Dynamical behaviour of the mass tensor in the lattice

Abstract

We study with the semiclassical model the dynamics of a particle moving in a orthorrombic lattice by the hopping mechanism in the presence of a static Coulomb field and a rapidly oscillating homogeneous field. Based on results for the anisotropic Kepler problem, where a chaotic trajectory arises due to different masses of the particle in the X and Y directions, and since this result is also obtained in the rectangular lattice at the continuum limit, the application of the oscillating field allows reparameterizing the effective masses so that they become equal in both directions. Thus, the result is an elliptic trajectory in the continuum limit of the lattice. This chaos-order transition is confirmed by the calculation of Lyapunov exponents for the different trajectories. It is also found that the application of an external magnetic field in the Z direction of the lattice yields chaos control which can be generalized beyond the simple case of the circular trajectory shown in this work. The previous scheme of chaos control was extended for trajectories in 3D in the continuum limit of the orthorrombic lattice which is also equivalent to the far region approximation where the particle moves with respect to the Coulomb field origin. In this case it found that the chaos control technique permits to observe two different dynamical regimes which, although consistent with Bohr-van Leeuwen theorem, suggests that an observable way to distinguish one from the other is to be sought.

The paper investigates whether a rapidly oscillating electric field can suppress chaotic semiclassical motion of a charged particle hopping on an orthorhombic lattice around a Coulomb impurity. The mechanism is to tune anisotropic effective masses until the continuum dynamics reduces to the isotropic Kepler problem. Starting from a nearest-neighbour tight-binding dispersion, high-frequency time averaging gives effective hoppings $\tilde A=AJ_0(\alpha)$ and $\tilde B=BJ_0(\beta)$, where $\alpha$ and $\beta$ encode the drive components. In the continuum limit, the effective mass ratio becomes $\tilde m_x/\tilde m_y=\xi J_0(\beta)/J_0(\alpha)$. For the studied anisotropy $\xi=2.94$, choosing $J_0(\alpha)/J_0(\beta)=2.94$ restores equal masses and elliptical trajectories. Numerical trajectories, angular-momentum conservation and reduced finite-time Lyapunov estimates support this transition; one example uses $\alpha=2$ and $\beta\approx2.262$. A simulation of the explicitly driven dynamics checks the averaging approximation at a drive-to-orbit frequency ratio of 2314. Away from the continuum limit, a heuristic magnetic-field extension is tested numerically for circular motion. Three-dimensional simulations with 1000 noninteracting particles compare chaotic near-impurity motion with approximately elliptical motion farther away, finding different magnetic-field time traces but approximately zero mean field at the impurity. The treatment is semiclassical rather than fully quantum. The Lyapunov algorithm does not produce nonpositive exponents for the elliptical cases. Finite-lattice control beyond circular trajectories and an experimentally observable distinction between the three-dimensional regimes are left open.