This is a symplectic midpoint solver for spin systems. These systems are symplectic differential equations derived from the Lie–Poisson structure of a product of spheres. The papers "Symplectic integrators for spin systems", "A minimal-coordinate symplectic integrator on spheres" and "Geometry of discrete-time spin systems" gives the details of the setting and the method, as well as a theoretical background.
The method and the implementation are a joint work of R. I. McLachlan, K. Modin and O. Verdier.
m0: An initial condition: an Nx3 matrix where each row is a vector of length one.gradient: A gradient which returns the gradient of a Hamiltonian at each such points.
You may also optionally need
strengths: A N vector of strength which determines the symplectic form on the product of spheres.
If such a strenghts vector is not provided, the strengths are assumed to be one for every sphere.
The code is then as follows:
import midpoint
import sphere
dt = .01 # time step
nb_steps = 10 # number of steps
vector = sphere.radially_constant(gradient, strengths)
generator = midpoint.run(midpoint.get_increment(vector), m0=m0.ravel(), dt=dt, nb_steps=nb_steps)
# run the solver
ms = [m.reshape(-1,3) for m in generator]Now the solution ms is a list of Nx3 matrices which correspond to the solutions at each time step.
There are two common spin systems which are already available in this package.
The Heisenberg spin chain system for N points is initialized as follows:
from spinsys import spinchain
gradient = spinchain.get_gradient(1/N**2)The strength vector may be omitted in the call of radially_constant:
vector = sphere.radially_constant(gradient)The gradient is obtained from the strengths vector:
from spinsys import vortex
import numpy as np
strengths = np.array([1., 1., -1.])
gradient = vortex.get_gradient(strengths)