Phase transition between two-component and three-component ground states of spin-1 Bose-Einstein condensates
Liren Lin, I-Liang Chern
TL;DR
This work provides a rigorous mathematical proof of the spin-1 antiferromagnetic Bose-Einstein condensate phase transition from a two-component (2C) ground state to a three-component (3C) ground state under increasing quadratic Zeeman energy $q$. The authors reduce the spinor problem to a constrained amplitude formulation and introduce a mass-density redistribution principle that nonincreases the kinetic energy, enabling precise comparisons between 2C and 3C configurations. They establish the existence of a critical function $q_c(M)$ such that the 2C ground state is unique for $q<q_c(M)$ and all ground states are 3C for $q>q_c(M)$, with detailed analysis of the boundary behavior and auxiliary bounds. The results align with numerical simulations and experiments, and shed light on the structure of ground states across the phase boundary, including the behavior of the two-component minimizer and the openness of the threshold case. The paper also highlights open questions about monotonicity, continuity, and uniqueness at $q=q_c(M)$, pointing to future mathematical and computational exploration.
Abstract
For an antiferromagnetic spin-1 Bose-Einstein condensate under an applied uniform magnetic field, its ground state $(ψ_1,ψ_0,ψ_{-1})$ undergoes a phase transition from a two-component state ($ψ_0 \equiv 0$) to a three-component state ($ψ_j\ne 0$ for all $j$) at a critical value of the magnetic field. This phenomenon has been observed in numerical simulations as well as in experiments. In this paper, we provide a mathematical proof based on a simple principle found by the authors: a redistribution of the mass densities between different components will decrease the kinetic energy.