Topological bands for ultracold atoms

Abstract

There have been significant recent advances in realizing band structures with geometrical and topological features in experiments on cold atomic gases. We provide an overview of these developments, beginning with a summary of the key concepts of geometry and topology for Bloch bands. We describe the different methods that have been used to generate these novel band structures for cold atoms, as well as the physical observables that have allowed their characterization. We focus on the physical principles that underlie the different experimental approaches, providing a conceptual framework within which to view these developments. However, we also describe how specific experimental implementations can influence physical properties. Moving beyond single-particle effects, we describe the forms of inter-particle interactions that emerge when atoms are subjected to these energy bands, and some of the many-body phases that may be sought in future experiments. 2 VI. Outlook 40 A. Turning to atomic species from the Lanthanide family 40 B. Topological lattices without light 40 C. Other topological insulators and topological metals 41 D. Far-from-equilibrium dynamics 42 E. Invariants in Floquet-Bloch systems 43 F. Open systems 45 VII. Summary 47 Acknowledgments 47 A. Topological bands in one dimension 47 1. Edges states in the SSH model 47 2. Gauge invariance and Zak phase 48 3. Time-reversal symmetry of the SSH model48 4. Adiabatic pumping for the Rice-Mele model 48 5. The Kitaev model for topological superconductors 49 B. Floquet systems and the Magnus expansion 50 C. Light matter interaction 52 D. Berry curvature and unit cell geometry 53 References 54 B A B J J J J J J J J J J J J J

Creutz ladder in a resonantly shaken 1D optical lattice

“…Motivated further by their novel transport properties, the study of topological phases constitutes one of the frontiers in modern condensed matter physics. Ultracold atoms in optical lattices, featuring tunneling amplitude engineering and tunable interaction strength, provide a unique platform for realizing and exploring such exotic topological states [3,4]. Along with the steady development of experimental techniques, many topological model systems have been recently realized, which include the Harper-Hofstadter Hamiltonian in 2D rectangular lattices [5,6], the Haldane model in a 2D hexagonal lattice [7], and various Hall and topological ladder systems based on additional synthetic dimensions such as internal atomic states [8][9][10][11][12][13] and lattice orbital states [14].…”

Topological magnon insulator and quantized pumps from strongly-interacting bosons in optical superlattices

“…Using current ultracold-atom technologies [13,14], one can modify the optical-lattice potential so as to modulate the parameters J , As for the standard RM model [10], one can construct a two-dimensional (artificial) Brillouin zone spanned by the momentum k 0,…”

“…It exhibits a variety of topological features, including topological solitons [4], fractional charges [5], quantized Zak phases [6], degenerate zero-energy topological edge states [7], and a characteristic 'chiral' displacement [8]. Adding staggered on-site energies to this model yields the Rice-Mele (RM) model [9], which realizes a quantized (Thouless) pump upon varying its parameters in a cyclic and adiabatic manner [10,11]. The SSH and RM models have both been experimentally realized using non-interacting single-component ultracold bosonic atoms trapped in an optical superlattice [12], where quantized and fractional Zak phases were measured.…”

“…in view of realizing a closed loop in parameter space[10,11]. To this purpose, we write the spin-exchange couplings as J where θ defines a dynamical parameter.…”

High-degeneracy points protected by site-permutation symmetries

“…Furthermore, lattices fulfilling the minimum in-gredients to the HD points, and robust against long range inter-sites couplings, have been found. The generality of the presented discussion allows for an interpretation in the context of different quasiparticle systems: electronic, photonic, phononic, magnonic, and cold-atoms [29][30][31][32]. Here, we focus on electronic spinless tight-binding (TB) models at the Γ point (k = 0) to show that pseudospin-1 and -2 Dirac fermions emerge in the HD points.…”