The Spherical Midpoint Method

Klas Modin

Joint work with

Robert McLachlan

Massey University

Olivier Verdier

Western Norway University

Outline

Classical spin systems
Spherical midpoint method and its properties
"Toy" examples
Applications in atomistic spin dynamics

Classical spin systems

\displaystyle \dot w_i = w_i\times \frac{\partial H}{\partial w_i}

\displaystyle \dot w_i = w_i\times \frac{\partial H}{\partial w_i}

Phase space $(S^2)^n$

Symplectic structure on $S^2$ $\Omega_w(u,v)=\mathrm{det}(w,u,v)$

+\cdots +

+\cdots +

Motivation for spin systems

Abundance in physics
(e.g. atomistic spin dynamics)
Simplest Kähler manifold
Simplest coadjoint orbit
Simplest inexact symplectic form

Some example spin systems

Free rigid body $\dot w = w\times I^{-1}w$

Heisenberg spin chain
(discrete Landau-Lifshitz) $\dot w_i = w_i\times (w_{i-1}+w_{i+1})$

Fluid particle tracking on sphere $\dot w = \xi(t,w), \quad \xi(t,\cdot)\in \mathfrak{X}_\mu(S^2)$

Point vortex dynamics on spheres
(Jupiter's great red spot)

Known symplectic integrators

Splitting methods
$S^2$ coadjoint orbit of reduced system on $T^*SU(2)$
RATTLE $\Rightarrow$ 9 variables
Variational Lie $\Rightarrow$ 8 variables
Collective symplectic integrator
$\Rightarrow$ 4 variables

T^*SU(2)

T^*SU(2)

\mathfrak{su}(2)^*

\mathfrak{su}(2)^*

Drawbacks: many auxiliary variables, complicated, large error constants

Minimum variables (=3)

and symplectic?

Classical midpoint method on $\mathbb{R}^3$
Riemannian midpoint method on $S^2$

$\Rightarrow$ not symplectic

$\Rightarrow$ not symplectic

Candidates:

Spherical midpoint method

\displaystyle\frac{W_i - w_i}{h} = \frac{W_i+w_i}{|W_i+w_i|}\times \frac{\partial H}{\partial w_i}\left(\frac{W_1+w_1}{|W_1+w_1|},\ldots,\frac{W_n+w_n}{|W_n+w_n|} \right)

\displaystyle\frac{W_i - w_i}{h} = \frac{W_i+w_i}{|W_i+w_i|}\times \frac{\partial H}{\partial w_i}\left(\frac{W_1+w_1}{|W_1+w_1|},\ldots,\frac{W_n+w_n}{|W_n+w_n|} \right)

Main result

Second order
Equivariant w.r.t. $SO(3)^n$
Symplectic
Preserves single-spin quadratic invariants
Self-adjoint

S^3 \simeq

S^3 \simeq

Hopf fibration

S^2

S^2

\pi

\pi

Extended Hopf fibration

\mathbb{R}^3_*

\mathbb{R}^3_*

\pi

\pi

T^*\mathbb{R}^2_*

T^*\mathbb{R}^2_*

\pi

\pi

Classical
midpoint

Riemannian
midpoint

Simple interpretation

\displaystyle\frac{W - w}{h} = \frac{W+w}{|W+w|}\times \frac{\partial H}{\partial w}\left(\frac{W+w}{|W+w|} \right)

\displaystyle\frac{W - w}{h} = \frac{W+w}{|W+w|}\times \frac{\partial H}{\partial w}\left(\frac{W+w}{|W+w|} \right)

\displaystyle X_H(w) = w\times \frac{\partial H}{\partial w}(w)

\displaystyle X_H(w) = w\times \frac{\partial H}{\partial w}(w)

\displaystyle \Rightarrow W-w = h X_H\left(\frac{W+w}{|W+w|}\right)

\displaystyle \Rightarrow W-w = h X_H\left(\frac{W+w}{|W+w|}\right)

\displaystyle \bar X_H(w) = X_H\left(\frac{w}{|w|} \right)

\displaystyle \bar X_H(w) = X_H\left(\frac{w}{|w|} \right)

\displaystyle \Rightarrow W-w = h \bar X_H\left(\frac{W+w}{2}\right)

\displaystyle \Rightarrow W-w = h \bar X_H\left(\frac{W+w}{2}\right)

Classical midpoint method applied to $\bar X_H$

Example 1: free rigid body

\dot w = w \times I^{-1}w

\dot w = w \times I^{-1}w

Example 2: irreversible rigid body

H(w)= \frac{1}{2} w\cdot I(w)^{-1}w

H(w)= \frac{1}{2} w\cdot I(w)^{-1}w

Example 3: development of chaos

H(w)= \frac{1}{2} w\cdot I^{-1}w + \varepsilon \sin(t)w^1

H(w)= \frac{1}{2} w\cdot I^{-1}w + \varepsilon \sin(t)w^1

Example 4: Landau-Lifshitz PDE

\dot w = w\times w'', \qquad w\colon \mathbf{R}\times S \to S^2

\dot w = w\times w'', \qquad w\colon \mathbf{R}\times S \to S^2

Example 5: atomistic spin dynamics

Atoms configured in lattice
Density functional theory (DFT) for spin couplings gives Hamiltonian
Use classical spin dynamics with large $n$

Simulate magnetic properties in condensed matter physics:

THANKS!

References:

Symplectic integrators for spin systems, Phys. Rev. E, 89:061301, 2014

arXiv:1402.4114

A minimal-variable symplectic integrator on spheres, Math. Comp., 86, 2325-2344, 2016

arXiv:1402.3334

Geometry of discrete-time spin systems, J. Nonlin. Sci., 26(5):1507-1523, 2016

arXiv:1505.04035

The Spherical Midpoint Method Klas Modin

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