Chiral Second-Order Topological Insulator

• Chiral second-order topological insulators are higher-order phases with an insulating bulk and gapped surfaces, but protected corner or hinge states emerge via mass inversion.
• They utilize mass inversion at boundaries to produce localized zero modes, offering versatile experimental realizations in photonics, electronics, and metamaterials.
• The theoretical framework employs diagnostics like winding numbers, nested Wilson loops, and quadrupole indices to precisely characterize higher-order topology.

Chiral second-order topological insulators are higher-order topological phases in which the bulk is insulating and the conventional first-order boundary is gapped, while codimension-two states remain protected at corners in two dimensions or along hinges in three dimensions. In the literature, the qualifier “chiral” is used in two complementary senses: for Cartan-class-AIII systems with chiral symmetry and broken time-reversal and charge-conjugation symmetries, and for three-dimensional phases with unidirectional chiral hinge transport after time-reversal breaking. In both settings, the decisive mechanism is a sign structure of boundary Dirac masses, so that a mass domain wall at a corner or hinge binds a robust lower-dimensional mode (Poata et al., 2023, Langbehn et al., 2017).

1. Symmetry setting and nomenclature

A central two-dimensional setting is Cartan class AIII, where chiral symmetry is present, time-reversal symmetry $\mathcal{T}$ and charge-conjugation symmetry $\mathcal{C}$ are broken, and zero-energy mid-gap states are robust against perturbations consistent with chiral symmetry. In the model analyzed for polygonal flakes, the chiral symmetry is written as $\{H,\sigma_z\tau_x\}=0$, and the protected states are zero-energy corner modes localized at selected vertices of a finite sample (Poata et al., 2023).

A recurrent source of ambiguity is that “chiral” does not always denote chiral symmetry. In three dimensions, the same adjective often refers to one-dimensional hinge modes that propagate only in one direction. These chiral hinge modes are the defining feature of a three-dimensional second-order topological insulator with gapped surfaces, and they may appear in class A settings with broken time-reversal symmetry, in crystalline constructions using reflection symmetry, or in models obtained by breaking chiral symmetry in a parent chiral-symmetric higher-order phase (Langbehn et al., 2017, Okugawa et al., 2019, Fu et al., 2021).

Another important distinction is between symmetry as a diagnostic and symmetry as a constructive device. The chiral-symmetry-only constructions of second-order topology require no crystalline symmetries, whereas reflection symmetry can be employed to systematically generate explicit models of second-order topological insulators and superconductors. In the reflection-based constructions, the protected corner or hinge states continue to exist if reflection symmetry is broken, provided the bulk and boundary gaps remain open (Okugawa et al., 2019, Langbehn et al., 2017).

2. Boundary mass inversion and corner physics in two dimensions

The most direct low-energy description of a two-dimensional chiral second-order topological insulator is an edge theory of massive Dirac fermions. For an edge with orientation angle $\alpha$, the projected Hamiltonian is

$$H_{\alpha,\mathrm{eff}}= \begin{pmatrix} -A\hat{k}_\parallel & iM(\alpha)e^{-i\alpha}\ -iM(\alpha)e^{i\alpha} & A\hat{k}_\parallel \end{pmatrix},$$

and a zero-energy corner state appears at the intersection of two edges if and only if the corresponding masses have opposite signs,

$$\mathrm{sign}[M(\alpha_1)] \neq \mathrm{sign}[M(\alpha_2)].$$

Its decay length along an edge is

$R_\alpha=\left|\frac{A}{M(\alpha)}\right|.$

This is the Jackiw-Rebbi mechanism in higher-order form: the corner is a mass domain wall, and $M(\alpha)\to 0$ drives delocalization from a corner state into an edge state (Poata et al., 2023).

Two explicit class-AIII models illustrate how geometry and symmetry fix the corner spectrum. In the inversion-symmetric model, the added mass term gives

$$M_{\mathrm{I}}(\alpha)=C\sin\!\left(\alpha-\frac{\pi}{4}\right),$$

and the number of corner states in a convex polygon is always two. In the model preserving the combined $\hat{C}_4^z\hat{T}$ symmetry, the edge mass is

$\mathcal{C}$0

so convex polygons can host $\mathcal{C}$1, $\mathcal{C}$2, or $\mathcal{C}$3 corner states, always in an even number. For triangular flakes, rotation changes the position of the corner states in the first model and can change both their position and their number in the second; for square flakes in the $\mathcal{C}$4-symmetric model, there is an orientation for which the corner states extend along the whole perimeter (Poata et al., 2023).

A more general chiral-symmetric lattice construction writes the Hamiltonian as

$\mathcal{C}$5

with second-order invariant

$\mathcal{C}$6

where $\mathcal{C}$7 and $\mathcal{C}$8 are the winding numbers of the constituent one-dimensional chiral systems. This formulation makes the bulk-corner correspondence explicit: topological edges in both directions imply zero modes at their intersections (Okugawa et al., 2019).

An alternative two-dimensional route uses stacked Chern insulators with opposite chiralities. In the bilayer Haldane construction, counterpropagating edge states are gapped by interlayer hopping, but the induced edge mass changes sign at corners, again yielding Jackiw-Rebbi zero modes. In that setting, a Jacobian-transformed nested Wilson loop diagnoses the bulk-corner correspondence, and the corner sector exhibits a filling anomaly with fractional charge $\mathcal{C}$9 in the hexagonal geometry (Shang et al., 2020).

3. Three-dimensional chiral hinge phases and their constructions

In three dimensions, a second-order topological insulator is a bulk insulator with gapped surfaces and one-dimensional gapless modes at the hinges. For chiral second-order phases, these hinge states are unidirectional and can form distinct flowing patterns, including double-loop and single-loop configurations. The bulk-hinge correspondence is captured by a pair of bulk indices: a quadrupole index that fixes the existence and direction of four chiral hinge modes along a given axis, and a slab Chern number that determines whether the hinge modes form closed loops carrying a three-dimensional quantum anomalous Hall response (Fu et al., 2021).

One route to these phases stacks two-dimensional chiral-symmetric second-order topological insulators along a third direction. In the chiral-symmetric limit, the three-dimensional systems support flat hinge bands or second-order topological semimetal phases depending on how the winding numbers vary with $\{H,\sigma_z\tau_x\}=0$0. Breaking chiral symmetry gaps the surface Dirac points; when adjacent surfaces acquire opposite Chern numbers, a propagating chiral hinge state appears at their interface (Okugawa et al., 2019).

A second route starts from a conventional first-order topological phase and gaps its surfaces by symmetry-allowed masses. In the reflection-symmetric class-A model, the surface Dirac theory admits a unique mass term that is odd under reflection, so reflection-related facets acquire masses with opposite signs. Their intersection is therefore a hinge domain wall supporting a one-dimensional chiral mode. This mechanism also underlies the statement that a three-dimensional second-order topological insulator with broken time-reversal symmetry shows a Hall conductance quantized in units of $\{H,\sigma_z\tau_x\}=0$1 (Langbehn et al., 2017).

A third route uses stacks of alternating Chern insulators. In a Chern-insulator stack with layers A and B carrying opposite Chern numbers and with dimerized interlayer couplings $\{H,\sigma_z\tau_x\}=0$2 and $\{H,\sigma_z\tau_x\}=0$3, the net bulk Chern vector is $\{H,\sigma_z\tau_x\}=0$4, yet the nontrivial regime $\{H,\sigma_z\tau_x\}=0$5 leaves dangling outermost layers whose chiral edge states become chiral hinge states of the three-dimensional structure. In the photonic implementation, the corresponding quadrupole indices are $\{H,\sigma_z\tau_x\}=0$6, $\{H,\sigma_z\tau_x\}=0$7, and $\{H,\sigma_z\tau_x\}=0$8 (Xia et al., 2024).

The coupled-wire construction extends the catalog further. A three-dimensional array of weakly coupled nanowires, subject to rotating magnetic fields and spatially modulated interwire tunnelings, can gap the bulk and all surfaces while leaving one or more chiral hinge states propagating along a closed hinge path. Depending on the commensurability between the rotating-field period and the Fermi wavelength, the same framework realizes integer and fractional chiral second-order topological insulators; in the interacting fractional regime, the hinge excitation carries charge $\{H,\sigma_z\tau_x\}=0$9 for a positive odd integer $\alpha$0 (Pinchenkova et al., 8 Apr 2025).

4. Topological diagnostics and observables

The diagnostic language of chiral second-order topology is correspondingly diverse. In chiral-symmetry-only lattice models, the winding number of a one-dimensional off-diagonal block,

$\alpha$1

enters multiplicatively into the second-order invariant $\alpha$2. This formula emphasizes that the higher-order phase is assembled from lower-dimensional chiral topological building blocks (Okugawa et al., 2019).

For stacked Chern constructions, nested Wilson loops provide a bulk characterization. In the bilayer Haldane-based second-order topological insulator, the determinant of the nested Wilson loop gives

$\alpha$3

where $\alpha$4 is the second Stiefel-Whitney number. The same work stresses that the Wannier spectrum can lack a Wannier gap, so the higher-order phase coexists with fragile topology rather than the fully stable nested-Wilson-loop structure familiar from quadrupole models (Shang et al., 2020).

In three-dimensional chiral hinge systems, the quadrupole index and slab Chern number form a natural pair. For hinges directed along $\alpha$5, the quadrupole moment

$\alpha$6

defines

$\alpha$7

while the slab Hall conductance is

$\alpha$8

The pair $\alpha$9 distinguishes double-loop hinge patterns from single-loop patterns and links the hinge network to a three-dimensional quantum anomalous Hall effect (Fu et al., 2021).

A broader topological ancestry is supplied by dimensional reduction. Descendants of a four-dimensional chiral Chern insulator yield two-dimensional second-order phases whose corner charge is quantized: for a nontrivial region enclosing the relevant Dirac or monopole structure, $$H_{\alpha,\mathrm{eff}}= \begin{pmatrix} -A\hat{k}_\parallel & iM(\alpha)e^{-i\alpha}\ -iM(\alpha)e^{i\alpha} & A\hat{k}_\parallel \end{pmatrix},$$0, whereas the trivial region gives $$H_{\alpha,\mathrm{eff}}= \begin{pmatrix} -A\hat{k}_\parallel & iM(\alpha)e^{-i\alpha}\ -iM(\alpha)e^{i\alpha} & A\hat{k}_\parallel \end{pmatrix},$$1. This relation ties corner charge accumulation to the first and second Chern numbers of the higher-dimensional ancestor and places chiral second-order topology within a hierarchy of topological pumps (Petrides et al., 2019).

5. Disorder, magnetic field, and transport

The disorder response of chiral second-order topological insulators is now understood to be structured rather than uniformly destructive. In the three-dimensional chiral second-order topological insulator studied with random scalar disorder preserving $$H_{\alpha,\mathrm{eff}}= \begin{pmatrix} -A\hat{k}_\parallel & iM(\alpha)e^{-i\alpha}\ -iM(\alpha)e^{i\alpha} & A\hat{k}_\parallel \end{pmatrix},$$2, the side-surface density of states becomes nonzero at $$H_{\alpha,\mathrm{eff}}= \begin{pmatrix} -A\hat{k}_\parallel & iM(\alpha)e^{-i\alpha}\ -iM(\alpha)e^{i\alpha} & A\hat{k}_\parallel \end{pmatrix},$$3, the bulk density of states becomes nonzero at $$H_{\alpha,\mathrm{eff}}= \begin{pmatrix} -A\hat{k}_\parallel & iM(\alpha)e^{-i\alpha}\ -iM(\alpha)e^{i\alpha} & A\hat{k}_\parallel \end{pmatrix},$$4, the transition from chiral second-order topological insulator to diffusive metal occurs at $$H_{\alpha,\mathrm{eff}}= \begin{pmatrix} -A\hat{k}_\parallel & iM(\alpha)e^{-i\alpha}\ -iM(\alpha)e^{i\alpha} & A\hat{k}_\parallel \end{pmatrix},$$5, and the Anderson-insulating transition occurs at $$H_{\alpha,\mathrm{eff}}= \begin{pmatrix} -A\hat{k}_\parallel & iM(\alpha)e^{-i\alpha}\ -iM(\alpha)e^{i\alpha} & A\hat{k}_\parallel \end{pmatrix},$$6. The averaged conductance remains quantized at $$H_{\alpha,\mathrm{eff}}= \begin{pmatrix} -A\hat{k}_\parallel & iM(\alpha)e^{-i\alpha}\ -iM(\alpha)e^{i\alpha} & A\hat{k}_\parallel \end{pmatrix},$$7 up to $$H_{\alpha,\mathrm{eff}}= \begin{pmatrix} -A\hat{k}_\parallel & iM(\alpha)e^{-i\alpha}\ -iM(\alpha)e^{i\alpha} & A\hat{k}_\parallel \end{pmatrix},$$8, and an averaged conductance plateau of $$H_{\alpha,\mathrm{eff}}= \begin{pmatrix} -A\hat{k}_\parallel & iM(\alpha)e^{-i\alpha}\ -iM(\alpha)e^{i\alpha} & A\hat{k}_\parallel \end{pmatrix},$$9 emerges in the diffusive metallic phase (Shen et al., 2023).

Structural disorder can also induce rather than merely degrade higher-order topology. In amorphous three-dimensional systems without crystalline symmetry, the second-order phase is characterized by the winding number of the quadrupole moment,

$$\mathrm{sign}[M(\alpha_1)] \neq \mathrm{sign}[M(\alpha_2)].$$0

with $$\mathrm{sign}[M(\alpha_1)] \neq \mathrm{sign}[M(\alpha_2)].$$1 in the second-order phase and $$\mathrm{sign}[M(\alpha_1)] \neq \mathrm{sign}[M(\alpha_2)].$$2 in the trivial phase. The same real-space framework predicts quantized longitudinal conductance $$\mathrm{sign}[M(\alpha_1)] \neq \mathrm{sign}[M(\alpha_2)].$$3 for the chiral amorphous phase and $$\mathrm{sign}[M(\alpha_1)] \neq \mathrm{sign}[M(\alpha_2)].$$4 for the helical time-reversal-symmetric amorphous phase, together with hinge-localized local density of states (Wang et al., 2020).

Magnetic fields expose another layer of boundary physics. In strong fields, surfaces pierced by flux develop Landau levels, while the chiral hinge modes mediate spectral flow between neighboring surfaces. Full lattice calculations show that the lowest Landau level lies closer to zero energy than the simple massive-Dirac surface theory predicts, so that inside the surface gap there are energy regions with either one or two chiral hinge modes propagating in either direction; the differential conductance is then quantized to either one or two conductance quanta (Levitan et al., 2020). A refined surface theory including $$\mathrm{sign}[M(\alpha_1)] \neq \mathrm{sign}[M(\alpha_2)].$$5 terms explains why the $$\mathrm{sign}[M(\alpha_1)] \neq \mathrm{sign}[M(\alpha_2)].$$6 Landau level falls inside the surface Dirac gap and why the $$\mathrm{sign}[M(\alpha_1)] \neq \mathrm{sign}[M(\alpha_2)].$$7 levels become asymmetric around zero energy under magnetic field, even though the zero-field model remains particle-hole symmetric (Levitan et al., 2021).

6. Realizations, analog platforms, and extensions

Experimental realization has advanced most directly in photonics. A six-layer stack of gyromagnetic photonic crystals with alternating layer magnetization and dimerized interlayer coupling realizes the chiral hinge states predicted for a Chern-insulator stack. Microwave measurements and full-wave simulations show gapless hinge propagation localized on the outermost layers, and the hinge state propagates around a metallic obstacle and around sharp corners with minimal backscattering (Xia et al., 2024).

Electronic and metamaterial generalizations display distinct higher-order signatures. Topological circuitry simulations based on stacked Chern insulators with opposite chiralities reproduce second-order topology, bulk-corner correspondence, and fragile topology in the absence of a Wannier gap (Shang et al., 2020). In a breathing square lattice of magnetic vortices, the Chern number vanishes for all parameter choices, while a quantized $$\mathrm{sign}[M(\alpha_1)] \neq \mathrm{sign}[M(\alpha_2)].$$8 Berry phase distinguishes trivial and nontrivial regimes. The resulting higher-order phase supports three corner states at $$\mathrm{sign}[M(\alpha_1)] \neq \mathrm{sign}[M(\alpha_2)].$$9 GHz, $R_\alpha=\left|\frac{A}{M(\alpha)}\right|.$0 GHz, and $R_\alpha=\left|\frac{A}{M(\alpha)}\right|.$1 GHz, protected by a generalized chiral symmetry of the quadripartite lattice (Li et al., 2020).

Bosonic analogues in magnetism emphasize that the higher-order mechanism is not confined to fermionic band insulators. A stacked ferromagnetic honeycomb model with alternating Dzyaloshinskii-Moriya interaction and sublattice-resolved interlayer dimerization yields a second-order topological magnon insulator with chiral hinge magnons protected by the bulk and surface gaps and tunable by boundary termination (Mook et al., 2020). Superconducting proximity adds a further extension: second-order-topological-insulator/superconductor heterostructures with broken time-reversal symmetry realize second-order topological superconductors hosting Majorana corner modes in two dimensions and chiral Majorana hinge modes in three dimensions (Yan, 2019).

Recent materials-oriented work has broadened the concept again. In the chiral altermagnet K[Co(HCOO)$R_\alpha=\left|\frac{A}{M(\alpha)}\right|.$2], a three-dimensional chiral second-order topological phase is predicted with $R_\alpha=\left|\frac{A}{M(\alpha)}\right|.$3-wave spin-split bands, alternating spin-up and spin-down hinge modes along the boundaries of hexagonal nanotubes, and anomalous Hall and magneto-optical responses whose signs reverse between left- and right-handed enantiomers. This suggests that lattice chirality can act as a control knob for higher-order topological transport (Xie et al., 18 Aug 2025).

Chiral second-order topological insulators therefore occupy a broad but coherent category: the protected states are corner zero modes or chiral hinge channels, the operative boundary mechanism is mass inversion, the bulk characterization ranges from winding numbers and nested Wilson loops to quadrupole indices and slab Chern numbers, and the phenomenology spans crystalline, amorphous, interacting, photonic, magnetic, and altermagnetic realizations.