---
title: Second-Order Topological Phases
url: https://www.emergentmind.com/topics/second-order-topological-phases
type: topic
---

# Second-Order Topological Phases

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 [1805.07555], 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 [1801.10053] and for chiral-symmetry-protected (class AIII) second-order phases [1907.01153]. 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 [1801.10053]. 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 [1907.01153].
- **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 [1805.07555].
- **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 [2110.15549].

In non-Hermitian systems, point-gap winding numbers and non-Bloch extensions of Wilson loops supersede traditional line-gap invariants [2411.06898, 2601.12760]. 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 $(P_x,P_y)=(1/2,1/2)$ [1805.07555].
- **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 [2008.03611, 2310.17992]), quantum anomalous Hall bilayers [2406.01037], and Kekulé-distorted graphene [2110.15549].
- **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 [1907.01153, 2406.01037, 2307.04334]. 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 [1810.04067, 2601.12760].

## 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 [2110.15549, 2009.09167, 2406.01037].
- **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 [1805.07555, 2601.12760].
- **Bosonic/magnonic:** Breathing kagome magnon insulators are explicitly shown to exhibit corner-localized bosonic modes, controlled via Dzyaloshinskii–Moriya interaction and exchange anisotropy [1911.12620].
- **Superconducting:** SOTSCs with Majorana corner modes arise from a variety of mechanisms, including phase-tunable Josephson arrays, $d+id$-wave orbital pairing [2310.17992], or via coupled-proximity/TI arrays [1904.02437, 2008.03611].
- **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 [2208.00939].
- **Quasicrystalline and chiral altermagnetic systems:** Forbidden rotation symmetries in quasicrystals enable SOTIs/SOTSMs with 8, 12, or more corner/hinge states [2307.04334], and chirality in altermagnets directly locks and reverses hinge state spin and resulting Hall/Kerr/Faraday effects [2508.12770].

Floquet-driving, especially when combined with non-Hermiticity, allows for unprecedented tunability, dynamically generating second-order phases and hybrid-order band-touching phenomena [2004.06283, 2106.03067, 2406.08532].

## 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 [2411.06898, 1810.04067, 2601.12760].
- **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 [2008.03611, 2310.17992, 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 [2406.08532, 2004.06283].
- **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 [2601.12760, 2009.09167].
  
## 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 [2411.06898].
- Hybrid-order topological semimetals and Weyl phases, where higher- and first-order invariants coexist, producing bulk nodes, hinge arcs, and surface Fermi arcs [2106.03067, 2307.04334].
- Intrinsically interacting and bosonic SOTIs/SOTSCs, including extension to mixed-symmetry-protected and higher-hierarchy phases [2208.00939, 1807.09781].
- Chirality-programmable second-order phases in chiral altermagnetic crystals, with electrically and optically switchable boundary modes [2508.12770].
- Floquet engineering of topological hierarchy and transitions between first-, second-, and hybrid-order phases [2406.08532, 2004.06283].

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.

Source: https://www.emergentmind.com/topics/second-order-topological-phases