Designing Topological Bands in Reciprocal Space

Motivated by new capabilities to realize artificial gauge fields in ultracold atomic systems, and by their potential to access correlated topological phases in lattice systems, we present a new strategy for designing topologically nontrivial band structures. Our approach is simple and direct: it amounts to considering tight-binding models directly in reciprocal space. These models naturally cause atoms to experience highly uniform magnetic flux density and lead to topological bands with very narrow dispersion, without fine-tuning of parameters. Further, our construction immediately yields instances of optical Chern lattices, as well as band structures with Chern numbers of magnitude larger than one.

Figure 1

One primitive unit cell of the reciprocal-space tight-binding model corresponding to the triangular lattice optical flux lattice with $N$ internal states [Eq. (3)], showing the optical couplings $-\mathcal{V}{e}^{i\varphi }$. The anticlockwise sum of the phases $\varphi$ around any one of the triangular plaquettes is $\pi /N$ modulo $2\pi$.

Figure 2

Spatial dependence of (a) the energy and (b) the magnetic flux density of the lowest energy dressed state of the triangular optical flux lattice [Eq. (3)] for $N=4$, within a unit cell containing $N_{\varphi }=1$ flux quantum. Both are highly uniform, becoming increasingly so as $N\to \infty$.

Figure 3

Low energy density of states for (a) the triangular optical flux lattice [Eq. (3)] with $N=4$ internal states and $\mathcal{V}=E_{\mathrm{R}}$; the spectrum closely reproduces the Landau level spectrum, all bands shown having a Chern number $\mathcal{C}=1$. (b) A related model (still with $N=4$ and $\mathcal{V}=E_{\mathrm{R}}$) in which the phases on the optical couplings are modified as described in the text to form a lowest energy band with a Chern number $\mathcal{C}=2$.