Non-Hermitian Band Topology and Edge States in Atomic Lattices

TL;DR

The paper develops a non-Hermitian topological framework for bipartite atomic lattices with long-range radiative coupling, deriving an effective two-band Hamiltonian in the single-excitation subspace. It reveals Dirac-like low-energy dynamics governed by a complex Fermi velocity $v_F$ and analyzes topological invariants via winding numbers in 1D and Chern numbers in 2D, including synthetic gauge-field breaking of time-reversal symmetry. The authors demonstrate bulk-edge correspondence through explicit edge-state solutions at domain walls in both the SSH and honeycomb models, and provide numerically stable lattice-sum methods using theta-function transforms and Ewald summation. The results show that Dirac physics and topological edge modes persist under non-Hermiticity and long-range coupling as long as lattice symmetry is preserved, with potential applications to waveguide QED and circuit QED platforms.

Abstract

We investigate the band structure and topological phases of one- and two-dimensional bipartite atomic lattices mediated by long-range dissipative radiative coupling. By deriving an effective non-Hermitian Hamiltonian for the single-excitation sector, we demonstrate that the low-energy dynamics of the system are governed by a Dirac equation with a complex Fermi velocity. We analyze the associated topological invariants for both the SSH and honeycomb models, utilizing synthetic gauge fields to break time-reversal symmetry in the latter. Finally, we explicitly verify the non-Hermitian bulk-edge correspondence by deriving analytical solutions for edge states localized at domain boundaries.

Figures (8)

• Figure 1: Lattice geometry and basis definitions. (a) Schematic of the Su-Schrieffer-Heeger (SSH) chain and (b) the 2D honeycomb lattice. The accompanying table summarizes the real-space basis vectors $\mathbf{a}_i$ and the corresponding reciprocal-space basis vectors $\mathbf{b}_j$ used for the honeycomb geometry. The internal displacement between $A$ and $B$ sublattices within a single unit cell is defined by the vector $\bm{\delta} = (-1/\sqrt{3}, 0)$ (purple arrow). Note that the basis vectors satisfy the orthonormality condition $\mathbf{a}_{i} \cdot \mathbf{b}_{j} = \delta_{ij}$.
• Figure 2: Band structure of the effective non-Hermitian SSH model for various values of the intracell separation $\delta$. Throughout this figure (and in all subsequent figures) we fix $\alpha_{A} = \alpha_{B} = 2.4$ and $\kappa_{A} = \kappa_{B} = 0.01$. Panel (a): $\delta = 0.2$; (b): $\delta = 0.4$; (c): $\delta = 0.5$; (d): $\delta = 0.8$. In each subfigure, the upper panel shows the real part of the band structure, while the lower panel shows the corresponding imaginary part.
• Figure 3: Winding number and schematic plots of the complex function $r(\beta) = h_{1}(\beta) + \mathrm{i} h_{2}(\beta)$ in the complex plane. (a) Winding number as a function of $\delta$; (b)–(g) Trajectories of $r(\beta)$ for various values of $\delta$: (b) $\delta = 0.2$; (c) $\delta = 0.35$; (d) $\delta = 0.4$; (e) $\delta = 0.5$; (f) $\delta = 0.65$; (g) $\delta = 0.8$. The red dot indicates the origin in the complex plane.
• Figure 4: Edge state of the effective non-Hermitian SSH model. The intracell atomic separation varies smoothly according to $\delta(x) = 0.02 \tanh(x) + 0.5$. All other parameters are the same as in previous figures. The boundary condition is set as $\psi_{A}(0) = 1$ and $\psi_{B}(0) = 0$. (a) Spatial profile of the intracell separation $\delta(x)$; (b) Corresponding winding number as a function of position; (c) The edge state $\psi_{A}(x)$ is localized near the boundary at $x = 0$, while $\psi_{B}(x)$ remains zero throughout the system.
• Figure 5: Band structure of the non-Hermitian honeycomb lattice. (a) Real (upper panel) and imaginary (lower panel) parts of the band structure without a synthetic gauge field, clearly showing the formation of Dirac cones at the $K$ and $K^{\prime}$ points; (b) band structure with a synthetic gauge field, using $t_{2} = 5 \times 10^{-3}$ and $\phi = \pi / 2$; (c) band structure along the high-symmetry path $\Gamma \rightarrow \Sigma \rightarrow M \rightarrow K \rightarrow \Lambda$ in the Brillouin zone. Solid and dashed lines correspond to cases with and without the gauge field, respectively. In all panels, the band energies are shifted by $h_{0}$, the energy of the vacuum state.