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

Martin Hermann, Tatjana Stykel, Mahima Yadav

TL;DR

The paper develops a geometric framework for computing ground states of rotating multicomponent Bose-Einstein condensates by formulating the problem on a quotient manifold to remove phase-induced non-uniqueness. It introduces Riemannian gradient-type methods with two metrics—an energy-adaptive metric and a Lagrangian-based metric—and proves a unified local convergence theory, including an auxiliary phase-alignment step to handle non-uniqueness. The energy-adaptive method enjoys monotone energy decay and global convergence, while the Lagrangian-based method accelerates local convergence by leveraging second-order information; both are supported by an inf-sup stability analysis of the Hessian on horizontal spaces. Numerical experiments on 2- and 3-component models corroborate the theory, showing faster convergence for LagrRGD and highlighting practical trade-offs in omega tuning and step sizes. The results provide a robust, scalable approach to ground-state computation in rotating multicomponent quantum fluids and open avenues for extending Riemannian optimization to broader constrained energy problems.

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.

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

TL;DR

The paper develops a geometric framework for computing ground states of rotating multicomponent Bose-Einstein condensates by formulating the problem on a quotient manifold to remove phase-induced non-uniqueness. It introduces Riemannian gradient-type methods with two metrics—an energy-adaptive metric and a Lagrangian-based metric—and proves a unified local convergence theory, including an auxiliary phase-alignment step to handle non-uniqueness. The energy-adaptive method enjoys monotone energy decay and global convergence, while the Lagrangian-based method accelerates local convergence by leveraging second-order information; both are supported by an inf-sup stability analysis of the Hessian on horizontal spaces. Numerical experiments on 2- and 3-component models corroborate the theory, showing faster convergence for LagrRGD and highlighting practical trade-offs in omega tuning and step sizes. The results provide a robust, scalable approach to ground-state computation in rotating multicomponent quantum fluids and open avenues for extending Riemannian optimization to broader constrained energy problems.

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.

Paper Structure

This paper contains 21 sections, 17 theorems, 113 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Constrained energy minimization and existence of a ground state
  3. Mathematical model
  4. Existence of a ground state
  5. Properties of the derivatives of the energy functional and nonlinear eigenvector problem
  6. The first-order derivative
  7. The second-order derivative
  8. Nonlinear eigenvector problem
  9. Quotient geometry
  10. Quasi-uniqueness of ground states within the quotient manifold framework
  11. Riemannian gradient-type methods and their convergence analysis
  12. Riemannian gradient descent method
  13. Local convergence analysis
  14. Energy-adaptive Riemannian gradient descent method
  15. Lagrangian-based gradient descent method
  16. ...and 6 more sections

Key Result

Theorem 2.2

Let Assumptions A1-- A3 be fulfilled. Then there exists a ground state ${\bm \varphi}_* \in \mathcal{O}\mathcal{B}_N^{\,\mathbb{C}}(p,H)$ which is a global minimizer of the constrained minimization problem eq:min.

Figures (8)

  • Figure 7.1: Model 1: component densities of the computed local minimizer.
  • Figure 7.2: Model 1: (top) convergence histories of the residuals; (bottom) contraction and convergence rates for different methods with $\tau=1$.
  • Figure 7.3: Model 1: convergence histories of the residuals for eaRGD and LagrRGD with $\omega=0.95$ for different step size strategies.
  • Figure 7.4: Model 2: component densities of the computed local minimizer.
  • Figure 7.5: Model 2: convergence histories of the residuals for eaRGD and LagrRGD with $\omega=0.95$ for different step size strategies. For LagrRGD-AD, we also include a moving mean with a window of 100 steps in each direction.
  • ...and 3 more figures

Theorems & Definitions (38)

  • Remark 2.1: Phase invariance of $\mathcal{E}$ and non-uniqueness of ground state
  • Theorem 2.2: Existence of a ground state
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 4.1
  • ...and 28 more