Geometric numerical integration of semi-classical Hamiltonain lattice dynamics

Abstract

In this work we provide a brief overview of recently proposed symplecticity-preserving symmetric splitting methods for semi-classical Hamiltonian dynamics of charge transfer by intrinsic localized modes in nonlinear crystal lattice models [1]. Without loss of generality, we consider one-dimensional crystal lattice models described by classical Hamiltonian dynamics, whereas charge particle is modeled as quantum particle within the tight-binding approximation. Canonical Hamiltonian equations for the coupled lattice-charge dynamics are derived. Structurepreserving splitting methods are constructed by splitting the total Hamiltonian into the sum of Hamiltonians which individual dynamics can be solved exactly. Exactly charge conserving symplectic splitting methods are also proposed which require only one solution of a linear system of equations per time step. Developed computationally efficient non-dissipative methods provide new means for long-time simulations of charge transfer by nonlinear lattice excitations.