584 research outputs found
Quantum hydrodynamics for supersolid crystals and quasicrystals
Supersolids are theoretically predicted quantum states that break the
continuous rotational and translational symmetries of liquids while preserving
superfluid transport properties. Over the last decade, much progress has been
made in understanding and characterizing supersolid phases through numerical
simulations for specific interaction potentials. The formulation of an
analytically tractable framework for generic interactions still poses
theoretical challenges. By going beyond the usually considered quadratic
truncations, we derive a systematic higher-order generalization of the
Gross-Pitaevskii mean field model in conceptual similarity with the
Swift-Hohenberg theory of pattern formation. We demonstrate the tractability of
this broadly applicable approach by determining the ground state phase diagram
and the dispersion relations for the supersolid lattice vibrations in terms of
the potential parameters. Our analytical predictions agree well with numerical
results from direct hydrodynamic simulations and earlier quantum Monte-Carlo
studies. The underlying framework is universal and can be extended to
anisotropic pair potentials with complex Fourier-space structure.Comment: 18 pages, 10 figures; supplementary information available on reques
Moving monotonicity formulae for minimal submanifolds in constant curvature
We discover new monotonicity formulae for minimal submanifolds in space
forms, which imply the sharp area bound for minimal submanifolds through a
prescribed point in a geodesic ball. These monotonicity formulae involve an
energy-like integral over sets which are, in general, not geodesic balls. In
the Euclidean case, these sets reduce to the moving-centre balls introduced by
the second author in [Zhu18].Comment: 11 pages, 1 figure; comments welcome
The prescribed point area estimate for minimal submanifolds in constant curvature
We prove a sharp area estimate for minimal submanifolds that pass through a
prescribed point in a geodesic ball in hyperbolic space, in any dimension and
codimension. In certain cases, we also prove the corresponding estimate in the
sphere. Our estimates are analogous to those of Brendle and Hung in the
Euclidean setting.Comment: 21 pages, 2 figures; v2: moved some discussion on monotonicity to new
preprint, other minor changes; comments welcome
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