Qualitative and Quantitative Analysis of Riemannian Optimization Methods for Ground States of Rotating Multicomponent Bose-Einstein Condensates

TL;DR

This paper develops Riemannian optimization algorithms to efficiently compute ground states of rotating multicomponent Bose-Einstein condensates, providing convergence analysis and demonstrating superior performance of a Lagrangian-based method.

Contribution

It introduces a unified convergence framework for Riemannian gradient methods on quotient manifolds and compares two specialized algorithms for this quantum physics problem.

Findings

Lagrangian-based method converges faster than energy-adaptive scheme.

Global convergence established only for the energy-adaptive method.

Numerical experiments confirm theoretical convergence rates.

Abstract

We develop and analyze Riemannian optimization methods for computing ground states of rotating multicomponent Bose-Einstein condensates, defined as minimizers of the Gross-Pitaevskii energy functional. To resolve the non-uniqueness of ground states induced by phase invariance, we work on a quotient manifold endowed with a general Riemannian metric. By introducing an auxiliary phase-aligned iteration and employing fixed-point convergence theory, we establish a unified local convergence framework for Riemannian gradient descent methods and derive explicit convergence rates. Specializing this framework to two metrics tailored to the energy landscape, we study the energy-adaptive and Lagrangian-based Riemannian gradient descent methods. While monotone energy decay and global convergence are established only for the former, a quantified local convergence analysis is provided for both methods. Numerical experiments confirm the theoretical results and demonstrate that the Lagrangian-based method, which incorporates second-order information on the energy functional and mass constraints, achieves faster local convergence than the energy-adaptive scheme.