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Riemannian optimisation methods for ground states of multicomponent Bose-Einstein condensates

R. Altmann, M. Hermann, D. Peterseim, T. Stykel

TL;DR

This work develops a comprehensive Riemannian-geometry framework for computing ground states of multicomponent Bose–Einstein condensates by formulating constrained energy minimisation on the infinite-dimensional generalised oblique manifold. It establishes existence and uniqueness (up to scaling) of ground states, and connects the problem to a nonlinear eigenvector formulation, enabling gradient- and Newton-type optimisation on manifolds with multiple metric choices, including energy-adaptive and Lagrangian-based preconditioners. The paper proves global convergence for the energy-adaptive Riemannian gradient descent method and analyzes local convergence, while demonstrating substantial computational gains through numerical experiments in 1D and 2D. The results show that Riemannian methods, especially with problem-tailored metrics and alternating strategies, robustly identify ground states across discretisations and potential landscapes, highlighting a path toward efficient simulations of complex multicomponent quantum systems.

Abstract

This paper addresses the computation of ground states of multicomponent Bose-Einstein condensates, defined as the global minimiser of an energy functional on an infinite-dimensional generalised oblique manifold. We establish the existence of the ground state, prove its uniqueness up to scaling, and characterise it as the solution to a coupled nonlinear eigenvector problem. By equipping the manifold with several Riemannian metrics, we introduce a suite of Riemannian gradient descent and Riemannian Newton methods. Metrics that incorporate first- or second-order information about the energy are particularly advantageous, effectively preconditioning the resulting methods. For a Riemannian gradient descent method with an energy-adaptive metric, we provide a qualitative global and quantitative local convergence analysis, confirming its reliability and robustness with respect to the choice of the spatial discretisation. Numerical experiments highlight the computational efficiency of both the Riemannian gradient descent and Newton methods.

Riemannian optimisation methods for ground states of multicomponent Bose-Einstein condensates

TL;DR

This work develops a comprehensive Riemannian-geometry framework for computing ground states of multicomponent Bose–Einstein condensates by formulating constrained energy minimisation on the infinite-dimensional generalised oblique manifold. It establishes existence and uniqueness (up to scaling) of ground states, and connects the problem to a nonlinear eigenvector formulation, enabling gradient- and Newton-type optimisation on manifolds with multiple metric choices, including energy-adaptive and Lagrangian-based preconditioners. The paper proves global convergence for the energy-adaptive Riemannian gradient descent method and analyzes local convergence, while demonstrating substantial computational gains through numerical experiments in 1D and 2D. The results show that Riemannian methods, especially with problem-tailored metrics and alternating strategies, robustly identify ground states across discretisations and potential landscapes, highlighting a path toward efficient simulations of complex multicomponent quantum systems.

Abstract

This paper addresses the computation of ground states of multicomponent Bose-Einstein condensates, defined as the global minimiser of an energy functional on an infinite-dimensional generalised oblique manifold. We establish the existence of the ground state, prove its uniqueness up to scaling, and characterise it as the solution to a coupled nonlinear eigenvector problem. By equipping the manifold with several Riemannian metrics, we introduce a suite of Riemannian gradient descent and Riemannian Newton methods. Metrics that incorporate first- or second-order information about the energy are particularly advantageous, effectively preconditioning the resulting methods. For a Riemannian gradient descent method with an energy-adaptive metric, we provide a qualitative global and quantitative local convergence analysis, confirming its reliability and robustness with respect to the choice of the spatial discretisation. Numerical experiments highlight the computational efficiency of both the Riemannian gradient descent and Newton methods.

Paper Structure

This paper contains 31 sections, 17 theorems, 139 equations, 3 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Constrained energy minimisation and existence of a ground state
  3. Energy-related mathematical model
  4. Properties of the derivative of the energy
  5. Existence of a ground state
  6. Nonlinear eigenvector problem and uniqueness of the ground state
  7. Critical points and nonlinear eigenvector problem
  8. Uniqueness of the ground state
  9. Second-order derivatives and necessary optimality condition
  10. Generalised oblique manifold
  11. Tangent and normal spaces
  12. Riemannian gradients and Hessians
  13. Choices of metric
  14. --metric
  15. Energy-adaptive metric
  16. ...and 16 more sections

Key Result

Proposition 2.2

Let Assumption A1 be fulfilled and let ${\bm \varphi}\in H$. Then the bilinear form $a_{\bm \varphi}$ defined in eq:aphi is symmetric, bounded, and satisfies a Gårding inequality.

Figures (3)

  • Figure 8.1: Two-component BEC: potential and components of the ground state for $\beta = 10, 100, 1000$ (from left to right). The potential is rescaled by a factor of $0.0025$ for plotting purposes.
  • Figure 8.2: Two-component BEC: convergence history of the residuals for $\beta = 10, 100, 1000$ (from left to right). For $\beta = 100$, the non-alternating versions of the eaRGD and LgrRGD methods are shown as well.
  • Figure 8.3: Three-component BEC: random and periodic potentials and components of the ground states computed with the alternating eaRGD method. For the potentials, we do not depict the additional trapping potential that is used to enforce the homogeneous Dirichlet boundary conditions.

Theorems & Definitions (39)

  • Remark 2.1: Scaling invariance of $\mathcal{E}$ and non-uniqueness of ground state
  • Proposition 2.2: Properties of $a_{\bm \varphi}$
  • proof
  • Remark 2.3: Coercivity of $a_{\bm \varphi}$ for non-negative $K$
  • Theorem 2.4: Existence of a ground state
  • proof
  • Remark 2.5: Non-negative ground state
  • Lemma 3.1: Eigenvalues and eigenfunctions of $\mathcal{A}_{{\bm \varphi},j}$
  • proof
  • Proposition 3.2: Characterisation of Lagrange multipliers as eigenvalues of components
  • ...and 29 more