## 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 +
$+\cdots +$

## 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

## 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

1. Second order
2. Equivariant w.r.t. $$SO(3)^n$$
3. Symplectic
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

1. Atoms configured in lattice

2. Density functional theory (DFT) for spin couplings gives Hamiltonian

3. 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

By Klas Modin

# Spherical midpoint method

Presentation given 2016-07-09 at the Foundations of Computational Mathematics Conference in Barcelona.

• 1,974