Hierarchical Bulk–Edge Correspondence in TIs

• Hierarchical bulk–edge correspondence is a framework that maps bulk topological invariants to boundary states like edges, hinges, and corners using effective network theories.
• It employs spectral flow techniques, scattering matrices, and Toeplitz operator algebras to quantitatively link bulk data with the emergence of localized soliton modes.
• The approach applies across different spatial dimensions and symmetry classes, offering a versatile toolset for designing and analyzing higher-order topological phases.

Hierarchical bulk–edge correspondence is a set of organizing principles and mathematical frameworks unifying the derivation of localized boundary phenomena, such as edge, hinge, and corner states, from bulk topological invariants in free-fermion topological insulators. The correspondence operates at multiple levels, from the direct relationship between bulk and (codimension-1) edge topology to the emergence of higher-order (codimension-2 and beyond) boundary-localized states controlled by a structured network of interface masses, symmetry constraints, and scattering junctions. The approach integrates effective edge network theories, spectral flow arguments, and operator algebraic techniques to connect bulk topological data to the existence, properties, and quantization of boundary states in arbitrary spatial dimension and symmetry class.

1. Effective Edge Network Theory

The effective edge network theory provides a quantitative description of coupled 1D helical or chiral edge modes formed at the boundaries of topological insulators (TIs). For a given 1D edge or hinge segment $i$, the system is described by a pair of counterpropagating helical modes, $\Psi^{T}=(\psi_{\alpha},\psi_{\beta})$, governed by the Hamiltonian

$$H_{\rm edge} = \sum_{i}\int dx_i\, \Psi_i^{\dagger}(x_i)\left[-iv\,\partial_{x_i}\,\sigma_z + {\cal M}_i(\theta_i)\right]\Psi_i(x_i).$$

The Dirac mass term,

$${\cal M}_i(\theta_i) = m(\cos\theta_i\,\sigma_x + \sin\theta_i\,\sigma_y),$$

gaps the edge modes for nonzero $m$, with the mass angle $\theta_i$ encoding bulk topological and crystalline data. At multi-leg junctions (vertices), boundary conditions or scattering matrices (e.g., for a Y-junction, the phase $\Lambda = \sum_{i=1}^3(\theta_i+\alpha_i)$) define the spectrum of possible bound states. The resulting energies and fractional charges of localized modes, including their robustness and dependence on mass-angle configurations, are accessible via closed-form expressions, such as

$$E/m = \cos\left(\frac{\Lambda+2\pi n}{l}\right),\quad N_s = -\frac{1}{2\pi}\sum_{i=1}^l\theta_i,$$

for an $l$-leg network (Wang et al., 2018).

2. Mapping Bulk Invariants to Boundary Parameters

The translation from bulk to boundary data relies on the identification of bulk invariants (e.g., Chern numbers, crystalline symmetry indicators) and their physical implementation at boundary segments. In systems with crystalline symmetries, permissible Dirac mass terms on each edge are fixed by symmetry constraints. For example, in a square-lattice second-order TI, mirror symmetries quantize the possible mass angles: edges aligned along $x$ directions ($[{\cal M},\sigma_x]=0$) are restricted to $\theta=0,\pi$, while those along $y$ follow $\theta=\pi/2,3\pi/2$. The boundary thus inherits a unique pattern of mass angles $\{\theta_i\}$ from the bulk. Domain walls (kinks) arise at vertices where adjacent segments have distinct $\theta_i$, enforcing the existence of localized modes (Wang et al., 2018).

In inversion-symmetric insulators, $Z_4$ indicators ($\mu_1$, $\kappa_1$) are derived from parity counts at TRIM points and uniquely determine the presence of higher-order topological insulating phases. These indicators match, under slab reduction, to edge Chern or $Z_2$ invariants, establishing a precise mapping from 3D bulk symmetry data to lower-dimensional (hinge or edge) phenomena (Takahashi et al., 2019).

3. Hierarchical Structure: Bulk–Network–Soliton

Bulk–edge correspondence in higher-order topological phases is inherently hierarchical, proceeding in three steps:

• (i) Bulk Invariants $\to$ Edge Network: Topological numbers and band-structure eigenvalues specify which boundary facets host Dirac edge modes and what mass angle $\theta_i$ characterizes each.
• (ii) Edge Network $\to$ Vertex Kinks: The abstract network of gapped edges forms a graph in which vertices act as multi-leg scattering centers. The pattern of adjacent mass angles around each vertex determines the existence of protected domain walls.
• (iii) Vertex Kinks $\to$ Solitons: Each domain wall or multi-leg junction admits bound (“soliton”) modes whose spectrum and fractionalized charge are analytic in the mass-angle differences, with explicit formulas derived from matching or scattering-matrix conditions.

This framework enables a direct, model-independent mapping: $$\text{bulk topological data} \quad\to\quad \text{boundary-mass network} \quad\to\quad \text{localized soliton modes},$$ applicable to classical examples such as the 1D Su–Schrieffer–Heeger (SSH) chain, 2D second-order TIs, and 3D topological fullerenes (Wang et al., 2018).

4. Spectral Flow and Dimensional Reduction

The spectral-flow technique (cutting approach) demonstrates a unifying logic connecting bulk indicators to lower-dimensional boundary modes. For 3D inversion-symmetric HOTIs, adiabatic transformation from a periodic system ($\lambda=+1$) to an open slab ($\lambda=0$) and further to an anti-periodic system ($\lambda=-1$) tracks in-gap state evolution at the boundary. Parity exchanges at high-symmetry momentum points relate the number of in-gap (edge or hinge) states in lower-dimensional slabs to bulk $Z_4$ symmetry indicators. Formally, spectral flow relates differences in band inversion between boundary conditions to the parity or Chern class of the resulting 2D slab, which in turn predicts the existence and connectivity of hinge or edge states. This analysis applies identically to 1D end modes, 2D Chern edge states, and 3D hinge states, revealing a single mechanism behind fractional end charges, edge currents, and higher-order boundary states (Takahashi et al., 2019).

5. Operator Algebras and the Toeplitz Bridge

A rigorous dimension-independent formalism is achieved using Toeplitz algebra for both bulk and edge Hamiltonians. Every edge Hamiltonian of a tight-binding model on a half-space can be realized as an operator in the Toeplitz algebra generated by the unilateral shift. The master identity,

$\Tr[X, Y] = \frac{i}{2\pi} \tr\int_{S^1} \varphi(X)\, d\varphi(Y),$

relates the (regularized) trace of commutators in the edge (Toeplitz) algebra to integrals over the bulk Brillouin torus, providing a direct algebraic “bridge” between the edge mode index and the bulk topological number. In 1D chiral (class AIII) systems, the Fredholm index of the edge Hamiltonian equals the bulk winding number; in 2D Chern insulators, the bulk Chern number is the net spectral flow of 1D edge Dirac bands. For even higher dimensions, the negative sum of emergent boundary Chern numbers recovers the bulk topological invariants, under nondegeneracy conditions for the zero mode manifolds. This approach replaces K-theory by a concrete calculus intrinsic to operator algebras and differential forms (Zhou et al., 25 Oct 2024).

6. Examples: Second-Order Topological Insulators and Topological Fullerenes

In a 2D square-lattice second-order TI with mass sequence: $${\cal M}_{AB}(0) = +m\sigma_x,\; {\cal M}_{BC}(\pi/2) = +m\sigma_y,\; {\cal M}_{CD}(\pi) = -m\sigma_x,\; {\cal M}_{DA}(3\pi/2) = -m\sigma_y,$$ corners correspond to single-kink Dirac problems with $\Delta\theta = \pm\pi/2$ and yield corner modes trapped at energy $E = \pm 1/\sqrt{2}$ with fractional charge $-e/4$, robust to moderate symmetry-breaking perturbations (Wang et al., 2018).

For 3D polyhedral topological fullerenes, e.g., the tetrahedral case (Frank index $f=3$), three helical edge pairs ($l=3$) at each vertex, with mass angles $(0, 0, \pi/2)$, give corner modes with energies $E = \cos(\pi/6)$ and fractionalization $N_s = -1/4$. External flux insertion through a corner shifts the mass angles and yields an analytic, quantized energy–flux response consistent with numerical observations (Wang et al., 2018).

7. Unifying Principles and Scope

The hierarchical bulk–edge correspondence encapsulates both first-order and higher-order (e.g., bulk–hinge, bulk–corner) phenomena in topological band insulators, formulated through a common language of edge network theory, spectral flow, and operator algebraic frameworks. It reconciles disparate models, symmetry classes, and spatial dimensions by reducing the prediction of boundary states to mechanical calculations on effective edge networks or rigorous index theorems. The approach unravels the topological origin of localized soliton, hinge, or corner modes beyond traditional bulk–boundary duality, providing a flexible toolset for characterizing and designing new phases of matter (Wang et al., 2018, Takahashi et al., 2019, Zhou et al., 25 Oct 2024).