---
title: 3D Third-Order Topological Insulator
url: https://www.emergentmind.com/topics/third-order-topological-insulator
type: topic
---

# 3D Third-Order Topological Insulator

A third-order topological insulator is a three-dimensional higher-order topological insulator whose protected boundary manifestation is pushed from ordinary two-dimensional surfaces and one-dimensional hinges down to zero-dimensional corners. In the general hierarchy, a \(d\)-dimensional \(n\)th-order topological insulator has \((d-n)\)-dimensional boundary states; in \(3\)D this yields surface states for first order, hinge states for second order, and corner states for third order [2007.03935]. In the literature, this codimension-\(3\) boundary topology is realized and diagnosed through several distinct but related frameworks, including quantized Wannier centers and polarization, octupole-related Wilson-loop structures, Berry-phase invariants, and chiral real-space indices [1903.01194][2305.19209].

## 1. Boundary codimension and the 3D hierarchy

The defining feature of a third-order topological insulator in three spatial dimensions is that the topological boundary response terminates at corners. Surfaces are codimension \(1\), hinges are codimension \(2\), and corners are codimension \(3\). In this sense, a 3D third-order phase is the highest-order topological phase available in ordinary three-dimensional space [2007.03935][2403.00316].

| 3D topological order | Boundary manifestation | Boundary codimension |
|---|---|---|
| First-order | 2D surface states | 1 |
| Second-order | 1D hinge states | 2 |
| Third-order | 0D corner states | 3 |

In the strongest form of the definition, the bulk is gapped, the codimension-\(1\) and codimension-\(2\) boundaries are also gapped, and only corner-localized states remain. Several papers make this explicit by contrasting third-order insulators with 3D higher-order semimetals: a higher-order Weyl semimetal may carry hinge states and Fermi arcs while remaining bulk-gapless, but that is a second-order semimetal rather than a third-order insulator [2007.03935]. The distinction is structural rather than semantic: third-order topology in 3D refers to corner physics of an insulating bulk.

The corner manifestation is not unique at the level of microscopic realization. Depending on symmetry and model class, the corner sector may appear as ordinary zero-energy corner modes, Kramers-paired helical corner states, or fractional corner charge. The common feature is not the microscopic boundary observable by itself, but the codimension-\(3\) bulk-boundary correspondence.

## 2. Bulk diagnostics and topological invariants

No single invariant exhausts the subject. Instead, the literature uses several parallel diagnostics, each adapted to a particular symmetry class and construction. In Wannier-center formulations, the relevant quantity is the bulk polarization
\[
P_i=-\frac{1}{V}\int_{\mathrm{BZ}} A_i\,d^3k,
\qquad
A_i=-i\langle \psi|\partial_{k_i}|\psi\rangle,
\]
and a nontrivial phase is identified when the Wannier center is pinned to a non-atomic high-symmetry position rather than a lattice site [1903.01194][1801.00437]. In the anisotropic-diamond and acoustic-diamond settings, the topological regime places the Wannier center at \(\left(\frac12,\frac12,\frac12\right)\), which predicts where corner or surface states appear under a chosen termination [1801.00437][1903.01194].

A different route is the octupole framework of the 3D Benalcazar–Bernevig–Hughes model and its descendants. In the disorder-induced third-order topological Anderson insulator, the bulk topology is described as a quantized octupole moment, while the practical disorder-space invariant is built from quantized boundary quadrupole moments:
\[
Q=8|q_{xy}q_{xz}q_{yz}|.
\]
Because each boundary quadrupole is quantized by chiral symmetry, \(Q\) is quantized to \(0\) or \(1\) for each disorder realization [2305.19209]. This is a higher-order analogue of bulk-boundary-corner correspondence: nontrivial bulk octupole-related topology induces gapped quadrupolar surfaces and, ultimately, corner states.

Other models use different invariants. In the breathing-pyrochlore Hubbard construction, the noninteracting third-order phase is characterized by a many-body \(\mathbb Z_4\) spin-Berry phase \(\gamma\), with \(\gamma=\pi\) in the higher-order phase and \(\gamma=0\) in the trivial band insulator [2105.09568]. In the electronic candidate Tl\(_4X\)Te\(_3\) (\(X=\)Pb, Sn), standard \(\mathbb Z_2\) and generalized glide \(\mathbb Z_4\) indicators are trivial, but second- and third-order Wilson-loop calculations yield a nontrivial time-reversal polarized octupole polarization
\[
(\tilde{p}_{x,+}^{+z,+y},\tilde{p}_{y,+}^{+x,+z},\tilde{p}_{z,+}^{+y,+x})
=
\left(\frac12,\frac12,\frac12\right),
\]
which diagnoses a helical third-order phase [2108.07946].

Chiral constructions admit still another real-space language. In the 3D spin-orbit-coupled circuit metamaterial, the relevant higher-order invariant is a multipole chiral number
\[
N=\frac{1}{2\pi i}\mathrm{Tr}\log(\bar{Q}_{xyz}^A\bar{Q}_{xyz}^{B\dagger}),
\]
with \(N=2\) in the topological regime and \(N=0\) in the trivial regime [2508.19531]. In the SSH-stacking family, a series of Bott indices reconstructs the corner-chirality pattern across ten distinct chiral-symmetric third-order models [2410.18016]. Taken together, these works show that third-order topology is not tied to one universal invariant, but to a family of bulk and boundary diagnostics adapted to symmetry, dimensional reduction, and model algebra.

## 3. Canonical lattice constructions

Two influential early model classes are the breathing pyrochlore lattice and the anisotropic diamond lattice. In the breathing pyrochlore model, a third-order topological insulator occurs for
\[
-1<t_a/t_b<1/2,
\]
with bulk index
\[
P_6=4(p_x^2+p_y^2+p_z^2)=3
\]
in the topological phase and \(P_6=0\) in the trivial phase [1709.08425]. In a tetrahedral geometry, the model hosts four corner-localized states and a \(1/4\) fractional charge at each corner when one electron occupies the fourfold corner-state manifold. The same paper emphasizes that the topological phase has no topological 2D surface states and no topological 1D hinge states; its decisive boundary manifestation is purely corner-based.

Ezawa’s minimal anisotropic-diamond model shows that a 3D third-order phase does not require a multiband octupole construction. The Hamiltonian is a two-band chiral-symmetric model on the diamond lattice with two hopping parameters \(t_a\) and \(t_b\), and the topological condition is
\[
|t_a|<\frac{t_b}{3}.
\]
In that regime, the Wannier center sits at
\[
(p_x,p_y,p_z)=\left(\frac12,\frac12,\frac12\right),
\]
and a rhombohedral sample hosts corner zero modes localized at two corners, each carrying a \(1/2\) fractional charge [1801.00437]. This construction is important because it shows that third-order topology can arise from anisotropic dimerization and quantized Wannier-center displacement alone.

A later generalization frames 3D third-order topology as SSH stacking in three orthogonal directions. The family of chiral-symmetry-protected models is written as
\[
H_d(\mathbf{k}_d)=\sum_{s=1}^d h_s(k_s),\qquad
h_s(k_s)=M_s(k_s)\Gamma_{sa}^{(d)}+\lambda_s\sin k_s\,\Gamma_{sb}^{(d)},
\]
with \(M_s(k_s)=t_s+\lambda_s\cos k_s\) [2410.18016]. By enumerating the generalized Pauli-matrix realizations, the construction yields ten distinct 3D models, including the 3D Benalcazar–Bernevig–Hughes model. Exact corner-state wavefunctions are built as products of the end-state envelopes of three SSH-like directions, and boundary projection shows that some surfaces realize 2D second-order topology while some hinges realize 1D SSH topology. The resulting picture is explicitly hierarchical:
\[
\text{3D bulk}\to \text{2D topological surfaces}\to \text{1D topological hinges}\to \text{0D corners}.
\]

These model families establish that third-order topology in 3D is broader than a single octupole prototype. The literature contains octupole-type constructions, Wannier-center constructions, and chiral SSH-stacking families, all converging on codimension-\(3\) corner boundary states.

## 4. Experimental realizations and candidate materials

The first direct realization of a 3D third-order topological insulator was reported in an anisotropic-diamond acoustic metamaterial [1903.01194]. In that system, cylindrical acoustic resonators implement lattice sites and thin waveguides implement hoppings. The topological phase occurs for
\[
\left|\frac{t_1}{t_2}\right|<\frac13,
\]
and the realized sample used \(r_{c1}=2\ \mathrm{mm}\), \(r_{c2}=7.8\ \mathrm{mm}\), corresponding to \(t_1/t_2\approx 0.076\). In a rhombohedron-like sample with 52 resonators, two in-gap modes appear around \(2891\)-\(2900\ \mathrm{Hz}\), sharply localized at the two corners predicted by the Wannier-center mismatch. A tetrahedron-like termination in the same platform supports the corresponding surface-state pattern on the \((111)\) face, confirming that the boundary response is termination-dependent but bulk-determined.

A distinct classical realization appears in a three-dimensional breathing cuboid lattice of magnetic vortices [2103.04043]. There the collective gyrotropic dynamics of vortex cores are mapped to an effective tight-binding problem derived from the Thiele equation. The topological regime is controlled by geometric dimerization:
\[
\frac{d_1}{d_2}>1,\qquad \frac{h_1}{h_2}>1.
\]
Finite lattices then exhibit bulk, surface, hinge, and corner modes, but only the corner states remain stable under the tested disorder and defect perturbations; their frequency stays pinned at \(\omega_0/2\pi=0.939\ \mathrm{GHz}\). This work is notable because it realizes third-order topology in a classical magnetic soliton platform rather than an acoustic or electronic one.

Electronic candidate materials were proposed in Tl\(_4X\)Te\(_3\) (\(X=\)Pb, Sn), whose (001) surface has wallpaper group \(p4m\) and supports hidden hourglass, fourfold-Dirac, and Möbius surface fermions [2108.07946]. Although the conventional topological indices are trivial,
\[
\nu_0;(\nu_1,\nu_2,\nu_3)=0;(0,0,0),
\qquad
(\chi_x,\chi_y)=(0,0),
\]
nested Wilson loops diagnose a nontrivial time-reversal polarized octupole structure. In a \(6\times 6\times 6\) finite calculation, the system hosts 16 nearly degenerate in-gap corner states, corresponding to Kramers pairs at eight corners. The authors presented this family as the first realistic electronic-material realization of a third-order topological insulator.

A more recent synthetic realization uses a three-dimensional circuit metamaterial with engineered 3D spin-orbit coupling [2508.19531]. The undimerized limit \(t_1=t_2\) realizes an ideal Weyl semimetal, while the dimerized regime \(t_1/t_2<1\) is a third-order topological insulator. In the topological circuit sample, \(C_1=330\ \mathrm{pF}\), \(C_2=3.3\ \mathrm{nF}\), so \(C_1/C_2=0.1\), and the resonant frequency is near \(f_0=1.19\ \mathrm{MHz}\). The phase is characterized by multipole chiral number \(N=2\), and the finite sample hosts \(4N=8\) pairs, equivalently 16 zero-energy corner modes, i.e. two corner modes per geometric corner. This doubled corner degeneracy distinguishes the realization from the canonical BBH expectation of one corner mode per corner.

## 5. Disorder, interactions, and correlated descendants

Third-order topology in 3D is not restricted to clean noninteracting band structures. In the disordered 3D BBH octupole model, a clean trivial insulator at \(\gamma=1.1\) becomes a third-order topological Anderson insulator once symmetry-preserving disorder in the intracell hoppings is introduced [2305.19209]. The disorder-induced phase is gapped, has a quantized octupole-related invariant, and exhibits corner-localized zero-energy states. The phase sequence with increasing disorder is
\[
\text{GI}\to \text{TOTAI}\to \text{DM}\to \text{AI},
\]
with critical disorder strengths
\[
W_c^{\mathrm{II}}=2.55(20),\qquad
W_c^{\mathrm{III}}=3.54(3),\qquad
W_c^{\mathrm{IV}}=24(2).
\]
The transition into the topological phase is captured analytically by self-consistent Born approximation through the renormalized parameter
\[
\gamma'=\gamma-\sigma,
\]
with the same topological criterion as in the clean BBH model: topological for \(\gamma'<1\), trivial for \(\gamma'>1\).

Interactions can also preserve the third-order sector while changing the nature of the corner excitation. In the breathing-pyrochlore Hubbard model, the noninteracting limit is a 3D third-order topological insulator with zero-energy corner states localized at the four corners of a tetrahedral cluster and characterized by \(\mathbb Z_4\) spin-Berry phase \(\gamma=\pi\) [2105.09568]. With repulsive Hubbard interaction \(U>0\), the phase remains adiabatically connected to the noninteracting third-order phase, but the boundary phenomenology becomes Mott-like: corner charge excitations are gapped, while corner spin excitations remain gapless. The resulting interacting phase is a higher-order topological Mott insulator, and its transition to an ordinary Mott insulator occurs when the bulk spin gap alone closes while the charge gap stays open.

These extensions are conceptually important because they show that third-order topology survives beyond the clean free-fermion limit, but the corner degree of freedom need not remain a single-particle electronic zero mode. Disorder can induce the phase, and interactions can transmute the corner response from charge to spin without destroying the underlying higher-order structure.

## 6. Surface mass engineering, hybrid orders, and common misconceptions

A surface-effective-theory perspective recasts third-order topology as a mass-texture problem. For a generic 3D third-order topological insulator or superconductor, the surface Hamiltonian can be written as
\[
\mathcal{H}_{\mathrm{surf}}^{3\mathrm{rd}}
=
h_1(\mathbf{k})\Gamma_1+h_2(\mathbf{k})\Gamma_2
+B_{\theta,\varphi}\Gamma_3+M_{\theta,\varphi}\Gamma_4,
\]
where \((B_{\theta,\varphi},M_{\theta,\varphi})\) is a two-component surface mass field [2403.00316]. Introducing a loop around a corner turns the problem into a synthetic \(k^2\times S^1\) space, and the emergent Chern–Simons term becomes
\[
\tilde{\theta}=\pi\,\mathcal{W}_{\mathbf h}\mathcal{W}_{\mathbf m}.
\]
For a single surface Dirac cone, \(\mathcal{W}_{\mathbf h}=1\), so the corner zero mode is controlled by the winding \(\mathcal{W}_{\mathbf m}\) of the mass field around the corner. In this language, a third-order corner is a vortex or intersection defect of two mutually anticommuting surface masses. The same framework distinguishes insulators and superconductors at the boundary level: the 3D third-order topological insulator hosts Dirac corner modes, whereas the corresponding superconductor hosts Majorana corner modes.

The taxonomy can be richer than a pure third-order phase. A 3D inversion-symmetric unconventional hybrid-order topological insulator may support second-order helical edge states and third-order corner states simultaneously in one band gap, with a bulk-edge-corner correspondence rather than a purely corner-only higher-order response [2507.22666]. This does not collapse the notion of third-order topology; rather, it shows that the third-order sector can coexist with other higher-order sectors in a single gapped phase.

Two confusions recur in the literature. First, a third-order topological insulator is not a higher-order semimetal. A 3D higher-order topological Weyl semimetal can host Weyl points, Fermi arcs, and hinge states, but that is a bulk-gapless phase with codimension-\(2\) boundary modes, not a codimension-\(3\) insulating phase [2007.03935]. Second, the phrase should not be conflated with the “third-order Hall effect,” where “third-order” refers to a Hall response cubic in electric field rather than to higher-order bulk-boundary correspondence [2209.06867].

Across these constructions, the stable content of the subject is precise. A third-order topological insulator is a 3D insulating topological phase whose definitive protected boundary response occurs at corners. What varies from model to model is the diagnostic language—Wannier centers, octupole polarization, Berry phase, Bott index, real-space chiral number, or surface Chern–Simons response—and the microscopic character of the corner sector, which may appear as fractional corner charge, zero-energy Dirac modes, or helical Kramers pairs.

Source: https://www.emergentmind.com/topics/third-order-topological-insulator