Topological properties of a dense atomic lattice gas

We investigate the existence of topological phases in a dense two-dimensional atomic lattice gas. The coupling of the atoms to the radiation field gives rise to dissipation and a nontrivial coherent long-range exchange interaction whose form goes beyond a simple power law. The far-field terms of the potential—which are particularly relevant for atomic separations comparable to the atomic transition wavelength—can give rise to energy spectra with one-sided divergences in the Brillouin zone. The long-ranged character of the interactions has another important consequence: it can break the standard bulk-boundary relation in topological insulators. We show that topological properties such as the transport of an excitation along the edge of the lattice are robust with respect to the presence of lattice defects and dissipation. The latter is of particular relevance as dissipation and coherent interactions are inevitably connected in our setting.

Figure 1

(a) Internal atomic level structure. Time-reversal symmetry is broken via a magnetic field of strength $\mathrm{\Delta }/{\mu }_B$ perpendicular to the lattice and the off-resonant coupling of $\left|+\right.〉$ to an auxiliary state $|2\rangle$ with $\varepsilon \approx \mathrm{\Omega }/2\delta \ll 1$. (b) Hamiltonian coefficients $A_r$ [Eq. (6), solid line] and $B_r$ [Eq. (7), dashed line] vs $r/\lambda$. (c) and (d) Real-space configurations and first Brillouin zone for SL and HL, respectively. The red circle represents the points of divergence and discontinuity of the spectrum for $a/\lambda =0.1$. (e) and (f) Dispersion relations for the SL and the HL, respectively, which exhibit the existence of a one-sided divergence at $\left|\mathbf{k}\right|=2\pi /\lambda$.

Figure 2

(a) Phase diagram in the $\mathrm{\Delta }-\varepsilon$ plane for the SL. (b) Phase diagram as a function of $\mathrm{\Delta }$ for the HL. The numbers in the square brackets indicate the Chern numbers and are ordered from the lowest to highest energy bands. (c) and (d) Spectrum in the infinite strip geometry for the SL $[N_y=15$ atoms, $\varepsilon =0.17, \mathrm{\Delta }/\mathrm{\Gamma }=-0.05$, red cross in (a)] and HL $[N_y=20$ atoms, $\mathrm{\Delta }/\mathrm{\Gamma }=-5$, bearded zigzag boundary conditions [71], red line in (b)]. Note that with these boundary conditions $b=\sqrt{3}a$ denotes the spacing between neighboring unit lattice sites in the $y$ direction. The size of the points indicate the weight of the given eigenmode on one (red) or the other (blue) edge of the strip (schematically shown on the right).

Figure 3

(a) Energy vs quasimomentum of a finite HL of hexagonal shape with the total of 486 sites (dots) together with the spectrum of the infinite strip geometry [gray lines, same parameters as in Fig. 2]. The blue (red) dots represent states with less (more) than 95% spatial support on the edge of the lattice (see text for details). (i)–(iii) show the spatial configuration of the indicated eigenstates. (b) and (c) show the excitation dynamics (direction of travel given by the arrows) and the steady state (with defects and dissipation) under the driving of a single atom on the edge (indicated by the star in the leftmost panels) with detuning $-2\mathrm{\Gamma }$ and $-10\mathrm{\Gamma }$, respectively.