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}
w˙i=wi×∂H∂wi\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)T^*SU(2)
\mathfrak{su}(2)^*
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)
Wi−wih=Wi+wi∣Wi+wi∣×∂H∂wi(W1+w1∣W1+w1∣,…,Wn+wn∣Wn+wn∣)\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
  4. Preserves single-spin quadratic invariants
  5. Self-adjoint
S^3 \simeq
S3≃S^3 \simeq

Hopf fibration

S^2
S2S^2
\pi
π\pi

Extended Hopf fibration

\mathbb{R}^3_*
R∗3\mathbb{R}^3_*
\pi
π\pi
T^*\mathbb{R}^2_*
T∗R∗2T^*\mathbb{R}^2_*
\pi
π\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)
W−wh=W+w∣W+w∣×∂H∂w(W+w∣W+w∣)\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)
XH(w)=w×∂H∂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)
⇒W−w=hXH(W+w∣W+w∣)\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)
XˉH(w)=XH(w∣w∣)\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)
⇒W−w=hXˉH(W+w2)\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
w˙=w×I−1w\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)=12w⋅I(w)−1wH(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)=12w⋅I−1w+εsin⁡(t)w1H(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
w˙=w×w′′,w​:R×S→S2\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