Condensed Matter Theory

Abstract:

We devise tractable models of unitary quantum many-body dynamics on tree graphs, as a first step toward a deeper understanding of dynamics in non-Euclidean spaces. To this end, we first demonstrate how to construct strictly local quantum circuits that preserve the symmetries of trees, such that their dynamical light cones grow isotropically. For trees with coordination number 𝑧, such circuits can be built from 𝑧-site gates. We then introduce a family of gates for which the dynamics are exactly solvable; these satisfy a set of constraints that we term “tree-unitarity.” Notably, tree-unitarity reduces to the previously established notion of dual-unitarity for 𝑧 =2, when the tree reduces to a line. Among the unexpected features of tree-unitarity is a trade-off between “maximum butterfly velocity” dynamics of out-of-time-order correlators and the existence of nonvanishing correlation functions in multiple directions, a tension absent in one-dimensional dual-unitary models and their Euclidean generalizations. We connect the existence of (a wide class of) solvable dynamics with nonmaximal butterfly velocity directly to a property of the underlying circuit geometry called 𝛿-hyperbolicity, and argue that such dynamics can only arise in non-Euclidean geometries. We give various examples of tree-unitary gates, discuss dynamical correlations, out-of-time-order correlators, and entanglement growth, and show that the kicked Ising model on a tree is a physically motivated example of maximum-velocity tree-unitary dynamics.

Abstract:

We develop a continuum theory to describe the collective dynamics of deformable epithelial cells, distinguishing the force-generating active filaments in the cells from their shape. The theory demonstrates how active flows driven by active filaments can create nematic domains and topological defects in the cell shape field. We highlight the role of the filament flow-aligning parameter, λQ, a rheological quantity that determines the response of the filaments to velocity gradients in the active flows, and plays a significant, to date unappreciated, role in determining the pattern of extensional and compressional active flows. In a contractile cell layer, local flows are expected to align elongated cells perpendicular to the active filaments. However, with increasing λQ, long-range correlations in the active turbulent flow field lead to extended regions where this alignment is parallel, consistent with recent experiments on confluent Madin-Darby canine kidney (MDCK) cell layers. Further, we distinguish defects in the filament director field, which contribute to the active driving, and those in the shape director field, measured in experiments, which are advected by the active flows. By considering the shape-filament orientation, we explain the unexpected motion of +1/2 defects towards their head in contractile cell layers, consistent with recent experiments on epithelial layers examining stress around shape defects.

Abstract:

We coarse-grain a model of closely packed ellipses that can vary their aspect ratio to derive continuum equations for materials comprising confluent deformable particles such as epithelial cell layers. We show that contractile nearest-neighbor interactions between ellipses can lead to their elongation and nematic ordering. Adding flows resulting from active hydrodynamic stresses produced by the particles also affects the aspect ratio and can result in active turbulence. Our results, which agree well with multiphase field simulations of deformable isotropic cells, provide a bridge between models that explicitly resolve cells and continuum theories of active matter.