Second-Order Topological Phases

• Second-order topological phases are gapped bulk states with robust, symmetry-protected gapless modes localized at boundaries of co-dimension two.
• They rely on crystalline, chiral, and non-Hermitian symmetry protections to map bulk invariants to unique corner or hinge states.
• These phases are realized in electronic, photonic, bosonic, and superconducting platforms, offering promising avenues for quantum material applications.

Second-order topological phases are gapped bulk states which host gapless or midgap modes localized at boundaries of co-dimension two (corners in two-dimensional systems, hinges in three dimensions). Unlike first-order topological insulators and superconductors, where boundary states reside on (d–1)-dimensional surfaces and are completely determined by bulk invariants, second-order topology supports robust, symmetry-protected (d–2)-dimensional modes which have distinct physical signatures and require new classification frameworks. Second-order phases have been demonstrated or proposed in fermionic, bosonic, and photonic systems, in both Hermitian and non-Hermitian settings, and in both non-interacting and interacting models.

1. Foundational Models and Theoretical Frameworks

Prototypical second-order topological phases include the quadrupole insulator and its generalizations. Classic constructions utilize tight-binding models such as the "2D photonic SSH" model (Xie et al., 2018), where a square lattice with four-site unit cell supports a transition from a trivial phase with no edge or corner modes, to first-order (edge-mode) phases, to a "second-order" phase hosting one corner state per corner. The effective Hamiltonian takes the form: $$H(\mathbf{k})= \begin{pmatrix} 0 & t_a + t_b e^{+ik_x} & t_a + t_b e^{-ik_y} & 0\ t_a + t_b e^{-ik_x} & 0 & 0 & t_a + t_b e^{-ik_y}\ t_a + t_b e^{+ik_y} & 0 & 0 & t_a + t_b e^{+ik_x}\ 0 & t_a + t_b e^{+ik_y} & t_a + t_b e^{-ik_x} & 0 \end{pmatrix}$$ with $t_a$, $t_b$ controlling intra/inter-cell couplings.

Classification schemes have been developed for both symmetry-protected crystalline SOTIs/SOTSCs (Geier et al., 2018) and for chiral-symmetry-protected (class AIII) second-order phases (Okugawa et al., 2019). These frameworks clarify the symmetry indicators that distinguish second-order phases from first-order ones, and establish the mapping between bulk invariants and protected (co-)dimension-2 boundary states.

2. Symmetry Protection and Topological Invariants

The emergence and stability of second-order phases strongly depend on spatial and/or internal symmetries:

• Crystalline symmetries: Mirror, twofold rotation, inversion, and $C_n$ rotation symmetries can quantize the needed invariants and pin domain walls to corners or hinges (Geier et al., 2018). Minimal Dirac models with mirror or rotation symmetries support $\mathbb{Z}$, $\mathbb{Z}_2$, or $\mathbb{Z}_n$ indices, such as mirror Chern numbers or rotation eigenvalue differences.
• Chiral (sublattice) symmetry: In class AIII (chiral but neither $\mathcal{T}$ nor $\mathcal{P}$), analytically tractable models demonstrate that the product of winding numbers in $x$ and $y$ directions, $\nu_{2D} = w_x w_y$, is the second-order invariant (Okugawa et al., 2019).
• Polarization and nested Wilson loops: In 2D, quantized bulk polarization $\vec{P}=(P_x,P_y)$, and edge polarization via the nested Wilson loop, are key invariants. For photonic SSH, corner charge $Q_c=4P_xP_y$ completely dictates the existence of corner states (Xie et al., 2018).
• Berry phases and Wannier bands: Cluster models and Kekulé-distorted honeycomb models utilize twist Berry phases (e.g., quantized $\mathbb{Z}_2$ or $\mathbb{Z}_6$ Berry phases on finite clusters) to identify topological filling anomalies and fragile topology (Bunney et al., 2021).

In non-Hermitian systems, point-gap winding numbers and non-Bloch extensions of Wilson loops supersede traditional line-gap invariants (Tanaka et al., 2024, Yang et al., 19 Jan 2026). The location and count of corner/hinge states may require diagnosing winding in the complex energy plane, rather than real spectral invariants.

3. Boundary and Corner/Hinge States

The hallmark of a second-order phase is the appearance of boundary-localized states at corners (2D) or hinges (3D), which are not connected to traditional 1D or 2D surface modes:

• Edge and corner criteria (photonic SSH): Edge states require $P_i=1/2$ (topological polarization), with corner states requiring $(P_x,P_y)=(1/2,1/2)$ (Xie et al., 2018).
• Jackiw–Rebbi domain wall principle: Corner/hinge states result at the intersection of two edges/surfaces where the mass term of the edge/hinge Dirac theory changes sign. This applies to superconductors (Majorana modes trapped at corners (Plekhanov et al., 2020, Wang et al., 2023)), quantum anomalous Hall bilayers (Liu et al., 2024), and Kekulé-distorted graphene (Bunney et al., 2021).
• 3D stacks and higher order: Stacking 2D SOTIs with alternating interlayer couplings produces 3D SOTIs with hinge states, or SOTSMs with dispersing hinge Fermi arcs (Okugawa et al., 2019, Liu et al., 2024, Chen et al., 2023). The number and chirality of hinge modes track the second-order invariant.

Non-Hermitian settings generate new behaviors: corner/hinge states can accumulate only at a single boundary through a non-Hermitian skin effect, violating bulk-corner correspondence unless point-gap winding is properly incorporated (Liu et al., 2018, Yang et al., 19 Jan 2026).

4. Materials Platforms and Realizations

Second-order topology is found or proposed in a variety of platforms:

• Electronic: Bilayer (Kane–Mele, BHZ) graphene, topological quantum wells, and honeycomb-lattice models with engineered bond or gauge field textures realize both strong and fragile second-order phenomena (Bunney et al., 2021, Shang et al., 2020, Liu et al., 2024).
• Photonic/phononic: 2D and 3D photonic SSH or Kagome crystals demonstrate direct midgap corner modes, with non-Hermitian variants exhibiting corner-mode splitting and bulk-mode skin accumulation (Xie et al., 2018, Yang et al., 19 Jan 2026).
• Bosonic/magnonic: Breathing kagome magnon insulators are explicitly shown to exhibit corner-localized bosonic modes, controlled via Dzyaloshinskii–Moriya interaction and exchange anisotropy (Sil et al., 2019).
• Superconducting: SOTSCs with Majorana corner modes arise from a variety of mechanisms, including phase-tunable Josephson arrays, $d+id$-wave orbital pairing (Wang et al., 2023), or via coupled-proximity/TI arrays (1904.02437, Plekhanov et al., 2020).
• Interacting systems: 1D models with correlated hopping that preserve inversion and time-reversal, but not chiral, produce gapped edge states and an even-degenerate entanglement spectrum, the hallmark of an interacting SOTI (Montorsi et al., 2022).
• Quasicrystalline and chiral altermagnetic systems: Forbidden rotation symmetries in quasicrystals enable SOTIs/SOTSMs with 8, 12, or more corner/hinge states (Chen et al., 2023), and chirality in altermagnets directly locks and reverses hinge state spin and resulting Hall/Kerr/Faraday effects (Xie et al., 18 Aug 2025).

Floquet-driving, especially when combined with non-Hermiticity, allows for unprecedented tunability, dynamically generating second-order phases and hybrid-order band-touching phenomena (Pan et al., 2020, Wang et al., 2021, Wu et al., 2024).

5. Diagnostics, Bulk–Boundary Correspondence, and Experiments

Detection and diagnostics of second-order phases leverage multiple approaches:

• Theoretical diagnostics: Calculation of quantized polarization, nested Wilson loops, Berry phases, and real-space symmetry indicators remains central for non-interacting systems. In non-Hermitian systems, point-gap winding must be used (Tanaka et al., 2024, Liu et al., 2018, Yang et al., 19 Jan 2026).
• Experimental probes: Corner or hinge states are observable directly through STM (zero-bias peaks at corners), local impedance or microwave spectroscopy (bosonic/photonic corners), or conductance quantization in superconducting devices (Plekhanov et al., 2020, Wang et al., 2023, 1904.02437).
• Disorder robustness: Many SOTIs/SOTSCs exhibit robustness to moderate disorder; some realize a second-order Anderson phase where disorder induces the topological regime (1904.02437).
• Dynamics and Floquet probes: Measurements of quench dynamics and time-averaged Floquet observables have been proposed for direct detection of bulk topological charges and quadrupole-like invariants (Wu et al., 2024, Pan et al., 2020).
• Breakdown and restoration of bulk–boundary correspondence: In non-Hermitian and fragile (Wannier-band crossing) cases, correspondence can fail or be anomalous, requiring finer analysis (Yang et al., 19 Jan 2026, Shang et al., 2020).

6. Future Directions and Generalizations

Ongoing research explores novel settings and generalizations:

• Non-Hermitian second-order phases with exceptional point-gap topology and associated non-Bloch invariants (Tanaka et al., 2024).
• Hybrid-order topological semimetals and Weyl phases, where higher- and first-order invariants coexist, producing bulk nodes, hinge arcs, and surface Fermi arcs (Wang et al., 2021, Chen et al., 2023).
• Intrinsically interacting and bosonic SOTIs/SOTSCs, including extension to mixed-symmetry-protected and higher-hierarchy phases (Montorsi et al., 2022, Dubinkin et al., 2018).
• Chirality-programmable second-order phases in chiral altermagnetic crystals, with electrically and optically switchable boundary modes (Xie et al., 18 Aug 2025).
• Floquet engineering of topological hierarchy and transitions between first-, second-, and hybrid-order phases (Wu et al., 2024, Pan et al., 2020).

As the classification and realization landscape expands—incorporating nontrivial crystalline, chiral, non-Hermitian, and interaction-driven mechanisms—second-order topology is poised to provide a robust and flexible paradigm for localized, symmetry-protected states, with broad implications for quantum materials, photonics, magnonics, and topological quantum computation.