Quasicrystalline Analogue of the Haldane Model

TL;DR

This work proposes a topological quasicrystal model realizable in cold-atom systems, formulated in reciprocal space within an optical-flux-lattice framework. It shows that symmetry-protected Dirac cones in the weak-coupling limit gap out under a time-reversal-symmetry-breaking perturbation, yielding a Chern insulator with $ obreak \\mathcal{C}=1$, analogous to the Haldane model but in a quasiperiodic lattice. Approximant analyses confirm a robust topological phase across parameter space and reveal a direct link between QBZ area and the number of states below the gap, while also uncovering narrow Chern bands that may host strongly-correlated physics. The results motivate experimental exploration with two-photon Raman schemes in cold atoms and raise theoretical questions about localization and interaction-driven states in topological quasicrystals.

Abstract

We present a model for a topological quasicrystalline system which is suitable for realisation in cold-atom experiments. We define the model in terms of complex momentum-space couplings which break time-reversal symmetry (TRS), and detail how it may be experimentally realised using two-photon Raman couplings. In the weak-potential limit, we study the model analytically by calculating the bandstructure over a `quasi-Brillouin zone' (QBZ). We find symmetry-protected Dirac cones, which are gapped by a TRS-breaking term, resulting in a Chern number $\mathcal{C}=1$. This provides a direct analogy to the Haldane model, but now in a quasicrystalline setting. We also infer the number of states below the topological gap from the QBZ area. We verify our analysis with numerical calculations of periodic approximants to our system, constructing a phase diagram in parameter space which shows a topological region extending beyond the weak-potential regime. We also find examples of narrow Chern bands with the potential for hosting strongly-correlated physics. Our work raises questions about the nature of localisation and strongly-correlated states in Chern bands in quasiperiodic systems.

Figures (11)

• Figure 1: Couplings in momentum space for (a) the "dual Haldane model" and (b) our quasiperiodic model. Upon hopping anticlockwise around a triangular plaquette, the atom acquires a phase $\Phi$ from the complex coupling elements. The quasi-Brillouin zone (see Sec. \ref{['Sec:PlaneWave']}) is shown dotted in black.
• Figure 2: (a) Laser wavevectors and electric field directions. The $\boldsymbol{\epsilon}_l^{\sigma +}$ (light blue) and $\boldsymbol{\epsilon}_l^{\pi}$ (yellow) lasers propagate in the $z$-direction, out of the plane. The circularly-polarised $\boldsymbol{\epsilon}_l^{\sigma +}$ electric fields (not depicted) rotate in the $xy$-plane; all other lasers are linearly polarised along the directions indicated. (b) Illustration of the energy levels and transitions contributing to the Hamiltonian. The coloured arrows indicate absorption or emission of a photon from the corresponding laser in (a). The $\boldsymbol{\epsilon}_l^{\sigma +}$/$\boldsymbol{\epsilon}_l^{\mathbf{G}}$ (light/dark blue) process generates the $H_{U}$ terms, and the $\boldsymbol{\epsilon}_l^{\pi}$/$\boldsymbol{\epsilon}_l^{\mathbf{g}}$ (yellow/red) process generates the $H_{V}$ terms.
• Figure 3: Couplings of corners in the QBZ. The $H_U$ and $H_V$couplings are shown on different diagrams (a) and (b) for clarity. The set of 8 states with opposite spin to those indicated here are coupled by different $U_l$ and $V_l$ terms; there is no coupling between the two sets.
• Figure 4: (a) Bandstructure across the QBZ, calculated in the basis of the 16 plane-wave states degenerate at the $\text{K}$ point, for $U/E_R=0.03,\,V=W=0$. The path in the QBZ is shown inset. All bands here are doubly degenerate. The relevant states are those highlighted in yellow which tend to the free-particle dispersion (black dotted curve) as $U\to0$. (b) Bandstructure in the vicinity of the $\text{K}$ point, along the line $\Gamma \text{K}$. The bands are calculated for $H_U$ only (dark blue), $H_U+H_V$ with $V/E_R=0.002$ (red), and $H_U+H_W$ with $W/E_R=0.002$ (light blue).
• Figure 5: Illustration of the construction of the approximant vectors, for $N_{\rm a}=3$. The exact quasicrystal $\mathbf{G}_l$ are shown in yellow, and the approximant $\widetilde{\mathbf{G}}_l$ and $\widetilde{\mathbf{g}}_l$ in blue and red respectively. The approximant BZ is the black square centred on the origin; the approximant QBZ is the blue (irregular) octagon; and the quasicrystal QBZ is the yellow (regular) octagon.