LPTMC Seminars

Pattern formation by turbulent cascades

Fully developed turbulence is a universal and scale-invariant chaotic state characterized by an energy cascade from large to small scales at which the cascade is eventually arrested by dissipation. Here we show how to harness these seemingly structureless turbulent cascades to generate patterns. Pattern formation entails a process of wavelength selection, which can usually be traced to the linear instability of a homogeneous state. By contrast, the mechanism we propose here is fully nonlinear. It is triggered by the non-dissipative arrest of turbulent cascades: energy piles up at an intermediate scale, which is neither the system size nor the smallest scales at which energy is usually dissipated. Using a combination of theory and large-scale simulations, we show that the tunable wavelength of these cascade-induced patterns can be set by a non-dissipative transport coefficient called odd viscosity, ubiquitous in chiral fluids ranging from bioactive to quantum systems. Odd viscosity, which acts as a scale-dependent Coriolis-like force, leads to a two-dimensionalization of the flow at small scales, in contrast with rotating fluids in which a two-dimensionalization occurs at large scales. Apart from odd viscosity fluids, we discuss how cascade-induced patterns can arise in natural systems, including atmospheric flows, stellar plasma such as the solar wind, or the pulverization and coagulation of objects or droplets in which mass rather than energy cascades.

Monopole Quantum Numbers and Projective Representations in Stable and Fragile Topological Crystalline Insulators

Over the past 15 years, a dizzying array of noninteracting topological insulator (TI) and topological crystalline insulator (TCI) phases have been theoretically predicted and identified in real materials. While the TI states are well understood, the TCI states – which comprise the majority of topological materials in nature – exhibit more complicated classification groups and boundary states and carry more ambiguous response signatures. For earlier variants of interacting symmetry-protected topological states (SPTs), both the classification and response were clarified through the many-body quantum numbers of the 0D collective excitations bound to crystal and electromagnetic defects, such as magnetic fluxes and monopoles. In particular, when 0D defects exhibit fractionalized quantum numbers, or more generally projective representations of the local many-body symmetry group, this can indicate the presence of quantized responses in the bulk that are governed by long-wavelength topological field theories that are stable to symmetric interactions. In this talk, I will introduce numerical methods for computing defect quantum numbers in stable and fragile TCI states via the reduced density matrix, revealing a deep connection between defect quantum numbers and the entanglement spectrum. Surprisingly, we find that when crystal symmetries are included in the local symmetry group, defects can appear to transform projectively even in Wannierizable (fragile) insulators, casting doubt on the suitability of magnetic monopoles for characterizing the TCI states present in real 3D materials. Our results represent a crucial step towards describing TCIs beyond tight-binding models and frameworks like “higher-order topology,” and facilitate more direct connections between free-fermion TCIs and interacting SPTs.

Topological Pathways to Two-Dimensional Quantum Turbulence

I shall present a combined experimental and theoretical investigation of the formation and decay kinetics of vortices in a two-dimensional turbulent superfluid. Fundamental topological conservation laws require that the formation and annihilation of vortices also involve critical points of the velocity field, namely nodes and saddles. Identifying the simplest bifurcations underlying these processes enables to develop an effective kinetic model that closely aligns with experimental observations, and shows that different mechanisms are responsible for vortex number growth and decay.

Slides (pdf)