Exploring the free-energy landscape of a rotating superfluid

The equilibrium state of a superfluid in a rotating cylindrical vessel is a vortex crystal -- an array of vortex lines which is stationary in the rotating frame. Experimental realisations of this behaviour typically show a sequence of transient states before the free-energy minimising configuration is reached. Motivated by these observations, we construct a new method for a systematic exploration of the free-energy landscape via gradient-based optimisation of a scalar loss function. Our approach is inspired by the pioneering numerical work of Campbell & Ziff (Phys. Rev. B 20, 1979), and makes use of automatic differentiation (AD) which crucially allows us to include entire solution trajectories in the loss. We first use the method to converge thousands of low-free-energy relative equilibria for vortex numbers in the range \(10 \leq N \leq 30\), which reveals an extremely dense set of mostly saddle-like solutions. As part of this search, we discover new continuous families of relative equilibria (in the unbounded domain) which are often global minimisers of the free energy. These continuous families all consist of crystals arranged in a double-ring configuration, and we assess which state from the family is most likely to be observed experimentally by computing energy-minimising pathways from nearby local minima -- identifying a common entry point into the family. Finally, we develop an approach to compute homoclinic orbits and use it to examine the dynamics in the vicinity of the minimising state by converging connections for low-energy saddles.