Hofstadter Butterfly Evolution in the Space of Two-Dimensional Bravais Lattices

Abstract

The self-similar energy spectrum of a particle in a periodic potential under a magnetic field, known as the Hofstadter butterfly, is determined by the lattice geometry as well as the external field. Recent realizations of artificial gauge fields and adjustable optical lattices in cold atom experiments necessitate the consideration of these self-similar spectra for the most general two-dimensional lattice. In a previous work, we investigated the evolution of the spectrum for an experimentally realized lattice which was tuned by changing the unit cell structure but keeping the square Bravais lattice fixed. We now consider all possible Bravais lattices in two dimensions and investigate the structure of the Hofstadter butterfly as the lattice is deformed between lattices with different point symmetry groups. We model the optical lattice by a sinusoidal real space potential and obtain the tight binding model for any lattice geometry by calculating the Wannier functions. We introduce the magnetic field via Peierls substitution and numerically calculate the energy spectrum. The transition between the two most symmetric lattices, i.e. the triangular and the square lattice displays the importance of bipartite symmetry featuring deformation as well as closing of some of the major energy gaps. The transition from the square to rectangular and from the triangular to centered rectangular lattices are analyzed in terms of coupling of one-dimensional chains. We calculate the Chern numbers of the major gaps and Chern number transfer between bands during the transitions. We use gap Chern numbers to identify distinct topological regions in the space of Bravais lattices.

arXiv comment11 pages, 11 figures

References in corpus (18)

• Topological Photonics
• Experimental realisation of the topological Haldane model
• Hofstadter's butterfly in moire superlattices: A fractal quantum Hall effect
• Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal
• Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices
• Analogs of quantum Hall effect edge states in photonic crystals
• Realizing the Harper Hamiltonian with Laser-Assisted Tunneling in Optical Lattices
• Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms
• Creating, moving and merging Dirac points with a Fermi gas in a tunable honeycomb lattice
• Creation of effective magnetic fields in optical lattices: The Hofstadter butterfly for cold neutral atoms
• Chiral groundstate currents of interacting photons in a synthetic magnetic field
• The Fractional Quantum Hall effect in an array of quantum wires
• Measuring Topological Number of a Chern-Insulator from Quench Dynamics
• Trapped Fermi Gases in Rotating Optical Lattices: Realization and Detection of the Topological Hofstadter Insulator
• Measurement of Chern numbers through center-of-mass responses
• From the Hofstadter to the Fibonacci butterfly
• Evolution of the Hofstadter butterfly in a tunable optical lattice
• P-band in a rotating optical lattice

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• Floquet engineering the Hofstadter butterfly in the square lattice and its effective Hamiltonian
• Algebraic area enumeration of random walks on the honeycomb lattice