Non-Abelian Hyperbolic Band Theory from Supercells

Patrick M. Lenggenhager, Joseph Maciejko, and Tomáš Bzdušek

Phys. Rev. Lett. 131, 226401 – Published 1 December 2023

Abstract

Wave functions on periodic lattices are commonly described by Bloch band theory. Besides Abelian Bloch states labeled by a momentum vector, hyperbolic lattices support non-Abelian Bloch states that have so far eluded analytical treatments. By adapting the solid-state-physics notions of supercells and zone folding, we devise a method for the systematic construction of non-Abelian Bloch states. The method applies Abelian band theory to sequences of supercells, recursively built as symmetric aggregates of smaller cells, and enables a rapidly convergent computation of bulk spectra and eigenstates for both gapless and gapped tight-binding models. Our supercell method provides an efficient means of approximating the thermodynamic limit and marks a pivotal step toward a complete band-theoretic characterization of hyperbolic lattices.

Received 2 June 2023
Revised 7 September 2023
Accepted 25 October 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.226401

Physics Subject Headings (PhySH)

Crystal phenomena Electronic structure Topological insulators

2-dimensional systems Lattices

Bloch wave theory Discrete symmetries in condensed matter Group theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Patrick M. Lenggenhager ^1,2,3,4,*, Joseph Maciejko ^4,5,†, and Tomáš Bzdušek ^1,2,‡

¹Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
²Condensed Matter Theory Group, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland
³Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland
⁴Theoretical Physics Institute, University of Alberta, Edmonton, Alberta T6G 2E1, Canada
⁵Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada

^*plengg@pks.mpg.de
^†maciejko@ualberta.ca
^‡tomas.bzdusek@uzh.ch

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Vol. 131, Iss. 22 — 1 December 2023

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Images

Figure 1
Supercell construction for (a),(c),(e) the Euclidean ${4, 4}$ and (b),(d),(f) hyperbolic ${8, 8}$ lattice. (a),(b) Primitive cell (blue) and symmetrized 2-supercell (yellow). (c),(d) Compactified cells in real space. (e) Density of states $ρ$ of the nearest-neighbor hopping model on the ${4, 4}$ lattice as a function of energy $E$ showing the characteristic van Hove singularity. The inset shows the momentum-space dispersion for the primitive cell (solid blue line) and for the supercell (yellow dashed line). (f) Density of Abelian Bloch states of the nearest-neighbor hopping model on the ${8, 8}$ lattice for the primitive cell (blue), for the 2-supercell (yellow), and schematic extrapolation (for details see Supplemental Material [41]) to large supercells (red). The black arrow indicates a suppression near the band edges (see text).
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Figure 2
Symmetries of hyperbolic lattices. (a) ${8, 8}$ lattice (black lines) with the triangle group $Δ (2, 8, 8)$ as space group (indicated by gray/white triangles). The primitive cell (blue polygon) and 2-supercell (yellow polygon) and their edge identifications are shown: the edge $\bar{1}$ is related to 1 by the translation generator $γ_{1}$ ( ${\tilde{γ}}_{1}$ for the supercell). Edges related by composite translations are labeled by $α$ and $β$ [41]. (b) Fundamental Schwarz triangle (gray) with reflections $a$ , $b$ , $c$ across the edges of the triangle and rotations $x = a b$ , $y = b c$ , $z = c a$ around the vertices. (c) Square lattice with primitive cell, supercell, triangle group $Δ (2, 4, 4)$ , and reflection lines $a$ , $b$ , $c$ .
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Figure 3
Illustration of the spaces of $d$ -dimensional irreducible representations (IRs) of a sequence of translation subgroups $Γ^{(m)}$ corresponding to supercells with $N^{(m)}$ primitive cells. The spaces of 1D IRs are hypertori (illustrated as square and cube) with dimension growing linearly with $N^{(m)}$ , while the spaces of higher-dimensional IRs are more complicated (illustrated as balls). The IRs of $Γ^{(m)}$ of dimension $d$ subduce (blue arrow) representations of $Γ^{(m + 1)}$ of the same dimension and induce (yellow arrow) representations of $Γ^{(m - 1)}$ of higher dimension (see text).
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Figure 4
Density of states $ρ (E)$ as function of energy $E$ of (a) the nearest-neighbor (NN) model on the octagon-kagome lattice and (b) the Haldane model on the ${8, 3}$ lattice with NN hopping $h_{1} = 1$ , next-NN hopping $h_{2} = 1 / 6$ , flux $ϕ = π / 2$ , and sublattice mass $h_{0} = 1 / 3$ . The inset in (a) shows the depletion of $ρ (E)$ near the flat band at $E = 2$ .
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Figure 5
Benalcazar-Bernevig-Hughes model on the ${6, 4}$ lattice. (a) Model definition on the primitive cell (blue polygon) of the ${6, 4}$ lattice (black lines). There are four orbitals (black dots) at each site, coupled by intersite hoppings $h_{0}$ (magenta) and intrasite hoppings $h_{1}$ (green). Light blue shading of plaquettes bounded by magenta and green lines indicates $π$ fluxes. (b) Density of states for the indicated choices of $(h_{0}, h_{1})$ . The top, middle, and bottom subpanels correspond to the trivial, critical, and nontrivial phases, respectively.
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