Symplectic integrators for spin systems

Abstract

We present a symplectic integrator, based on the implicit midpoint method, for classical spin systems where each spin is a unit vector in R 3 . Unlike splitting methods, it is defined for all Hamiltonians and is O(3)-equivariant, i.e., coordinate-independent. It is a rare example of a generating function for symplectic maps of a noncanonical phase space. It yields a new integrable discretization of the spinning top.Symplectic integrators for the computer simulation of Hamiltonian dynamics are widely used in computational physics [1,2]. For canonical Hamiltonian systems, with phase space R 2N and canonical coordinates (q i , p i ), simple and effective symplectic integrators are known. For noncanonical systems, like spin systems with phase space (S 2 ) N , some symplectic integrators are known. These are, however, either (i) based on local coordinates and not rotationally invariant, (ii) defined only for special Hamiltonians, or (iii) excessively complicated with many auxiliary variables. Here we solve the computational physics problem of providing a globally-defined, rotationally invariant, minimal-variable symplectic integrator for general spin systems. The method is surprisingly simple and depends only on the vector field of the system at hand. It is a rare example of a generating function for symplectic maps on a noncanonical phase space: a noncanonical analogue of the Poincaré generating function of classical mechanics. The method produces new discrete-time physical models, such as a new completely integrable discrete spinning top, and unveils new directions for symplectic integrators, discrete physics, and symplectic geometry.Classical spin systems are a class of noncanonical Hamiltonian systems with phase space (S 2 ) N and symplectic form the sum of the standard area elements on each sphere. If the spheres are realized as s i 2 = 1, s i ∈ R 3 , and H is the Hamiltonian on (S 2 ) N arbitrarily extended to (R 3 ) N , the equations of motion take the formṡ Our main result is a new integrator for (1) given byThis spherical midpoint method is globally defined and preserves many structural properties of the exact flow.Before explaining these properties we review symplectic integrators for canonical and noncanonical systems. Symplectic integrators for canonical Hamiltonian systems fall into two main classes: explicit methods, based on splitting the Hamiltonian into integrable terms and composing their flows, and implicit methods, typically based on generating functions. (Discrete Lagrangians can generate both types of method.) The leapfrog or Störmer-Verlet method, almost universally used in molecular dynamics, is an example of an explicit method, whereas the classical midpoint methodforż = F (z) with z ∈ R 2N , is an example of an implicit method. The classical midpoint method (3) has a number of striking features: (i) it is defined for all Hamiltonians in a uniform way (splitting methods are only defined for separable Hamiltonians); (ii) it conserves quadratic invariants; (iii) it is equivariant with respect t...

Hamiltonians and canonical coordinates for spinning particles in curved space-time

“…Most symplectic integrators require that the equations are formulated in terms of pairs of canonical coordinates, i.e. a colection of phase-space coordinates χ i , π i , with i some labelling index, such that {χ i , π j } = δ i j (however, there do exist symplectic integrators for special classes of systems that require no such coordinates [53,52]).…”

A minimal-variable symplectic method for isospectral flows

“…In [17] it is shown that also (15) coincides with (16) when B(·) = ∇H(·), for some Hamiltonian function H : R 3 → R constant on the rays (which implies B(·) to be orthogonal to the rays). In this case, (16) is known to be symplectic, whereas this fails for general Hamiltonian H. Therefore, (14) can be seen as the second order correction of (16) to be symplectic for any Hamiltonian H.…”

Accelerating spin-space sampling by auxiliary spin dynamics and temperature-dependent spin-cluster expansion

“…While it is very difficult or impossible to find an effective symplectic time-reversible integrator for the semiclassical spin dynamics 48,49 , this is not a problem for the auxiliary spin-dynamics. We combine the velocity Verlet method 50 and a spin rotation scheme to obtain an efficient numerical integration algorithm that is timereversible, area-preserving and preserves spin length.…”