Simulating superfluids with many vortices

By Tom Sharp on September 7, 2016

Superfluidity is an exotic state of matter occurring in physical systems ranging from liquid helium to neutron stars to ultracold atomic gases. In a recent paper in Journal of Physics: Condensed Matter, a model for describing infinite vortex lattices was proposed. Luca Mingarelli introduces his work below.

Our work focuses on superfluids under rotation which are characterised by the presence of vortices carrying quantised circulation. The emerging vortices interact with each another and self-organise into lattices. How can one predict the vortex lattice configuration that minimises the energy? In our recent paper, we have developed a method addressing this problem by finding solutions to the Gross-Pitaevskii equation for infinite periodic vortex lattices.

The smallest periodic cell needed to achieve the Abrikosov lattice: during the propagation in imaginary time the ratio of the cell evolves as well, as to minimise the total energy. In this case one finds R=√3. Image taken from Mingarelli et al. 2016 J. Phys. Cond. Mat. 28 285201 © IOP Publishing, All Rights Reserved.

The key idea in our approach is a reformulation of the problem using its underlying symmetry. The symmetry of the problem is described not by the standard translation group, but rather by the so-called Magnetic Translation Group (MTG). To do so requires an extension of the commonly employed Fourier transform which we term the ‘Magnetic Fourier Transform’ (MFT). The MFT is needed to incorporate the correct twisted boundary conditions into the simulations, allowing periodic vortex lattices to be obtained with relative ease. With this in mind, we formulated a discrete model (a non-linear Hofstadter model) with the same MTG symmetry as the GPE. We solved this numerically using split-step spectral methods in conjunction with a discrete version of the MFT which we formulate based on an extension of the FFT. When propagating in imaginary time, the scheme is able to quickly find the ground state solutions which closely match known theoretical results based on lowest-Landau levels, as well as provide an accurate characterisation of lattice structures. Because we are considering periodic vortex lattices, unit cells containing only a small number of vortices can be used, making the computation extremely efficient.

While our method focuses on a single component system, we believe our methodology can be readily extended to and find wide application in exploring the richer multicomponent systems and systems under more general ‘synthetic’ gauge fields.

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