Topological states and flat bands in exactly solvable decorated Cayley trees

TL;DR

Decorated Cayley trees are analyzed as non-Euclidean analogs of Lieb, double-Lieb, kagome (Husimi), and star lattices to expose exact flat bands and their topological origins. Using a symmetry-adapted shell basis, the tight-binding problem on finite and infinite trees reduces to a small set of SSH-type one-dimensional blocks, enabling exact spectra and revealing flat bands arising from topological edge modes or boundary-defect localization. The flat-band states are often protected by SSH- or SSH3-type topology or by rank-nullity bounds, and in the infinite-tree limit these bands persist via a covering-graph interpretation of the lattice. Finite trees exhibit rich spectra with termination-dependent flat bands and edge-localized modes, suggesting robust platform candidates for experimental realization in photonic or circuit networks.

Abstract

We derive the full spectrum of decorated Cayley trees that constitute tree analogs of selected two-dimensional Euclidean lattices; namely of the Lieb, the double Lieb, the kagome, and the star lattice. The common feature of these Euclidean lattices is that their nearest-neighbor models give rise to flat energy bands interpretable through compact localized states. We find that the tree analogs exhibit similar flat or nearly flat energy bands at the corresponding energies. Interestingly, such flat bands in the decorated Cayley trees acquire an interpretation that is absent in their Euclidean counterparts: as edge states localized to the inner or the outer boundary of the tree branches. In particular, we establish an exact correspondence between the Lieb-Cayley tree and an ensemble of one-dimensional Su-Schrieffer-Heeger chains, which maps topological edge states on one side of the chains to flat-band states localized in the bulk of the tree, furnishing the flat energy band with a topological stability. Similar mapping to topological edge states or to states bound to edge defects in one-dimensional chains is shown for flat-band states in all the considered tree decorations. We finally show that the persistence of exact flat bands on infinite decorated trees (i.e., Bethe lattices) arises naturally from a covering interpretation of tree graphs. Our findings reveal a rich landscape of flat-band and topological phenomena in non-Euclidean systems, where geometry alone can generate and stabilize unconventional quantum states.

Figures (24)

• Figure 1: A Cayley tree with nearest and next-nearest neighbor hopping. Here we set the number of layers to $M=4$ and the branching factor to $K=2$. The tree is invariant under permutation of individual sub-branches and under cyclic rotations of the three main branches.
• Figure 2: A sketch demonstrating the notation of the position basis in the construction of (a) the shell-symmetric states and of (b) the shell-non-symmetric states. (Similar schematics appear in Ref. hamanaka_multifractal_2024)
• Figure 3: The bulk density of states of the NNN-hopping Calyey tree with connectivity $K=2$ with nearest-neighbor hopping $t_1 = 1$. For $t_2=0$ we recover the density of states of the simple Bethe lattice mahan_energy_2001. As we increase $t_2$ the spectrum becomes asymmetric. For $t_2 = 1$ we find a square-root singularity at the lower band-edge as well as a cusp (i.e., infinite slope of the density of states) inside the energy band eckstein_hopping_2005.
• Figure 4: Spectrum of a Cayley tree with $M=50$ layers and branching factor $K=2$. Nearest-neighbor hopping is fixed at $t_1=1$ and next-nearest-neighbor (NNN) hopping $t_2$ is tuned from $0$ to $1$. For $t_2=0$ we recover the spectrum of a simple Cayley tree with only NN hopping aryal_complete_2020. As we turn on NNN hopping $t_2 > 0$, the spectrum shifts to larger energies and loses its symmetrical arrangement around $E=0$.
• Figure 5: (a) The Euclidean Lieb lattice: an additional site is placed at the center of each edge of the square lattice. The unit cell contains three sites, labeled $A$, $B$, $C$. A typical compact localized state has support on the four sites highlighted in pink, with positive (negative) amplitude on the $B$ sites (on the $C$ sites) as indicates with the $\pm$ signs. (b) The Lieb decoration of a Cayley tree with branching factor $K=3$. The Cayley nodes of the decorated tree are colored in red, while the Lieb nodes are colored in blue.