Evolution of the hofstadter butterfly in a tunable optical lattice

Abstract

There are a limited number of exact solutions for quantum mechanical systems. It is critical to obtain solutions for complex systems. One of these unsolved equations was the famous Harpers equation, which was proposed in 1955[1]. It investigates the behavior of a particle in a periodic potential under a uniform magnetic field in two dimensions. Douglas Hofstadter, in 1976, obtained a numerical solution[2] for the first time, discovering a non-trivial energy spectrum as a function of magnetic ux. The spectrum is a fractal structure, the Hofstadter buttery, and depends purely on the lattice geometry. In other words, primitive lattice vectors and basis vectors determine the fractal energy spectrum under a uniform magnetic field. The experimental demonstration of such an energy spectrum requires a magnitude of thousands of Teslas magnetic field in the solid state systems since the area of a unit cell is on the order of a few square nanometers. Recently, two main developments in cold atom physics led the way to the realization of the Hofstadter buttery energy spectrum. The first one as the creation and manipulation of optical lattices. It provides a controllable environment with lattice constants up to a few hundred nanometers, which means the required magnetic field is now within experimental capabilities. The second development is the realization of synthetic gauge fields on optical lattices. One recent development we focus in this thesis is the creation of an adjustable lattice geometry[3]. The self-similar energy spectra for a uniform magnetic field depends purely on the lattice geometry. Recently, the Zurich group presented a unique chance to examine the connection between them. Particularly, we calculate the Hofstadter buttery for all lattice parameters which can be obtained by the Zurich group[3]. We then investigate the transition of the Hofstadter buttery from a checkerboard lattice to a honeycomb lattice, which includes the observation of the change in topological invariants, the Chern numbers of the self-similar energy spectra. For this purpose, we first present the theoretical building blocks utilized throughout the research. We show the step-by-step procedure to obtain the Hofstadter buttery, starting from the continuous Hamiltonian and projection onto a tight-binding Hamiltonian. We explicitly demonstrate the butteries for the square lattice and the honeycomb lattice. Next, we concentrate on the experiment carried out by the Zurich group, and obtain the Hofstadter butteries for all lattice geometries. The Hofstadter butteries are analysed in detail. There are three different regimes. In the first regime the spectrum is formed by two stacked square lattice Hofstadter butteries separated by a large energy gap. As the optical lattice evolves from the checkerboard to the honeycomb geometry, the second regime begins with the emergence of Dirac points for particular rational magnetic ux values _ = p=q, where p; q are mutually prime integers. In the third regime infinitely many sequential closings of adjacent bands around zero energy give the honeycomb lattice Hofstadter buttery as a limit. This closing process can be probed with current setups. We show that the existence of Dirac points at zero magnetic field does not imply its existence at a finite field. The topological properties of the energy spectrum can change with the applied magnetic field. We calculate the Chern numbers of the major gaps in the spectra and examine the exchange and the transfer of these topological invariants during the evolution of the lattice geometry. An analytic formula to determine the critical value for the emergence of Dirac points around zero energy is obtained in Eq.5.2