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Hamiltonian dynamics of classical spins

Slobodan Radoševi\' c, Sonja Gombar, Milica Rutonjski, Petar Mali, Milan Panti\' c, Milica Pavkov-Hrvojevi\' c

Abstract

We discuss the geometry behind classical Heisenberg model at the level suitable for third or fourth year students who did not have the opportunity to take a course on differential geometry. The arguments presented here rely solely on elementary algebraic concepts such as vectors, dual vectors and tensors, as well as Hamiltonian equations and Poisson brackets in their simplest form. We derive Poisson brackets for classical spins, along with the corresponding equations of motion for classical Heisenberg model, starting from the geometry of two-sphere, thereby demonstrating the relevance of standard canonical procedure in the case of Heisenberg model.

Hamiltonian dynamics of classical spins

Abstract

We discuss the geometry behind classical Heisenberg model at the level suitable for third or fourth year students who did not have the opportunity to take a course on differential geometry. The arguments presented here rely solely on elementary algebraic concepts such as vectors, dual vectors and tensors, as well as Hamiltonian equations and Poisson brackets in their simplest form. We derive Poisson brackets for classical spins, along with the corresponding equations of motion for classical Heisenberg model, starting from the geometry of two-sphere, thereby demonstrating the relevance of standard canonical procedure in the case of Heisenberg model.

Paper Structure

This paper contains 18 sections, 114 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Vectors, dual vectors and tensors
  3. Vectors and dual vectors
  4. Tensors
  5. Metric tensor
  6. Linear operators
  7. Tensor fields
  8. Symplectic form on $\mathbb{R}^2$
  9. Hamiltonian dynamics on two-sphere ${\rm S}^2$
  10. Geometry of two-sphere
  11. A simple quadratic model on ${\rm S}^{2}$
  12. Classical Heisenberg model
  13. Poisson algebra of classical spins
  14. Hamiltonian and phase space
  15. Classical spin waves
  16. ...and 3 more sections

Figures (3)

  • Figure 1: Tangent plane to a sphere and two basis vectors $e_\varphi$ and $e_\theta$ at a point $\rm M \in S^2$.
  • Figure 2: Phase space trajectory $(\varphi(t), p(t) = \cos \theta(t))$ for a simple quadratic Hamiltonian \ref{['S2QuadrHam']} and two sets of initial conditions $(\varphi_0 = 0.5, \cos \theta_0 = 0)$ and $(\varphi_0 = 0.8, \cos \theta_0 = 0)$.
  • Figure 3: Time evolution of phase space variables $\varphi(t), \theta(t)$ and $p(t) = \cos \theta (t)$ for initial conditions $(\varphi_0 = 0.5, \cos \theta_0 = 0)$.