Positive-density ground states of the Gross-Pitaevskii equation

Abstract

We consider the nonlinear Gross-Pitaevskii equation at positive density, that is, for a bounded solution not tending to 0 at infinity. We focus on infinite ground states, which are by definition minimizers of the energy under local perturbations. When the Fourier transform of the interaction potential takes negative values we prove the existence of a phase transition at high density, where the constant solution ceases to be a ground state. The analysis requires mixing techniques from elliptic PDE theory and statistical mechanics, in order to deal with a large class of interaction potentials.

arXiv commentFinal version to appear in "Probability and Mathematical Physics"

References in corpus (41)

• Observing the Rosensweig instability of a quantum ferrofluid
• Observation of the supersolid stripe phase in spin-orbit coupled Bose-Einstein condensates
• Supersolid formation in a quantum gas breaking continuous translational symmetry
• Observation of a dipolar quantum gas with metastable supersolid properties
• Three-dimensional Roton-Excitations and Supersolid formation in Rydberg-excited Bose-Einstein Condensates
• Transient supersolid properties in an array of dipolar quantum droplets
• Long-lived and transient supersolid behaviors in dipolar quantum gases
• Bosons in a Trap: A Rigorous Derivation of the Gross-Pitaevskii Energy Functional
• Exact Criterion for Determining Clustering vs. Reentrant Melting Behavior for Bounded Interaction Potentials
• Supersolid symmetry breaking from compressional oscillations in a dipolar quantum gas
• A superfluid-droplet crystal and a free-space supersolid in a dipole-blockaded gas
• Two-dimensional supersolidity in a dipolar quantum gas
• Why do ultrasoft repulsive particles cluster and crystallize? Analytical results from density functional theory
• Ground State Energy of the Low Density Bose Gas
• Defect-induced supersolidity with soft-core bosons
• Supersolid Vortex Crystals in Rydberg-dressed Bose-Einstein Condensates
• Supersolidity around a critical point in dipolar Bose Einstein condensates
• Elementary excitations of ultracold soft-core bosons across the superfluid-supersolid phase transition
• Freezing and clustering transitions for penetrable spheres
• Crystalline ground states for classical particles
• Mean-field and stability analysis of two-dimensional flowing soft-core bosons modeling a supersolid
• Spontaneous breaking of continuous translational invariance
• Observation of vortices in a dipolar supersolid
• Supersolid structure and excitation spectrum of soft-core bosons in 3D
• New Duality Relations for Classical Ground States
• Two-dimensional supersolidity in a planar dipolar Bose gas
• Nonclassical rotational inertia for a supersolid under rotation
• Periodic minimizers in 1D local mean field theory
• The Hartree-Fock phase diagram of the two-dimensional electron gas
• Hartree-Fock Ground State Phase Diagram of Jellium
• Metal-insulator transition in the Hartree-Fock phase diagram of the fully polarized homogeneous electron gas in two dimensions
• Metal-insulator transition in the two-dimensional fully polarized homogeneous electron gas from Hartree-Fock solutions
• Dynamical excitation of maxon and roton modes in a Rydberg-Dressed Bose-Einstein Condensate
• Excitations of a two-dimensional supersolid
• Ground state at high density
• Traveling waves for some nonlocal 1D Gross-Pitaevskii equations with nonzero conditions at infinity
• Properties of Hartree-Fock solutions of the three-dimensional electron gas
• Clusterization of weakly-interacting bosons in one dimension: an analytic study at zero temperature
• The characterization of ground states
• Classical Density Functional Theory: The Local Density Approximation
• 3D vortex approximation construction and $\epsilon$-level estimates for the Ginzburg-Landau functional