Rotating two-dimensional Bose--Einstein condensates in power-law traps: collapse, Thomas--Fermi scaling, and the inverse-square endpoint
Abstract
In two dimensions, the kinetic and contact energies scale identically under dilation, a trap V=rs/s as λ−s. We follow this balance through the power-law family −2≤ s≤4, s≠0, for both signs of the coupling g, with rotation, using scaling bounds, radial solutions, Bogoliubov spectra, and two-dimensional imaginary-time states. Attraction collapses the cloud at the Townes coupling g* in every trap, with width ℓ∝δ1/(s+2), δ=1−|g|/g*; repulsion spreads it to a Thomas--Fermi (TF) radius ∝ g1/(s+2). A rotating collapse leaves the axis once the rotation frequency Ω exceeds its dipole frequency, which softens only for s>2; in a quartic trap this happens at δ≃0.32 Ω6, within 3% of the Bogoliubov threshold; it then collapses on the minimum ring of the rotating-frame potential, which also empties a repulsive TF cloud at Ω=(3g/π)1/6. For repulsive vortex states the TF energy error splits exactly into a density-gradient and a flow part of comparable or larger size. At s=−2 only the cutoff sets a length: the energy is exactly covariant with it, the condensate falls to the center for both signs, and repulsion leaves a halo of density ∝1/(gr2) whose extent grows exponentially with g. Near collapse, split-step imaginary time has a first-order stationary bias invisible to residual tests.
Related
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BibTeX
@misc{nguyen2026rotating,
title = {{Rotating two-dimensional Bose--Einstein condensates in power-law traps: collapse, Thomas--Fermi scaling, and the inverse-square endpoint}},
author = {Dinh-Thi Nguyen and Bao-Duy Le},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04849},
eprint = {2611.04849},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04849},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rotating two-dimensional Bose--Einstein condensates in power-law traps: collapse, Thomas--Fermi scaling, and the inverse-square endpoint
AU - Dinh-Thi Nguyen
AU - Bao-Duy Le
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04849
UR - https://arxiv.org/abs/2611.04849
AB - In two dimensions, the kinetic and contact energies scale identically under dilation, a trap V=r^s/s as λ^−s. We follow this balance through the power-law family −2≤ s≤4, s≠0, for both signs of the coupling g, with rotation, using scaling bounds, radial solutions, Bogoliubov spectra, and two-dimensional imaginary-time states. Attraction collapses the cloud at the Townes coupling g_* in every trap, with width ℓ∝δ^1/(s+2), δ=1−|g|/g_*; repulsion spreads it to a Thomas--Fermi (TF) radius ∝ g^1/(s+2). A rotating collapse leaves the axis once the rotation frequency Ω exceeds its dipole frequency, which softens only for s>2; in a quartic trap this happens at δ≃0.32 Ω^6, within 3% of the Bogoliubov threshold; it then collapses on the minimum ring of the rotating-frame potential, which also empties a repulsive TF cloud at Ω=(3g/π)^1/6. For repulsive vortex states the TF energy error splits exactly into a density-gradient and a flow part of comparable or larger size. At s=−2 only the cutoff sets a length: the energy is exactly covariant with it, the condensate falls to the center for both signs, and repulsion leaves a halo of density ∝1/(gr^2) whose extent grows exponentially with g. Near collapse, split-step imaginary time has a first-order stationary bias invisible to residual tests.
ER -