Second-Order Fermi Arc States

• Second-order Fermi arc states are topologically protected one-dimensional modes localized at the hinges of 3D higher-order topological semimetals, emerging via mechanisms like Jackiw–Rebbi domain walls.
• They are realized in diverse systems, including quasicrystalline stacks and spin-3/2 Majorana models, with experimental validation from STM/STS and numerical diagonalization.
• Their robustness stems from bulk-hinge correspondence and higher-order invariants such as the second Stiefel-Whitney number and quadrupole moment, enabling anisotropic, hinge-localized conduction.

Second-order Fermi arc states are topologically protected one-dimensional modes that localize on the hinges—i.e., the intersection lines between facets—of three-dimensional (3D) higher-order topological semimetals. These states generalize conventional Fermi arcs, which appear on surfaces of Weyl semimetals due to first-order topological bulk-boundary correspondence, to situations where the nontrivial topology enforces gapless hinge or corner states. Unlike surface Fermi arcs, second-order Fermi arcs provide a direct manifestation of higher-order topology in both crystalline and quasicrystalline materials, as well as in engineered spin models. Their existence, characterization, symmetry protection, and experimental signatures have been established in several theoretical and experimental studies.

1. Conceptual Foundation and Definition

Second-order Fermi arc states arise in 3D systems that possess higher-order topological phases, typically as a result of quantized bulk invariants beyond conventional Chern or $\mathbb{Z}_2$ indices. In such systems, the bulk is metallic (with nodal points or lines), surfaces are generically gapped, but gapless states localize on lower-dimensional boundaries—i.e., the one-dimensional hinges between facets. These hinge-localized states are the "second-order Fermi arcs."

In crystalline Weyl semimetals, each Weyl node acts as a monopole of Berry curvature, and the resulting change in Chern number across momentum slices results in surface Fermi arcs. However, if the surface mass term changes sign across hinges (step edges), a Jackiw–Rebbi domain-wall mechanism guarantees the existence of robust one-dimensional chiral modes localized at the intersection—these are hinge Fermi arcs and constitute direct evidence of second-order bulk-boundary correspondence (Zheng et al., 2023). In more general higher-order topological semimetals—including those with quasicrystalline aperiodicity or emergent Majorana fermions—second-order Fermi arcs appear when symmetry-protected bulk topological invariants control strong boundary phenomena on hinges (Chen et al., 2023, Zhao et al., 2020).

2. Microscopic Models and Hamiltonians

Multiple microscopic models realize second-order Fermi arcs, including:

• Quasicrystalline Stacks:

The quasicrystalline second-order topological semimetal (SOTSM) model considered by Chen, Zhou, and Xu involves stacking two-dimensional second-order topological insulators (SOTIs), defined on non-crystalline Ammann–Beenker or Stampfli tilings, along a third direction using orbital-dependent interlayer hopping. The full 3D Hamiltonian is:

$$H_{3D} = \sum_{z} \big[ H_{2D}(M)_z + t_z \sum_j (c^\dagger_{j,z} s_0 \tau_3 c_{j,z+1} + h.c.) \big],$$

where $H_{2D}(M)$ represents the 2D SOTI on a quasicrystalline tiling. After Fourier transformation in $z$, the gap closes at certain $k_z^*$, resulting in 4-fold degenerate Dirac-like points. Between these, exactly $n$ zero-energy bands, flat in $k_z$, run along the 1D hinges—these are the second-order Fermi arcs (Chen et al., 2023).

• Spin-3/2 Majorana Quantum Spin Liquids:

An exactly solvable spin-3/2 model on a pentacoordinated graphite lattice, with bond-dependent quartic spin interactions, maps to free Majorana fermions coupled to a static $\mathbb{Z}_2$ gauge field. Nodal-line semimetal phases appear with nontrivial second Stiefel-Whitney and Berry $\mathbb{Z}_2$ invariants. Bulk-boundary correspondence guarantees the presence of zero-energy modes extended along hinges for specific momentum ranges, forming hinge Fermi arcs (Zhao et al., 2020).

• Weyl Semimetals:

In TaAs and related compounds, ab initio tight-binding Hamiltonians describe bulk Weyl point physics. When a sample features a step edge or hinge between different surface facets, the sign of the mass term changes across the hinge, leading to a localized, linearly dispersing 1D Fermi arc mode—experimentally realized as a robust feature in local density of states and directly observed via STM/STS (Zheng et al., 2023).

3. Bulk Topological Invariants and Symmetry Protection

Second-order Fermi arc states are robustly protected by specific symmetries and quantized invariants:

• Bulk-Hinge Correspondence:

In Weyl semimetals, the projection of the bulk Chern number difference between two surfaces onto the hinge ensures protected 1D modes when this difference is nonzero (Zheng et al., 2023).

• Higher-Order Topology:

In models with real Bloch bands and $\mathcal{P}T$ (off-centered spacetime inversion) symmetry, the second Stiefel-Whitney number $w_2$ classifies topologically distinct gapped 2D slices. For slices with $w_2=1$, the 2D Hamiltonian is a second-order topological insulator, which in 3D stacking manifests as hinge arcs for certain momentum ranges—protected against any perturbation that preserves symmetry and gap (Zhao et al., 2020).

• Quasicrystalline Systems:

In quasicrystalline SOTSMs, $n$-fold rotational symmetry (for $n=4,8,12$) not allowed in periodic crystals protects $n$-component hinge multiplets. The nested Wilson-loop-derived quadrupole moment $Q_{xy}(k_z)$ jumps as a function of $k_z$ precisely where hinge arcs appear, directly tying topology to boundary mode emergence (Chen et al., 2023).

4. Distinction from Conventional (First-Order) Fermi Arcs

Second-order Fermi arcs differ fundamentally from first-order, or surface, Fermi arcs in several respects:

• Dimensionality: They are strictly one-dimensional, localized on hinges, as opposed to two-dimensional surface arcs.
• Dispersion: In certain models (especially with strong symmetry or quasicrystallinity), hinge arcs are flat in momentum, i.e., dispersionless in $k_z$ (Chen et al., 2023), though dispersive arcs have also been observed (Zheng et al., 2023).
• Topological Protection: Surface arcs are guaranteed by the change in bulk Chern number; hinge arcs require higher-order topological invariants or mass domain walls between surfaces.
• Aperiodicity and $n$-fold Multiplicity: In quasicrystals, the number of hinge arcs $n$ need not be four, and can be $8$ or $12$ depending on the underlying rotational symmetry of the tiling, a feature forbidden in periodic lattices (Chen et al., 2023).
• Experimental Manifestation: Hinge arcs manifest as zero-bias peaks strictly confined to step edges or hinges, with decay lengths on the order of 1 nm, and are robust against surface disorder (Zheng et al., 2023). In contrast, surface arcs are extended and delocalized.

5. Experimental Observation and Numerical Evidence

• STM/STS of TaAs: Edge-selective dI/dV mapping under atomic-resolution conditions reveals a zero-bias conductance peak at atomic-step ledges, with an exponential decay into the adjacent terrace. The peak persists on multiple step heights and is robust to local defects, confirming topological origin (Zheng et al., 2023).
• Numerical Diagonalization: For both quasicrystalline (Chen et al., 2023) and spin 3/2 (Zhao et al., 2020) models, direct diagonalization of finite lattices under open boundary conditions reveals hinge-localized zero-energy states with the expected symmetry-multiplicity and bulk-hinge correspondence.

Table: Comparison of Second-order Fermi Arc Prototypes

Model Type	Symmetry / Invariant	Hinge Arc Features
TaAs Weyl semimetal (Zheng et al., 2023)	Bulk Chern, mass domain wall	1D chiral, linear-dispersing hinge arcs
Quasicrystal SOTSM (Chen et al., 2023)	$C_n^z T$, quadrupole $Q_{xy}(k_z)$	n-fold, flat (kz), strongly localized
Spin-3/2 Majorana QSL (Zhao et al., 2020)	$w_2,\nu_{1D}$, $\mathcal{P}T$	Dispersionless, hinge-localized Majorana

6. Physical Consequences and Experimental Probes

Second-order Fermi arc states have several experimentally accessible signatures:

• Local DOS: Zero-bias peaks tunable by changing chemical potential, localized at hinges according to spatial and symmetry selection rules (Chen et al., 2023, Zheng et al., 2023).
• Transport Anisotropy: Onset of highly anisotropic conduction when the chemical potential enters the hinge arc bands, with negligible bulk/surface background in gapped regions (Chen et al., 2023).
• Disclination Bound States: In quasicrystalline SOTSMs, topological defects such as wedge disclinations host zero-mode bound states for each $k_z$-slice carrying the higher-order topological invariant, manifesting the tie between geometry and hinge topology (Chen et al., 2023).

7. Theoretical and Material Scope

The emergence of second-order Fermi arc states decisively broadens the landscape of topological matter:

• Beyond Crystallinity: Quasicrystalline, aperiodic, and amorphous systems can host higher-order Fermi arcs through spatially nontrivial tilings and symmetry-protected mechanisms (Chen et al., 2023).
• Symmetry Classes: Both symmetry-enforced (from rotation, mirror, or inversion) and symmetry-indicated (from projections of bulk invariants) mechanisms are realized, inducing second-order Fermi arcs across a range of topological symmetry classes (Chen et al., 2023, Zhao et al., 2020).
• Correlated Spin Systems: Quantum spin liquids with Majorana fermions represent a strongly interacting platform for realizing and controlling hinge-localized Fermi arcs, with clear classification in terms of real band topology (Zhao et al., 2020).

A plausible implication is that higher-order topological semimetals, including second-order Fermi arc states, offer a robust and tunable platform for edge-mode engineering and may be relevant for future quantum devices. Their robustness to crystalline symmetry breaking, as in quasicrystals or at atomic-scale defects, underlines the universality of topological protection in condensed matter systems.