Topological states in two-dimensional optical lattices

📝 Abstract

We present a general analysis of two-dimensional optical lattice models that give rise to topologically non-trivial insulating states. We identify the main ingredients of the lattice models that are responsible for the non-trivial topological character and argue that such states can be realized within a large family of realistic optical lattice Hamiltonians with cold atoms. We focus our quantitative analysis on the properties of topological states with broken time-reversal symmetry specific to cold-atom settings. In particular, we analyze finite-size effects, multi-orbital phenomena that give rise to a variety of distinct topological states and transitions between them, the dependence on the trap geometry, and most importantly, the behavior of the edge states for different types of soft and hard boundaries. Furthermore, we demonstrate the possibility of experimentally detecting the topological states through light Bragg scattering of the edge and bulk states.

💡 Analysis

We present a general analysis of two-dimensional optical lattice models that give rise to topologically non-trivial insulating states. We identify the main ingredients of the lattice models that are responsible for the non-trivial topological character and argue that such states can be realized within a large family of realistic optical lattice Hamiltonians with cold atoms. We focus our quantitative analysis on the properties of topological states with broken time-reversal symmetry specific to cold-atom settings. In particular, we analyze finite-size effects, multi-orbital phenomena that give rise to a variety of distinct topological states and transitions between them, the dependence on the trap geometry, and most importantly, the behavior of the edge states for different types of soft and hard boundaries. Furthermore, we demonstrate the possibility of experimentally detecting the topological states through light Bragg scattering of the edge and bulk states.

📄 Content

Topological states in two-dimensional optical lattices Tudor D. Stanescu,1, 2 Victor Galitski,1 and S. Das Sarma1 1Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA 2Department of Physics, West Virginia University, Morgantown, West Virginia 26506, USA We present a general analysis of two-dimensional optical lattice models that give rise to topologically non-trivial insulating states. We identify the main ingredients of the lattice models that are responsible for the non-trivial topological character and argue that such states can be realized within a large family of realistic optical lattice Hamiltonians with cold atoms. We focus our quantitative analysis on the properties of topological states with broken time-reversal symmetry specific to cold-atom settings. In particular, we analyze finite-size effects, multi-orbital phenomena that give rise to a variety of distinct topological states and transitions between them, the dependence on the trap geometry, and most importantly, the behavior of the edge states for different types of soft and hard boundaries. Furthermore, we demonstrate the possibility of experimentally detecting the topological states through light Bragg scattering of the edge and bulk states. I. INTRODUCTION It has been shown recently that band structures of non-interacting lattice models and quadratic mean-field Hamiltonians can be classified according to the topolog- ical character of the wave functions associated with the bands. The most complete classification of this type of Hamiltonians in the physically relevant two and three dimensions was recently presented by Kitaev [1] and by Ryu et al. [2] who identified all distinct topological classes, which differ sharply depending on the presence or absence of particle-hole symmetry and time-reversal symmetry. The general interacting case was addressed by Volovik [3] using the Green function, rather than the Hamiltonian, as the object for the topological classifica- tion [4, 5]. With this understanding achieved, a question appears on how to realize various such topological states in physical systems. Until now a few promising solid- state materials have been identified that are expected to host certain topological phases. However, in solid-state settings, we are bound to work with the existing com- pounds provided by nature and we have no choice but to rely on serendipity in our search for physical realiza- tions of topological states, rather than on a controlled ”engineering” of appropriate lattice Hamiltonians that are guaranteed to host these exotic phases. On the other hand, optical lattices populated with cold atoms offer a very promising alternative avenue to build topological insulating states. Cold-atom systems pro- vide more control in constructing specific optical lattice Hamiltonians by allowing both tunable hoppings and interparticle interactions that can be adjusted as needed, hence opening the possibility of accessing interacting topological states such as topological Mott insulators. However, cold-atom settings bring in their own spe- cific challenges associated with the trapping potential, the effective vector potential responsible for the nontriv- ial topological properties, the soft boundaries, and also with the fact that cold atom experiments involve neutral particles and therefore make any transport measurement irrelevant or very difficult, thus bringing up the question of how to probe experimentally the topological character of these phases. Motivated by the opportunity of creat- ing topological insulating states with cold atoms and by the aforementioned challenges, we discuss in this arti- cle a general prescription for building certain types of topological optical lattice models and analyze in detail the properties of the emergent states in the presence of trapping potentials with different geometries. Until the discovery in the early 1980s of the quantum Hall effect [6, 7], the standard way of classifying quan- tum states of condensed matter systems was to consider the symmetries they break. The existence of extremely robust properties, such as the quantized Hall conduc- tance, was found to be linked to the nontrivial topologi- cal structure of the quantum Hall states. These states do not break any symmetry, hence cannot be described by the Landau symmetry breaking theory [8], but possess a more subtle organizational structure sometimes called topological order [9]. In two-dimensional systems, such as the quantum Hall fluids, the nontrivial topological structure is intrinsically connected with the existence of robust gapless edge modes. In the three-dimensional case it leads to robust gapless surface or interface modes, such as the interface midgap states in heterojunctions composed of semiconductors with opposite band-edge symmetry [10, 11]. In recent years a significant number of different models and solid state systems with topo- logically order

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