Flat bands and band-touching from real-space topology in hyperbolic lattices

Motivated by the recent experimental realizations of hyperbolic lattices in circuit quantum electrodynamics and in classical electric-circuit networks, we study flat bands and band-touching phenomena in such lattices. We analyze noninteracting nearest-neighbor hopping models on hyperbolic analogs of the kagome and dice lattices with heptagonal and octagonal symmetries. We show that two characteristic features of the energy spectrum of those models, namely, the fraction of states in the flat band as well as the number of touching points between the flat band and the dispersive bands, can both be captured exactly by a combination of real-space topology arguments and a reciprocal-space description via the formalism of hyperbolic band theory. Furthermore, using real-space numerical diagonalization on finite lattices with periodic boundary conditions, we obtain insights into higher-dimensional irreducible representations of the non-Euclidean (Fuchsian) translation group of hyperbolic lattices. First, we find that the fraction of states in the flat band is the same for Abelian and non-Abelian hyperbolic Bloch states. Second, we find that only Abelian states participate in the formation of touching points between the flat and dispersive bands.

Figure 1

Octagon-kagome lattice. The eight-sided Bolza cell (red), which contains 24 inequivalent sites (black dots), can be translated by the Fuchsian group generators ${\gamma }_j, j=1,...,4$ (blue), to generate the entire lattice. Crosses indicate sites on the cell boundary that are related by Fuchsian translations to sites that are already taken to belong to the displayed Bolza cell, i.e., the crossed sites belong to adjacent unit cells. The 96 gray and white triangles are fundamental domains for the action of the symmetry group, isomorphic to $\mathrm{GL}(2,{\mathbb{Z}}_3)⋊{\mathbb{Z}}_2$, of the Bolza surface.

Figure 2

The maximally localized CLS on the octagon-kagome lattice are single-octagon states, with nonvanishing amplitude that alternates between $+1$ and $-1$ (red and blue dots) on the eight sites of a single octagon of the lattice. The six different colors represent the six inequivalent single-octagon states belonging to one Bolza cell.

Figure 3

Two examples of flat-band states that extend along noncontractible loops on PBC clusters of the octagon-kagome lattice. The state extending along the green path is compatible with PBC on a single Bolza cell (see Fig. 1) and corresponds to one of the four noncontractible-loop states discussed in Sec. 3b1. The state on the orange path extends along a hyperbolic geodesic.

Figure 4

Example flat-band state that winds nontrivially around a single Bolza cell with TBC captured by 4-momentum $\mathit{k}=(k_1,k_2,k_3,k_4)$ (cf. Fig. 1 for the correspondence between momenta and the generators of Fuchsian translations). The green path is a geodesic on the Bolza surface that crosses each boundary an even number of times.

Figure 5

Spectrum of the octagon-kagome lattice. (a) HBT spectrum (red) from ${10}^4$ momentum points in the 4D Brillouin zone vs exact spectrum (grayscale density plot bounded by the green range) from 8544 PBC clusters with $N=24$ Bolza cells (576 sites), of which 1560 are non-Abelian. (b), (c) Abelian band dispersion along high-symmetry lines $\mathit{k}=k\mathit{n}$ in the 4D Brillouin zone. The chosen directions are $\mathit{n}\in \left\{\right(1,0,0,0),(1,1,0,0),(1,1,1,0\left)\right\}$ in (b), and $\mathit{n}\in \left\{\right(1,0,1,0),(1,1,1,1\left)\right\}$ in (c). Arrows indicate the touching of the flat bands at $E=-2t$ with a dispersive band at $\mathit{k}=\mathit{0}$.

Figure 6

Octagon-dice lattice. The Bolza cell and Fuchsian group generators are as in Fig. 1, but the cell now contains 22 inequivalent sites. The lattice is bipartite, with 16 three-coordinated sites (black symbols) and 6 eight-coordinated sites (orange dots) per Bolza cell. Crossed sites on the boundary of the displayed cell belong to adjacent Bolza cells. Note that the three-coordinated sites constitute the $\{8,3\}$ lattice (see also Fig. 7), which is itself bipartite as indicated by the black square vs triangle symbols.

Figure 7

Maximally localized CLS on the octagon-dice lattice. Octagons shaded in six distinct colors represent the six distinct CLS belonging to one Bolza cell, with nonvanishing amplitudes $\pm 1$ (red and blue dots) on three-coordinated sites only. For each CLS we define the amplitude to be positive (negative) on the sites indicated with a triangle (square) in Fig. 6. A superposition of all CLS with coefficients $1,\omega ,{\omega }^2$ as indicated produces zero, where $\omega =e^{\pm i2\pi /3}$.

Figure 8

Three flat-band states (indicated by shades of cyan/magenta/yellow) that wind around the Bolza cell along a noncontractible loop corresponding to the ${\gamma }_1$ direction. Each state has nonvanishing amplitude on exactly four of the six sites (green circles) along the highlighted string. The sum of all three states vanishes; therefore, only two of them are linearly independent. Analogous noncontractible-loop states can be constructed for nontrivial cycles corresponding to ${\gamma }_{2,3,4}$.

Figure 9

Example flat-band state on the octagon-dice lattice that winds nontrivially around a single Bolza cell with TBC captured by 4-momentum $\mathit{k}=(k_1,k_2,k_3,k_4)$ (see Fig. 6).

Figure 10

Spectrum of the octagon-dice lattice ($V_B/t=1$). (a) HBT spectrum (red) from $2\times {10}^4$ momentum points in the 4D Brillouin zone vs exact spectrum (grayscale density plot with support bounded by the green range) from 8544 PBC clusters with $N=24$ Bolza cells (528 sites), of which 1560 are non-Abelian. (b), (c) Abelian band dispersion along high-symmetry lines $\mathit{k}=k\mathit{n}$ in the 4D Brillouin zone. The chosen directions are $\mathit{n}\in \left\{\right(1,0,0,0),(1,1,0,0),(1,1,1,0\left)\right\}$ in (b), and $\mathit{n}\in \left\{\right(1,0,1,0),(1,1,1,1\left)\right\}$ in (c). Arrows indicate the touching of the flat bands at $E=0$ with two dispersive bands at $\mathit{k}=\mathit{0}$.

Figure 11

Heptagon-kagome lattice. The 14-sided Klein cell (red), which contains 84 inequivalent sites (black dots), can be translated by the Fuchsian group generators ${\gamma }_j, j=1,...,7$ (blue), to generate the entire lattice. The 336 gray and white triangles are fundamental domains for the action of the symmetry group of the Klein surface. The edges of the underlying $\{3,7\}$ tessellation are drawn in purple.

Figure 12

Left side: The maximally localized CLS on the heptagon-kagome lattice is the heptagon-pair state (left), associated to a single vertex of the heptagon-kagome lattice or, equivalently, to a single edge (dark green) of the $\{3,7\}$ lattice. The equal-weight superposition of the heptagon-pair CLS associated with edges of any triangle (green) of the $\{3,7\}$ lattice results in the zero vector, indicating that the constructed CLS form an overcomplete basis. Right side: One can construct noncontractible-loop states (dark green) as linear combinations of the heptagon-pair CLS. The cross symbols indicate cancellation of amplitudes from two heptagon-pair states; red and blue dots indicate amplitudes $\pm 1$; circled red and blue dots indicate amplitudes $\pm 2$.

Figure 13

The Klein quartic is tiled by 336 Schwarz triangles (displayed in white and gray) with interior angles $\pi /2, \pi /3$, and $\pi /7$. A pair of Schwarz triangles (highlighted blue or green triangles) is a fundamental domain for the action of the order-168 group $G\cong \mathrm{PSL}(2,{\mathbb{Z}}_7)$ of conformal automorphisms of the Klein quartic. Each site of the heptagon-kagome lattice (blue dot) belongs to a rhombus consisting of two such fundamental triangles related by a $\pi$ rotation around the site, and each three-coordinated site of the heptagon-dice lattice (green dot) belongs to an equilateral triangle consisting of three fundamental triangles related by a $2\pi /3$ rotation around the site.

Figure 14

Number of normal subgroups of index $N$ in the Fuchsian translation groups ${\mathrm{\Gamma }}_{\{8,8\}}$ (blue circles) and ${\mathrm{\Gamma }}_{\{14,7\}}$ (red circles). The solid lines are the analytical result (18) for subgroups of prime index.

Figure 15

Spectrum of the heptagon-kagome lattice. (a) HBT spectrum (red) from ${10}^4$ momentum points in the 6D Brillouin zone vs exact spectrum (grayscale density plot bounded by the green range) from $27995$ PBC clusters with $N=8$ Klein cells (672 sites), of which 2240 are non-Abelian. (b), (c) Abelian band dispersion along selected high-symmetry lines $\mathit{k}=k\mathit{n}$ in the 6D Brillouin zone. The chosen directions are $\mathit{n}=(1,0,0,0,0,0)$ in (b), and $\mathit{n}=(1,0,-1,0,1,0)$ in (c). Arrow indicates the energy gap that separates the flat bands at $E=-2t$ from the dispersive bands.

Figure 16

Heptagon-dice lattice. The Klein cell and the Fuchsian group generators are as in Fig. 11, but the cell now contains 80 inequivalent sites. The lattice is bipartite, with 56 three-coordinated sites (black dots) and 24 seven-coordinated sites (orange dots) per Klein cell.

Figure 17

The maximally localized CLS state on the heptagon-dice lattice is the star-triplet state, bound to three heptagonal faces of the underlying $\{7,3\}$ lattice, shown in red. Equivalently, the CLS is bound to a single triangular face of the $\{3,7\}$ lattice (green triangle). Red and blue dots denote probability amplitudes of alternating sign $\pm 1$. An equal-weight superposition of the star-triplet states bound to the seven triangles adjacent to a common seven-coordinated vertex (green dot) has canceling amplitudes, implying a linear dependence relation.

Figure 18

Construction of a flat-band noncontractible-loop state (with support on the magenta line) on the heptagon-dice lattice. The state is obtained as a weighted superposition of the star-triplet CLS (Fig. 17) centered on the eight highlighted triangles of the underlying $\{3,7\}$ lattice (the support of one such state is explicitly illustrated with the yellow outline). The star-triplet CLS pertaining to the red (blue) triangles are assigned weight $+1$ ($+2$) in the superposition, and the resulting amplitudes on the individual sites are specified by seven distinct symbols according to the legend at the bottom of the figure. The state thus obtained for a finite string (green path) of star-triplet CLS on the Poincaré disk results in a chain CLS with a pair of heptagons attached at each end (green dashed circles). Upon the Klein identification of the Klein cell boundary, the amplitudes on those heptagons interfere destructively and leave behind only the string that winds nontrivially around the Klein cell.

Figure 19

Spectrum of the heptagon-dice lattice ($V_B=0$). (a) HBT spectrum (red) from ${10}^4$ momentum points in the 6D Brillouin zone vs exact spectrum (grayscale density plot bounded by the green range) from $27995$ PBC clusters with $N=8$ Klein cells (640 sites), of which 2240 are non-Abelian. (b), (c) Abelian band dispersion along selected high-symmetry lines $\mathit{k}=k\mathit{n}$ in the 6D Brillouin zone. The chosen directions are $\mathit{n}=(1,0,0,0,0,0)$ in (b), and $\mathit{n}=(1,0,-1,0,1,0)$ in (c). Arrows indicate the energy gaps that separate the flat bands at $E=0$ from the dispersive bands at positive and negative energies.