Klas Modin PRO
Mathematician at Chalmers University of Technology and the University of Gothenburg
Numerical integration
of classical spin systems
Presentation by Klas Modin
UppASD Autumn School 2022
Motivation
Elastic pendulum, slightly damped
Which integrators preserve qualitative long-time behaviour?
Heun's method
(2nd order accurate)
"Geometric" method
(2nd order accurate)
Why?
Total energy over time
Long-time trajectories for geometric method
Motto: discretized system should retain geometric structure
Outline
- Classical spin systems
- Hamiltonian structure and symplecticness
- Review and comparison of numerical methods
Classical spin systems
\displaystyle \dot{\mathbf{s}}_i = \mathbf{s}_i\times \nabla_{\mathbf{s}_i} H(\mathbf{s}_1,\ldots,\mathbf{s}_n) +
f(\mathbf{s}_1,\ldots,\mathbf{s}_n, \dot{{W}})
Phase space \((S^2)^n\)
+\cdots +
\mathbf{s}_i = (s^x_i,s^y_i,s^z_i)\in S^2\subset \mathbb{R}^3
Hamiltonian
Non-conservative
Noise
Conservation laws
Spin lengths: \(\lVert \mathbf{s}_i \rVert = \) const
\displaystyle\frac{d}{dt} \lVert \mathbf{s}_i\rVert^2 = 2\, \mathbf{s}_i\cdot \dot{\mathbf{s}}_i =
2\, \mathbf{s}_i\cdot (\mathbf{s}_i\times \nabla_{\mathbf{s}_i}H) = 0
Total energy: \(H(\mathbf{s}_1,\ldots,\mathbf{s}_n) = \) const
\displaystyle\frac{d}{dt} H = \sum_{i=1}^n \nabla_{\mathbf{s}_i} H \cdot \dot{\mathbf{s}}_i =
\sum_{i=1}^n \nabla_{\mathbf{s}_i} H\cdot (\mathbf{s}_i\times \nabla_{\mathbf{s}_i}H) = 0
Single-spin system: rigid body
H(\mathbf s) = \mathbf s\cdot \mathbb{I}^{-1}\mathbf s
moments of inertia tensor
angular momentum
Single-spin system: rigid body
Conservation of energy: \(\mathbf{s}\cdot \mathbb{I}^{-1}\mathbf{s}\) = const
Conservation of total momentum: \(\lVert\mathbf{s}\rVert^2\) = const
Many-spin system: Heisenberg chain
H(\mathbf s_1,\ldots,\mathbf s_n) = J\sum_i \mathbf s_i\cdot \mathbf s_{i+1}
coupling constant
Are there other conservation laws?
Symplectic structure on \(S^2\) \[\Omega_\mathbf{s}(\mathbf u,\mathbf v)=\mathrm{det}(\mathbf s,\mathbf u,\mathbf v)\]
\mathbf s
\mathbf v
\mathbf u
Symplectic flow:
infinitesimal area preserved in time
A
\varphi^t(A)
Are there other conservation laws?
Symplectic structure on \((S^2)^n\) \[\Omega_{(\mathbf{s}_1,\ldots,\mathbf{s}_n)}(\mathbf u_1,\ldots,\mathbf u_1,,\mathbf v_1,\ldots,\mathbf v_n)=\sum_i\mathrm{det}(\mathbf s_i,\mathbf u_i,\mathbf v_i)\]
\mathbf s_1
\mathbf v_1
\mathbf u_1
\(+\cdots+\)
\mathbf s_n
\mathbf v_n
\mathbf u_n
What is a numerical integrator?
Exact flow:
(\mathbf s_1(t),\ldots,\mathbf s_n(t)) = \varphi^t(\mathbf s_1(0),\ldots,\mathbf s_n(0))
flow map
Numerical flow:
\Phi_h(\mathbf s_1,\ldots,\mathbf s_n) \approx \varphi^h(\mathbf s_1,\ldots,\mathbf s_n)
integrator map
Symplectic integrator
Symplectic integrator: \(\Phi_h\) preserves symplectic area form \(\Omega\)
A
\varphi^h(A)
\Phi_h(A)
But what does symplecticity entail?
Integrator \(\Phi_h\) symplectic \(\iff\) \(\Phi_h\) exact flow for modified Hamiltonian \(\tilde H_h\)
\(\Phi_h(\mathbf s_1,\ldots,\mathbf s_n) = \varphi_{\tilde H_h}^h(\mathbf s_1,\ldots,\mathbf s_n)\)
Consequences of symplecticness
This explains why symplectic integrators are superior!
- "Almost" conservation of tori for Hamiltonian perturbations of integrable systems (KAM theory)
- Convergence towards correct macroscopic equilibrium (e.g. temperature in MD)
Examples of integrators
Heun's method
\displaystyle {\mathbf s}_i^{k+1} = \mathbf s_i^{k} + \frac{h}{2}(\mathbf f_i + \tilde{\mathbf f}_i)
\mathbf f_i = \mathbf s_i^{k}\times \nabla_{\mathbf s_i} H( \mathbf s_1^{k},\ldots, \mathbf s_n^{k})
\tilde\mathbf f_i = (\mathbf s_i^{k}+h\mathbf f_i)\times \nabla_{\mathbf s_i} H( \mathbf s_1^{k}+h\mathbf f_1,\ldots, \mathbf s_n^{k}+h\mathbf f_n)
Spherical midpoint method
\displaystyle\frac{\mathbf s^{k+1}_i - \mathbf s^{k}_i}{h} = \frac{\mathbf s^{k+1}_i+\mathbf s^{k}_i}{|\mathbf s^{k+1}_i+\mathbf s^{k}_i|}\times
\nabla_{\mathbf s_i} H\left(\frac{\mathbf s^{k+1}_1+\mathbf s^{k}_1}{|\mathbf s^{k+1}_1+\mathbf s^{k}_1|},\ldots,
\frac{\mathbf s^{k+1}_n+\mathbf s^{k}_n}{|\mathbf s^{k+1}_n+\mathbf s^{k}_n|} \right)
Midpoint method
\displaystyle\frac{\mathbf s^{k+1}_i - \mathbf s^{k}_i}{h} = \frac{\mathbf s^{k+1}_i+\mathbf s^{k}_i}{2}\times
\nabla_{\mathbf s_i} H\left(\frac{\mathbf s^{k+1}_1+\mathbf s^{k}_1}{2},\ldots,
\frac{\mathbf s^{k+1}_n+\mathbf s^{k}_n}{2} \right)
Properties
| Heun | Midpoint | Spherical midpoint | |
|---|---|---|---|
| Explicit | yes | no | no |
| Spin lengths | no | yes | yes |
| Energy | no | if quadratic | modified |
| Symplectic | no | no | yes |
Example 1: rigid body
H(\mathbf s) = \mathbf s\cdot \mathbb{I}^{-1}\mathbf s
Example 2: complicated single spin
H(\mathbf s) = \mathbf s\cdot \mathbb{I}(\mathbf s)^{-1}\mathbf s
H(t)-H(0)
steps
Example 3: development of chaos
H(\mathbf s)= \frac{1}{2} \mathbf s\cdot \mathbb{I}^{-1}\mathbf s + \varepsilon \sin(t)s^x
Problem with implicit methods
Fix-point iterations or Newton iterations needed \(\Rightarrow\) intractable for large systems (too expensive)
Semi-implicit midpoint method
\displaystyle\frac{\tilde{\mathbf s}^{k+1}_i - \mathbf s^{k}_i}{h} = \frac{\tilde{\mathbf s}^{k+1}_i+\mathbf s^{k}_i}{2}\times
\nabla_{\mathbf s_i} H\left(\mathbf s^{k+1}_1,\ldots,
\mathbf s^{k}_n\right)
[Mentink et al, 2010]
\displaystyle\frac{\mathbf s^{k+1}_i - \mathbf s^{k}_i}{h} = \frac{\mathbf s^{k+1}_i+\mathbf s^{k}_i}{2}\times
\nabla_{\mathbf s_i} H\left(\frac{\tilde{\mathbf s}^{k+1}_1+\mathbf s^{k}_1}{2},\ldots,
\frac{\tilde{\mathbf s}^{k+1}_n+\mathbf s^{k}_n}{2} \right)
Interpretation: two iterations for midpoint method
| Symplectic | no |
| Explicit | yes |
| Spin lengths | yes |
| Energy | no (mostly) |
Summary
- Don't use "off-the-shelf" methods (Heun, RK, ...)
- Correct phase space geometry affects qualitative long-time behavior more than local accuracy
(no need for high order methods)
- Qualitatively, symplectic methods always best
(also for slightly damped systems)
- Symplectic methods are implicit \(\Rightarrow\) problem for large systems
- Semi-implicit methods tuned for spin-systems can be good trade-off
Slides available at: slides.com/kmodin
Numerical integration of classical spin systems
By Klas Modin
Numerical integration of classical spin systems
Presentation given 2022-10 at the UppASD Autumn School 2022 in Stockholm.
