Klas Modin PRO
Mathematician at Chalmers University of Technology and the University of Gothenburg
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×∂wi∂H
Phase space (S2)n
Symplectic structure on S2 Ωw(u,v)=det(w,u,v)
+\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 w˙=w×I−1w
Heisenberg spin chain
(discrete Landau-Lifshitz) w˙i=wi×(wi−1+wi+1)
Fluid particle tracking on sphere w˙=ξ(t,w),ξ(t,⋅)∈Xμ(S2)
Point vortex dynamics on spheres
(Jupiter's great red spot)
Known symplectic integrators
- Splitting methods
- S2 coadjoint orbit of reduced system on T∗SU(2)
RATTLE ⇒ 9 variables
Variational Lie ⇒ 8 variables
- Collective symplectic integrator
⇒ 4 variables
T^*SU(2)
T∗SU(2)
\mathfrak{su}(2)^*
su(2)∗
Drawbacks: many auxiliary variables, complicated, large error constants
Minimum variables (=3)
and symplectic?
- Classical midpoint method on R3
- Riemannian midpoint method on S2
⇒ not symplectic
⇒ 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)
hWi−wi=∣Wi+wi∣Wi+wi×∂wi∂H(∣W1+w1∣W1+w1,…,∣Wn+wn∣Wn+wn)
Main result
- Second order
- Equivariant w.r.t. SO(3)n
- Symplectic
- Preserves single-spin quadratic invariants
- Self-adjoint
S^3 \simeq
S3≃
Hopf fibration
S^2
S2
\pi
π
Extended Hopf fibration
\mathbb{R}^3_*
R∗3
\pi
π
T^*\mathbb{R}^2_*
T∗R∗2
\pi
π
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)
hW−w=∣W+w∣W+w×∂w∂H(∣W+w∣W+w)
\displaystyle X_H(w) = w\times \frac{\partial H}{\partial w}(w)
XH(w)=w×∂w∂H(w)
\displaystyle \Rightarrow W-w = h X_H\left(\frac{W+w}{|W+w|}\right)
⇒W−w=hXH(∣W+w∣W+w)
\displaystyle \bar X_H(w) = X_H\left(\frac{w}{|w|} \right)
XˉH(w)=XH(∣w∣w)
\displaystyle \Rightarrow W-w = h \bar X_H\left(\frac{W+w}{2}\right)
⇒W−w=hXˉH(2W+w)
Classical midpoint method applied to XˉH
Example 1: free rigid body
\dot w = w \times I^{-1}w
w˙=w×I−1w
Example 2: irreversible rigid body
H(w)= \frac{1}{2} w\cdot I(w)^{-1}w
H(w)=21w⋅I(w)−1w
Example 3: development of chaos
H(w)= \frac{1}{2} w\cdot I^{-1}w + \varepsilon \sin(t)w^1
H(w)=21w⋅I−1w+εsin(t)w1
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
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
A minimal-variable symplectic integrator on spheres, Math. Comp., 86, 2325-2344, 2016
Geometry of discrete-time spin systems, J. Nonlin. Sci., 26(5):1507-1523, 2016
The Spherical Midpoint Method Klas Modin
Spherical midpoint method
By Klas Modin
Spherical midpoint method
Presentation given 2016-07-09 at the Foundations of Computational Mathematics Conference in Barcelona.
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