Emergent order in Hydrodynamic Spin Lattices*

In this talk, we will introduce a hydrodynamic analog system that allows us to investigate simultaneously the wave-mediated self-propulsion and interactions of effective spin degrees of freedom in inertial and rotating frames. Millimetric liquid droplets can walk across the surface of a vibrating fluid bath, self-propelled through a resonant interaction with their own wave fields. By virtue of the coupling with their wave fields, these walking droplets, or 'walkers', extend the range of classical mechanics to include certain features previously thought to be exclusive to the microscopic, quantum realm. Walkers may be impelled to walk in small circles (corresponding to `spin states') in either clockwise or counterclockwise directions, through the influence of submerged circular wells.  When many such spin states are arranged in a regular 1D or 2D lattice geometry, a thin fluid layer between wells enables wave-mediated interactions between neighboring walkers. Through experiments and mathematical modeling, we demonstrate the spontaneous emergence of coherent droplet rotation dynamics for different types of lattices. For sufficiently strong pair-coupling, wave interactions between neighboring walkers may induce local spin flips leading to `ferromagnetic' or `antiferromagnetic' states, as arise when neighboring droplet pairs orbit in the same or opposite sense, respectively. Transitions between these two forms of collective order can be induced through variations in non-equilibrium driving, lattice geometry and Coriolis forces mimicking an external magnetic field. Theoretical predictions based on a generalized Kuramoto model derived from first principles rationalize our experimental observations, establishing HSLs as a generic paradigm for active phase oscillator dynamics.