we model epithelia as two-dimensional monolayers to study their collective behaviours.

Using a multi-phase field model for epithelia, we first study the contributions of polar activity and cell-cell adhesion to the rotation of pairs of cells in confinement. Then we dispense with polar activity in favour of dipolar active stresses in order to study bulk epithelia. We investigate the microphase separation of mixtures of extensile and contractile dipolar cells. Cell sorting of this type has been observed in experiment and is relevant to embryogenesis and morphogenesis, and we propose an active origin for the observations.

Then we focus on cell intercalations, which are responsible in part for tissue fluidisation and therefore collective migration. We model fluctuations in the number of cadherin proteins at adherens junctions between cells using an Ornstein-Uhlenbeck process. We vary the timescale and variance of the random process and find a region that promotes translational diffusion and neighbour rearrangements, and show that the orientational order in the system vanishes. We also find that the translational diffusion has a non-monotonic dependence on the timescale of the adhesion fluctuations.

Finally, we study the flow of epithelia confined to a channel. Plug and shear flow have been observed in experiment and have implications for gastrulation in the embryo. First, we recover macroscopic flow for contractile cells by enforcing cell orientations at the edge the channel. Then we reformulate the force balance in the multi-phase field model to include Stokesian dynamics and, therefore, internal friction. The improved model exhibits an oscillatory shear flow that becomes more persistent as the coefficient of internal friction is increased. This development helps to bridge the gap from microscopic models to continuum theories.

Abstract:

Abstract:

The Lieb-Schultz-Mattis (LSM) theorem implies that gapped phases of matter must satisfy non-trivial conditions on their low-energy properties when a combination of lattice translation and $U(1)$ symmetry are imposed. We describe a framework to characterize the action of symmetry on fractons and other sub-dimensional fractional excitations, and use this together with the LSM theorem to establish that X-cube fracton order can occur only at integer or half-odd-integer filling. Using explicit parton constructions, we demonstrate that "odd" versions of X-cube fracton order can occur in systems at half-odd-integer filling, generalizing the notion of odd $Z_2$ gauge theory to the fracton setting. At half-odd-integer filling, exiting the X-cube phase by condensing fractional quasiparticles leads to symmetry-breaking, thereby allowing us to identify a class of conventional ordered phases proximate to phases with fracton order. We leverage a dual description of one of these ordered phases to show that its topological defects naturally have restricted mobility. Condensing pairs of these defects then leads to a fracton phase, whose excitations inherit these mobility restrictions.