Higher-order Topological Type-II Hyperbolic Lattices

Technical Summary: Higher-Order Topological Type-II Hyperbolic Lattices

Problem Statement
While higher-order topological phases have been successfully extended from Euclidean to non-Euclidean hyperbolic lattices, existing research has focused exclusively on type-I hyperbolic lattices. In these systems, the degeneracy of topological corner states is strictly constrained by the rotational symmetry of the tessellated polygons (e.g., an octagon tiling yields a fixed degeneracy). Consequently, the ability to tune the degeneracy of these states is limited by the geometric constraints of the tiling. Furthermore, type-I lattices typically support topological edge states only on their outer boundaries. The existence and properties of type-II hyperbolic lattices—which possess both inner and outer boundaries and a tunable characteristic radius—remain unexplored in the context of higher-order topological phases. This work addresses the gap in understanding how the unique geometry of type-II lattices influences first-order edge states and second-order corner states, specifically investigating whether the characteristic radius offers a new degree of freedom for tuning topological degeneracy.

Methodology
The authors theoretically construct a higher-order topological insulator (HOTI) model by mapping the celebrated Bernevig–Hughes–Zhang (BHZ) model, known for describing Quantum Spin Hall Insulators (QSHIs), onto a type-II hyperbolic lattice.

1. Hamiltonian Construction: The system is described by a tight-binding Hamiltonian ($H_{QSHI}$) incorporating spin and orbital degrees of freedom. To transition from a first-order to a second-order topological phase, a Wilson mass term ($H_m$) is introduced. This term breaks time-reversal symmetry and gaps the 1D helical edge states, inducing mass domain walls that host zero-dimensional corner states.
2. Numerical Simulation: The Hamiltonian is diagonalized under open boundary conditions. The authors analyze the energy spectrum, local density of states (LDOS), and spatial intensity distributions to identify topological states.
3. Topological Invariants:
   • Spin Bott Index ($C_B$): Used to characterize the nontrivial band topology of the edge states in the absence of periodicity.
   • Real-space Quadrupole Moment ($Q_{xy}$): Employed to distinguish between the higher-order topological phase and the trivial phase, as momentum-space invariants are not well-defined in hyperbolic geometries.
4. Parameter Tuning: The study systematically varies the characteristic radius ($r_h$) (or inner radius) of the type-II lattice to observe its effect on the rotational symmetry parameter ($k$) and the resulting degeneracy of corner states.

Key Results

1. Dual-Boundary Edge States: Unlike type-I lattices which support edge states only on the outer boundary, the type-II hyperbolic lattice supports first-order topological helical edge states on both its inner and outer boundaries. These states exhibit double degeneracy and chiral propagation, with slightly different energy values due to the sublattice dependence on the polar angle $\theta_{ij}$.
2. Second-Order Corner States: By introducing the Wilson mass term, the system transitions into a higher-order topological phase hosting zero-energy corner states (ZECSs) localized at the corners of both the inner and outer boundaries.
3. Arbitrarily Tunable Degeneracy: The most significant finding is that the degeneracy of the corner states in type-II lattices is not fixed by the number of sides of the tessellated polygons ($p$). Instead, the degeneracy is directly proportional to the rotational symmetry parameter $k$, which can be arbitrarily tuned by adjusting the characteristic radius $r_h$ (governed by $r_h = e^{-2\pi/k}$).
   • The study demonstrates this by showing that for an octagon tessellation ($p=8$), changing $r_h$ from 0.604 to 0.715 increases the rotational symmetry from $k=8$ to $k=12$.
   • This results in a change in corner state degeneracy from 16-fold (8 inner + 8 outer) to 24-fold (12 inner + 12 outer).
4. Phase Transition: The transition from the higher-order topological phase to a trivial phase is confirmed by the real-space quadrupole moment $Q_{xy}$, which shifts from 0.5 to 0 as the Wilson mass magnitude $g$ crosses a critical threshold ($g_T \approx 1.45$).

Significance and Claims
The paper claims to present the first theoretical exploration of higher-order topological type-II hyperbolic lattices. Its primary contribution is establishing type-II lattices as a versatile platform where the degeneracy of topological corner states can be continuously and arbitrarily tuned via geometric parameters (the characteristic radius), overcoming the rigid constraints imposed by Euclidean symmetry and type-I hyperbolic geometries.

The authors suggest that the dual-boundary nature of these lattices, supporting counter-propagating edge states, could serve as an ideal platform for extending fundamental topological phenomena—such as Thouless pumping, Landau-Zener transitions, and Laughlin's pump—from Euclidean to non-Euclidean geometries. They note that such lattices are experimentally accessible via existing platforms including circuit quantum electrodynamics, electric circuits, nonreciprocal networks, and integrated nanophotonic chips. The work significantly expands the scope of hyperbolic topological physics by introducing a new class of tunable, higher-order topological phases.