Proofs of some simplified characterizations of the ground states of spin-1 Bose-Einstein condensates
Liren Lin, I-Liang Chern
TL;DR
This work provides rigorous justification for two simplified characterizations of the ground states in spin-1 Bose-Einstein condensates. By a simple kinetic-energy–reduction principle via mass redistribution among spin components, the authors prove that ferromagnetic spin-1 BECs obey the single-mode approximation (SMA), reducing the problem to a one-component energy $\mathbb{E}^s[f]$; conversely, for antiferromagnetic interactions with nonzero magnetization, the $m_F=0$ component vanishes, yielding a two-component reduction. The paper also analyzes degenerate cases, showing SMA can persist while ground states are not unique, and provides a unified framework to derive these characterizations directly from energy comparisons. The results give a rigorous foundation for numerically observed SMA and vanishing phenomena, enabling simplified, lower-dimensional analyses of ground-state structures in spin-1 BECs and clarifying the influence of magnetization and spin-exchange interactions on the ground-state composition.
Abstract
We justify some characterizations of the ground states of spin-1 Bose-Einstein condensates exhibited from numerical simulations. For ferromagnetic systems, we show the validity of the single-mode approximation (SMA). For an antiferromagnetic system with nonzero magnetization, we prove the vanishing of the $m_F=0$ component. In the end of the paper some remaining degenerate situations are also discussed. The proofs of the main results are all based on a simple observation, that a redistribution of masses among different components will reduce the kinetic energy.