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arXiv:1408.5323 · Physical Review A

Topological flat bands in optical checkerboardlike lattices

Tomi Paananen, T. Dahm

Published Mar 4, 2015• 11 citations
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Abstract

We present comparatively simple two-dimensional and three-dimensional checkerboardlike optical lattices possessing nontrivial topological properties accompanied by topological surface states. By simple tuning of the parameters, these lattices can have a topological insulating phase, a topological semimetallic phase, or a trivial insulating phase. This allows the study of different topological phase transitions within a single cold-atom system. In the topologically nontrivial phases, flat bands appear at the surfaces of the system. These surface states possess short localization lengths such that they are observable even in systems with small lattice dimensions.

DOI
10.1103/physreva.91.033604

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We present comparatively simple two-dimensional and three-dimensional checkerboardlike optical lattices possessing nontrivial topological properties accompanied by topological surface states. By simple tuning of the parameters, these lattices can have a topological insulating phase, a topological semimetallic phase, or a trivial insulating phase. This allows the study of different topological phase transitions within a single cold-atom system. In the topologically nontrivial phases, flat bands appear at the surfaces of the system. These surface states possess short localization lengths such that they are observable even in systems with small lattice dimensions.
We propose a feasible experimental scheme to realize a three-dimensional chiral topological insulator with cold fermionic atoms in an optical lattice, which is characterized by an integer topological invariant distinct from the conventional Z(2) topological insulators and has a remarkable macroscopic zero-energy flat band. To probe its property, we show that its characteristic surface states--the Dirac cones--can be probed through time-of-flight imaging or Bragg spectroscopy and the flat band can be detected via measurement of the atomic density profile in a weak global trap. The realization of this novel topological phase with a flat band in an optical lattice will provide a unique experimental platform to study the interplay between interaction and topology and open new avenues for application of topological states.
Weyl semimetals are three-dimensional, topologically protected, gapless phases which show exotic phenomena such as Fermi arc surface states or negative magnetoresistance. It is an open question whether interparticle interactions can turn the topological semimetal into a topologically nontrivial Mott-insulating phase. We investigate an experimentally motivated model for Weyl physics of cold atoms in optical lattices, with the main focus on interaction effects and topological properties, by means of dynamical mean-field theory. We characterize topological phases by numerically evaluating the Chern number via the Ishikawa-Matsuyama formula for interacting phases. Within our studies, we find that the Chern numbers become trivial when interactions lead to insulating behavior. For a deeper understanding of the Weyl-semimetal-to-Mott-insulator topological phase transition, we evaluate the topological properties of quasiparticle bands as well as so-called blind bands. Finally, we consider a system with an open boundary along one spatial direction in order to study correlation effects of surface states. In a narrow regime close to the topological phase transition, we find a correlation-induced state in which the surface becomes metallic while the bulk is semimetallic.
Topological states of matter are characterized by global topological invariants which change their value across a topological quantum phase transition. It is commonly assumed that the transition between topologically distinct noninteracting gapped phases of fermions is necessarily accompanied by the closing of the band gap as long as the symmetries of the system are maintained. We show that such a quantum phase transition is possible without closing the gap in the case of a three-dimensional topological band insulator. We demonstrate this by calculating the free energy of the minimal model for a topological insulator, the Bernevig-Hughes-Zhang model, and show that as the band curvature continuously varies, a jump between the band-gap minima corresponding to the topologically trivial and nontrivial insulators occurs. Therefore, this first-order phase transition is a generic feature of three-dimensional topological band insulators. For a certain parameter range we predict a reentrant topological phase transition. We discuss our findings in connection with the recent experimental observation of a discontinuous topological phase transition in a family of topological crystalline insulators.
We study a tight-binding model on the two-dimensional star lattice (also called the decorated honeycomb lattice) with short-range hoppings modulated by staggered fluxes. We observe a series of topological quantum phase transitions when tuning the hopping parameters, and find nontrivial topological bands with high Chern numbers. We also find in some parameter regions, the system exhibits the interesting topological flat bands with large flatness ratios of the band gaps over the bandwidths. We further illustrate the 1/21/21/2 fractional quantum Hall effect of interacting hard-core bosons filled in a lowest topological flat band of the star lattice.
Mathematical tools and concepts initially developed to study topological properties of geometrical objects have allowed for better understanding and interpretation of a number of phenomena in condensed matter physics, from the quantum Hall effect to topological insulators. It would be interesting to realize analogs of these phenomena with light scattered by cold atoms because atomic systems are simpler and allow for better control than complex solid-state compounds in which electrons interact with each other and with phonons, and where disorder is omnipresent. At the same time, mastering topological photonics of cold atoms may lead to interesting applications in quantum information science, for example. We study the opening of topological band gaps in two-dimensional honeycomb lattices of two-level atoms [1], the impact of disorder on their properties [2], as well as the constructive role that disorder can play by inducing a topologically nontrivial phase known as “topological Anderson insulator” [2,3]. Modifying the structure of the atomic lattice in an appropriate way allows for constructing a synthetic spin degree of freedom and thus for exploring the physics of spinful particles, such as the quantum spin Hall effect, for example. The analogy between topological phenomena in solid-state physics and in optics of cold atoms is not complete: atomic systems exhibit a number of new features due to the vector nature of light and to the differences in the impact of disorder on their properties.1. P. Wulles and S.E. Skipetrov, Topological photonic band gaps in honeycomb atomic arrays, arXiv:2308.134232. S.E. Skipetrov and P. Wulles, Topological transitions and Anderson localization of light in disordered atomic arrays, Phys. Rev. A 105, 043514 (2022)3. S.E. Skipetrov and P. Wulles, Photonic topological Anderson insulator in a two-dimensional atomic lattice, Comptes Rendus. Physique 24(S3), 39–54 (2023)
We propose to simulate and detect quantum anomalous Hall phase with ultracold atoms in a one-dimensional optical lattice, with the other synthetic dimension being realized by modulating spin-orbit coupling. We show that the system manifests a topologically nontrivial phase with two chiral edge states which can be readily detected in this synthetic two-dimensional system. Moreover, it is interesting that at the phase transition point there is a flat energy band and this system can also be in a topologically nontrivial phase with two Fermi zero modes existing at the boundaries by considering the synthetic dimension as a modulated parameter. We also show how to measure these topological phases experimentally in ultracold atoms. Another model with a random Rashba and Dresselhaus spin-orbit coupling strength is also found to exhibit topological nontrivial phase, and the impact of the disorder to the system is revealed.