Stable Topology in Exactly Flat Bands

Abstract

Topological flat bands (FBs) offer an ideal platform for realizing exotic topological phases, such as fractional Chern insulators, yet their realization with both exact flatness and stable topology in local lattice models has been long hindered by fundamental no-go theorems. Here, we overcome this barrier by demonstrating the existence of critical topological FBs (CTFBs) in finite-range hopping models. They saturate the no-go theorems via a unique structure of Bloch wavefunctions: While continuous over the whole Brillouin zone, the wavefunctions are non-analytic at isolated band touching points, thereby relaxing the inherent restrictions on the coexistence of exact flatness and stable topology. We establish a general principle to construct CTFBs, as well as their parent Hamiltonians, that carry desired topological invariants in given space groups. Explicit examples exhibiting Chern numbers, strong $\mathbb{Z}_2$ index, and crystalline-symmetry-protected invariants in two and three dimensions are provided. Furthermore, an automated algorithm identifies more than 50,000 robust, symmetry-indicated CTFBs. Filling such CTFBs yields short-range entangled topological states that exhibit power-law correlations. Crucially, all filled CTFB states admit exact tensor-network representations with finite bond dimensions, providing a tractable starting point for exploring strongly correlated topological matter.

Figures (10)

• Figure 1: Conceptual framework and realization of a $\mathbb{Z}_2$ CTFB. (a) Classification of exactly flat bands via wavefunction analyticity in finite-range lattice models. Gapped FB: The wavefunction $u(\mathbf{k})$ is smooth and must have trivial (or fragile) topology. Singular FB: Essential discontinuity of $u(\mathbf{k})$ at the touching point renders the topology ill-defined, as seen in frustrated models like the Kagome lattice. CTFB: $u(\mathbf{k})$ is continuous but non-analytic, allowing stable topological invariants well-defined. (b) Realization of a time-reversal-protected $\mathbb{Z}_2$ CTFB in layer group $p\bar{3}1m$. (c) The band structure with representative parameters. (d) The Wilson loop spectrum of the two-fold CTFB displaying the characteristic $\mathbb{Z}_2$ zigzag flow. (e) The entanglement spectrum $\xi(k_2)$ of $| \Omega \rangle$ obtained by spatial cuts parallel to the dashed line in (b). The presence of crossing edge modes confirms the bulk-boundary correspondence. (f) Entanglement entropy (EE) scaling with system size $L\times L$. The EE is calculated from $\xi(k_2)$ in (e), and scales linearly in $L$, confirming the area-law behavior. (g) Real space correlation functions showing a power-law decay of $r^{-4}$.
• Figure 2: CTFBs with various Chern numbers. Columns from left to right display the lattice structures, band structures with representative parameters, and Wilson loop spectra for (a) $C=1$, (b) $C=2$, and (c) $C=3$ constructions, respectively. The $C=1$ model respects the symmetry of wallpaper group $p4$, while the $C=2,3$ models respect $p6$. Red bonds represent arbitrary symmetry-allowed hoppings. In (c), dashed lines indicate additional next-nearest neighbor hoppings, which are likewise arbitrary but required to be non-vanishing. For all the cases, a large $\Delta$ is chosen such that bands corresponding to $\mathcal{BR}_{\widetilde{L}}$ appear at negative energies outside the shown energy window.
• Figure 3: 3D CTFB with strong $\mathbb{Z}_2$ index in space group $P\bar{4} 3m$. (a) and (b) are the lattice structure and Brillouin zone, respectively. (c) shows a band structure with representative parameters. (d) is the entanglement spectrum of the Fock state $| \Omega \rangle$ occupying CTFB and lower bands with spatial cut parallel to the $xy$ plane. The blue and red spectra represent the topological boundary modes on the top and bottom surfaces, respectively.
• Figure 4: Robustness of symmetry-indicated CTFBs versus singular FBs. (a-c) Evolution of the CTFB (defined in fig:Chern-main(a)) under a gap-opening perturbation $\Delta'$, which is chosen as the on-site energy of the $s$ orbital in $\mathcal{BR}_{L}$. Top panels display the band structures near $\mathrm{M}$. Bottom panels show the corresponding Wilson loop spectra, confirming that $C$ remains robust throughout the gap-opening process. (d-f) Evolution of the singular FB in Kagome lattice under a flux $\Phi$ inserted into the triangles, which respects the symmetry of $p6$. In the unperturbed case in (e), the band touching consists of distinct irreps, resulting in essential discontinuities that render $C$ ill-defined. Upon introducing $\Phi$, the resulting topology is sensitive to the sign of $\Phi$.
• Figure S1: Critical flat band with Chern number 1 in wallpaper group $p4$. (a) The nearest neighbor hopping terms $\langle \mathbf{R},\alpha| H_{\rm F} | 0, \beta \rangle$ between $\alpha=a$, $c_x$, $c_y$ and $\beta=b_+$, $b_0$. (b) The band structure, where the red characters are the irreps formed by Bloch states. (c) The Berry's curvature of the flat band. (d) The eigenvalue $\theta(k_x)$ of the flat band Wilson loop integrated over $k_y$, plotted as a function of $k_x$. In (b)-(d), parameters are chosen as $t_1=t_2=t_3=t_4=\Delta=1$.