Type-II Hyperbolic Lattices & Topological Modes

• Type-II hyperbolic lattices are defined by regular annular tessellations with an inner BTZ-like horizon and outer conformal boundary.
• They enable unique topological phases featuring higher-order corner modes, dual chiral edge states, and dynamic inter-edge transfer via APT and LZ mechanisms.
• These structures offer a versatile platform for exploring holographic duality, entanglement scaling, and practical implementations in metamaterials.

A type-II hyperbolic lattice is a discrete structure arising from regular tessellations of hyperbolic space, specifically corresponding to a spatial slice of the Euclidean BTZ (2 + 1) black hole at fixed time. Unlike type-I hyperbolic lattices, which discretize the Poincaré disk and possess a single boundary, type-II lattices are defined on the Poincaré ring (an annulus) and feature both an inner boundary (the discrete analogue of the BTZ horizon) and an outer boundary (the Dirichlet/physical edge), supporting a richer set of topological and geometric phenomena including higher-order and Chern phases, domain-localized zero modes, and dual sets of chiral edge states (Chen et al., 2023, Liu et al., 11 Jan 2026, Chen et al., 9 Mar 2025).

1. Geometric Definition and Embedding

Type-II hyperbolic lattices originate from tessellations of a one-sheeted hyperboloid

$-X_0^2 + X_1^2 + X_2^2 = -1, \quad X_0>0$

with constant negative curvature $K = -1/\ell^2$. Stereographic projection maps this manifold onto the complex plane via

$$z = \frac{X_1 + i X_2}{X_0 + 1}= r e^{i\theta}, \qquad r=|z|<1.$$

However, in type-II construction, the image is not the full disk but an annulus: the Poincaré ring, with explicit inner ($r_{\text{in}}=e^{-4\pi/(kP)}$) and outer ($r_{\text{out}}=1$) boundaries, where $P\simeq1.559$ is a geometric constant and $k$ controls discrete rotational symmetry. The canonical lattice is specified by a Schläfli symbol $\{p, q\}$, where $q$ regular $p$-gons meet at each vertex, generalized to $\{r_n, p, q\}$ to encode the inner/outer boundary ratio (Liu et al., 11 Jan 2026, Chen et al., 9 Mar 2025).

In the AdS/CFT context, the inner circle discretely realizes the BTZ black hole horizon, and the outer ring mimics the conformal boundary (Chen et al., 2023). The resulting geometry supplies two topologically and spectrally distinct edges, enabling phenomena forbidden in conventional single-boundary settings.

2. Lattice Tiling, Discrete Curvature, and Metric Properties

Type-II hyperbolic lattices use regular $\{p, q\}$ tessellations, for example $\{3, 7\}$ (triangular faces, coordination $q=7$), satisfying $(p-2)(q-2)>4$ to ensure negative curvature. The discrete Gaussian curvature at each vertex is

$K_{\rm vertex} = 2\pi - q \frac{\pi(p-2)}{p}$

guaranteeing uniform negative curvature in the thermodynamic limit (Chen et al., 2023, Chen et al., 9 Mar 2025).

Metric properties descend from the continuum BTZ slice in global coordinates $(\xi, \theta)$:

$ds^2 = \ell^2 (d\xi^2 + \cosh^2\xi d\theta^2),$

with $\xi=0$ the horizon, or, in exponential coordinates $r = e^{\xi}$,

$$ds^2 = \ell^2 \left( \frac{dr^2}{r^2} + r^2 d\theta^2 \right), \quad r\in[r_{\text{in}}, r_{\text{out}}].$$

Vertices fall on discrete radii corresponding to these boundaries, with the lattice constant and edge lengths determined by hyperbolic trigonometric relations.

Geodesics bend around the horizon (inner boundary) in contrast to the “radial” arcs of type-I lattices (which lack the inner circle) (Chen et al., 2023).

3. Topological Band Models and Higher-Order Modes

On the vertex set, two four-band tight-binding models have been constructed: the modified Bernevig-Hughes-Zhang (BHZ) model and the Benalcazar-Bernevig-Hughes (BBH) model. Both involve angle-dependent hopping guided by the hyperbolic geometry:

• Modified BHZ: Incorporates spin and orbital degrees of freedom, Wilson mass terms alternating around the ring, and yields domain walls at mass sign changes.
• BBH: Employs $\Gamma$-matrix structures to realize a quadrupole insulator.

The real-space (generalized) quadrupole moment $Q_{xy}$ is computed via a projected nested Berry-phase formula adapted to the ring geometry:

$$Q_{xy} = \left[\frac{1}{2\pi}\Im\ln\det(\Psi_{\rm occ}^\dagger \hat U \Psi_{\rm occ}) - Q_0 \right]\mod 1$$

with $\hat U$ a diagonal operator encoding the $(x_m y_m)$ coordinate of each vertex rescaled by the hyperbolic “area” of the ring (Liu et al., 11 Jan 2026).

Domain-wall sign changes in the Wilson mass produce localized, spectrally isolated zero-energy cornerlike modes pinned to both boundaries. The number and angular placement are tunable through model parameters (Wilson mass period $\eta$, phase offsets), with an even distribution between inner and outer edges, in contrast to type-I lattices where such states appear at a single boundary only.

Topological indices confirm robust higher-order phases: for the modified BHZ with $\eta=2$, eight zero energy modes (four per edge) emerge when $0.14\lesssim g\lesssim1.55$ and $Q_{xy}=0.5$; BBH transitions occur by varying the coupling $\lambda$, giving likewise a quantized quadrupole $Q_{xy}$ and corner states. These states are protected up to disorder strength $W_c\approx1$–2 (Liu et al., 11 Jan 2026).

4. Chiral Edge States and Dynamic Inter-Edge Transfer

Type-II hyperbolic Chern insulators (Qi-Wu-Zhang analogues) manifest topologically nontrivial bulk bands (Bott index $C_B=-1$) and support two spectrally distinct chiral edge state (CES) branches:

• An outer CES, localized at the outer boundary, propagating counterclockwise (CCW).
• An inner CES, pinned to the inner boundary, propagating clockwise (CW).

These CESs are exponentially localized to their respective boundaries and cross the bulk bandgap with opposite group velocities. The presence of both chiralities is rooted in the dual-boundary geometry, which does not arise for type-I (disk) lattices (Chen et al., 9 Mar 2025).

Type-II lattices enable dynamic transfer of chiral edge states between boundaries through two distinct mechanisms:

• Anti-Parity-Time (APT) Phase Transition: Coupling inner and outer CESs via a radial channel, a precisely tuned imaginary coupling leads to an exceptional point (EP) where modes coalesce and transfer between edges. This process occurs when the coupling equals the energy flow difference between edge states.
• Landau-Zener (LZ) Single-Band Pumping: A nonreciprocal phase in the coupling facilitates an adiabatic energy sweep, where slow driving (small “velocity”) pumps the edge state from one boundary to the other. Fast driving yields standard LZ tunneling, leaving the initial edge state intact (Chen et al., 9 Mar 2025).

These dynamic protocols are impossible in type-I geometries due to the absence of the inner edge.

5. Holographic Signatures, Entanglement Scaling, and AdS/CFT

Type-II hyperbolic lattices provide discretizations of BTZ–AdS$_{2+1}$ geometries suitable for holographic duality simulations:

• Bulk Entanglement Entropy (BEE): Subsystems anchored to a boundary interval $A$ are associated with “entanglement wedges” in the bulk. The entropy, computed from ground-state Gaussian data, scales with the geodesic length of the wedge’s minimal curve according to the Ryu–Takayanagi (RT) formula,

$$S_{\rm CFT}(A) = \frac{c}{3} \ln\left( \frac{\beta}{\pi} \sinh \frac{\pi \theta_A}{\beta} \right) = \frac{c}{6} \mathrm{Length}(\gamma_A)$$

with effective central charge $c_{\rm eff}$ extracted from fits.

• Boundary–Boundary Correlation: Two-point scalar correlators between vertices on an edge exhibit exponential decay with geodesic length, governed by the Klebanov–Witten relation for boundary scaling dimension $\Delta$:

$$\langle \phi(x) \phi(y) \rangle \sim \exp\left[ -\Delta d_{\text{hyp}}(x, y) \right], \quad \Delta(2-\Delta)=m^2\ell^2$$

Both observables are found to quantitatively match their continuum AdS/CFT counterparts, with $(c_{\rm eff}, \Delta_{\rm exp})$ converging to continuum values as lattice size increases. The BEE crosses over from logarithmic to linear scaling as the subsystem interval approaches half the ring, reflecting horizon-induced extensivity (Chen et al., 2023).

6. Robustness, Tunability, and Experimental Realizations

Type-II hyperbolic lattices exhibit several robust and tunable features:

• The number, position, and energy of cornerlike zero modes can be controlled via model parameters (Wilson mass period, angular offsets, the integer $k$ controlling annulus size).
• Disorder resilience is confirmed by tracking zero-mode stability and topological invariants ($Q_{xy}$) across random-on-site potentials for both BHZ and BBH-type models; quantized values persist up to moderate disorder strengths.
• Finite-size effects are mitigated by increasing $k$, which enlarges the ring and reduces boundary overlap.
• All platforms applicable to type-I lattices – circuit QED arrays, topolectrical circuits, photonic and acoustic metamaterials – are in principle adaptable to type-II. Angle-dependent couplings and ring geometry are the design requirements (Liu et al., 11 Jan 2026).

A table summarizing distinguishing features appears below:

Feature	Type-I Lattice	Type-II Lattice
Boundary count	1	2 (inner + outer)
Cornerlike zero modes	Single boundary only	Both edges (tunable)
Chiral edge states	Outer edge only	Inner & outer, dual
Inter-edge dynamics	Not possible	APT/LZ transfer
Holography	AdS$_{2+1}$ (disk)	BTZ–AdS$_{2+1}$ (ring)

Type-II lattices thus uniquely enable investigation of inter-edge phenomena, higher-order topology in non-Euclidean settings, and quantum-gravity analogues in engineered metamaterials (Chen et al., 9 Mar 2025, Chen et al., 2023, Liu et al., 11 Jan 2026).

7. Applications and Theoretical Significance

The existence of two dynamically linked boundaries, higher-order topological modes, and the direct correspondence with BTZ–AdS geometries position type-II hyperbolic lattices as platforms for:

• Probing nontrivial holographic dualities and entanglement structures in laboratory settings.
• Implementing dynamically switchable waveguide/circuit networks leveraging topologically protected edge transfer protocols immune to reflection and certain disorder types.
• Engineering higher-dimensional (3D “stacked”) hyperbolic topological phases, potentially enabling hinge or surface states without Euclidean analogues.
• Simulating quantum pumps, topological frequency combs, or robust zero-mode lasers using APT or LZ-based transfer mechanisms.
• Exploring gravity/gauge phenomena and boundary-driven quantum effects inaccessible in single-boundary hyperbolic models (Chen et al., 9 Mar 2025, Chen et al., 2023, Liu et al., 11 Jan 2026).

A plausible implication is that type-II lattices may serve as test beds for both fundamental AdS/CFT conjectures and practical device-oriented topological states, covering a spectrum of effects unique to genuine non-Euclidean “ring” geometries.