Glued tree lattices with only compact localized states

TL;DR

This work addresses constructing tight-binding models with exclusively flat bands and compact localized states, irrespective of translation symmetry. It develops two general construction strategies using complex-weighted glued trees: (i) edge replacement on arbitrary countable graphs and (ii) tilings of space by rhombi built from glued trees, yielding Euclidean and hyperbolic lattices such as lotus and dice-like families. A central theorem shows that if a Hamiltonian confines any initial state to a finite region, then every eigenstate can be expressed as a finite local superposition of compact localized states; at special flux values $\Phi$ in the flat-flux set $F^X$, all eigenstates become compact localized and the single-particle spectrum is entirely flat. These results extend flat-band physics to a broad class of geometries, including noneuclidean tilings, and point toward realizations in cold-atom or circuit-QED platforms where AB caging can be exploited for robust localization and potential interaction-induced phenomena.

Abstract

Flat band physics is a central theme in modern condensed matter physics. By constructing a tight--binding single particle system that has vanishing momentum dispersion in one or more bands, and subsequently including more particles and interactions, it is possible to study physics in strongly interacting regimes. Inspired by the glued trees that first arose in one of the few known examples of quantum supremacy, we define and analyze two infinite families of tight binding single particle Bose--Hubbard models that have only flat bands, and only compact localized states despite having any nonnegative number of translation symmetries. The first class of model that we introduce is constructed by replacing a sufficiently large fraction of the edges in a generic countable graph with glued trees modified to have complex hoppings. The second class arises from thinking of complex weighted glued trees as rhombi that can then be used to tile two dimensional space, giving rise to the familiar dice lattice and infinitely many generalizations thereof, of which some are Euclidian while others are hyperbolic.

Figures (7)

• Figure 1: Left: The rhombic lattice, a quasi--one dimensional lattice that has all flat bands at $\Phi = \pi$. An orange box encloses a single unit cell. The single particle rhombic lattice Hamiltonian eigenenergies are the eigenvalues of a matrix in $\mathbb C^{3 \times 3}$. Right: The density of states of the rhombic lattice at various values of $\Phi$. The fact that the colored points are finite in number at the vertical slice at $\Phi = \pi$ means that the bands of the rhombic lattice are flat at $\Phi = \pi$. Another fact that can be inferred from the figure above is that the spectrum at $\Phi = 0$ is gapless, and that the largest spectral gap at any value of $\Phi$ occurs at $\Phi = \pi$. The horizontal line at $E = 0$ means that there is always a flat band at zero energy.
• Figure 2: Various examples of $p$--nary trees. The $p$--nary trees with $p = 2,3,4$ are shown in of each row starting with shrubs and ending with depth to $d = 3$
• Figure 3: Eigenvalues of various glued trees sorted in increasing order. Horizontal axes are scaled by the total number of eigenvalues. Notice that approximately $\frac{p-1}{p+1}$ of the total eigenvalues of the $p$--nary glued trees are zero. Top Left: The eigenvalues of the binary ($2$--nary) glued tree of depth 2000. We remark that the curve depicted bears a striking resemblance to the Cantor ternary function. Bottom Left: Eigenvalue spectra of $p$-nary trees of depth 2000 for various values of $p$. The horizontal lines denote the largest eigenvalues of each tree, which approach $2 \sqrt p$. Top Right:Note that $\{n\}_{n = 2}^5 = \{2,3,4,5\}$, which is a notation defining a sequence. The eigenvalue spectra of upward cascading trees. Bottom Right: Eigenvalue spectra of downward cascading trees. Note that the degeneracy of the zero eigenvalue appears to be fixed by $x_1$ in the sense that approximately $\frac{x_1 - 1}{x_1 + 1}$ of eigenvalues are zero.
• Figure 4: Above we have pictured a lattice that we have proven to have only compact localized eigenstates at flat points, and a plot showing that bands at flat points are themselves flat. Left: Three unit cells of the one dimensional chain of glued trees grown from $X = \{2,3,2\}$ in the canonical gauge. An arrow between a pair of vertices denotes that $\langle i | H | j \rangle = e^{\mathrm i \frac{\Phi}{4}}$. Likewise four and six arrows between a pair of vertices denotes $\langle i| H | j \rangle = e^{\mathrm i\Phi}$ and $\langle i | H | j \rangle = e^{\frac{3}{2}\mathrm i \Phi}$ respectively. Right: Above plotted is the density of states of the Bloch Hamiltonian of the $X = \{2,3,2\}$ one--dimensional chain of glued trees. Flat points $\Phi$ in $F^{\{2,3,2\}}$ are labeled on the horizontal axis, where we see the total bandwidth vanish, indicating that all bands are dispersion free.
• Figure 5: Pictured above are finite chunks of various lotus lattices constructed from $3$--shrubs drawn on the Poincaré disk. A dark arrow between a pair of vertices $i$ and $j$ indicates that $\langle i | H |j\rangle = e^{\mathrm i \Phi}$, while a pair of dark arrows indicates that $\langle i | H | j \rangle = e^{2 \mathrm i\Phi}$. Plaquettes are colored according to the sign of phase wound counterclockwise around their boundary.

Theorems & Definitions (34)

• Definition 1: $p$--shrub
• Definition 4: Growth
• Definition 5: Glued tree
• Definition 6: $p$--nary
• Theorem 8
• proof
• proof
• Definition 10: Canonical embedding
• Definition 11: Weighted adjacency matrix
• Definition 12