Linked Fermi Arcs in Weyl Semimetals

• Linked Fermi arcs are interconnected surface-state structures in topological semimetals governed by both local Weyl-node chirality and global symmetry constraints.
• Researchers use tight-binding models, Wilson-loop analysis, and boundary-condition engineering to reveal arc–pocket networks and intervalley coupling effects.
• These insights inform experimental probes and material design, impacting quantum transport in electronic and photonic systems.

Searching arXiv for recent and foundational papers on linked Fermi arcs and closely related surface-arc topology in Weyl/Dirac systems. Linked Fermi arcs are surface-state structures in topological semimetals in which the open Fermi-arc contours are not independent objects but are constrained to coexist, reconnect, or intertwine with additional surface features by bulk topology, surface symmetry, or interface reconstruction. In time-reversal-invariant Weyl semimetals, the most precise usage refers to arc configurations whose connectivity is topologically interdependent with closed Fermi pockets arising from surface Dirac cones pinned to time-reversal invariant momenta, so that the arcs and pockets form a linked arc–pocket network (Lau et al., 2017). More broadly, the term also appears in settings where intervalley boundary coupling links previously disconnected surface-state rays into arcs (Devizorova et al., 2016), where opposite surfaces or twisted interfaces reconstruct arcs into closed loops or multi-surface networks [(Hosur, 2012); (Abdulla et al., 2021)], and where Floquet driving produces surface contours that become nontrivially linked on the compactified surface Brillouin-zone torus (Liu et al., 6 Jul 2025). Across these realizations, the unifying theme is that Fermi-arc connectivity is controlled not only by local Weyl-node chirality, but also by global constraints from symmetry, boundary conditions, topology, and external driving.

1. Topological setting and basic definitions

In a Weyl semimetal, bulk bands cross linearly at isolated Weyl nodes, each of which acts as a Berry-flux monopole with chirality

$$C=\frac{1}{2\pi}\oint_S \Omega(\mathbf{k})\cdot d\mathbf{S}.$$

Because the net charge over the Brillouin zone vanishes, Weyl nodes occur in opposite-chirality sets. On a crystal surface, this bulk topology is encoded by open constant-energy contours, the Fermi arcs, which terminate at the surface projections of bulk Weyl nodes of opposite chirality (Lau et al., 2017).

The adjective “linked” has several technically distinct meanings in the literature. In the time-reversal-invariant Weyl-semimetal setting of “Generic coexistence of Fermi arcs and Dirac cones on the surface of time-reversal invariant Weyl semimetals” (Lau et al., 2017), it denotes the coexistence and topological interdependence of Fermi arcs with closed Fermi pockets originating from surface Dirac cones pinned to surface TRIM. In “Fermi arcs formation in Weyl semimetals: the key role of intervalley interaction” (Devizorova et al., 2016), linked Fermi arcs are open contours obtained when intervalley interfacial coupling connects otherwise disconnected “rays” emitted from Weyl-node projections. In twisted-interface problems, the linkage refers to arc reconstruction, splitting, and reconnection into loops in the mini Brillouin zone (Abdulla et al., 2021). In Floquet systems, “linked” is used in the strict torus-topological sense: after compactifying the surface Brillouin zone, the arc-derived loops have nontrivial Gauss linking number (Liu et al., 6 Jul 2025).

This plurality of meanings does not imply inconsistency. Rather, it indicates that Fermi-arc connectivity is not uniquely fixed by the existence of Weyl nodes alone. This suggests that “linked Fermi arcs” is best understood as a family of phenomena in which surface open contours are subject to additional global constraints beyond simple opposite-chirality endpoint matching.

2. Time-reversal-invariant Weyl semimetals and arc–pocket linkage

The most developed topological formulation of linked Fermi arcs occurs in time-reversal-symmetric Weyl semimetals (Lau et al., 2017). In such systems, each Weyl node is accompanied by a time-reversal partner of the same chirality, so the total number of bulk nodes is $4n$. More importantly, the six time-reversal-invariant planes $k_i=0,\pi$, $i\in\{x,y,z\}$, are gapped away from the Weyl nodes and each carries an independent $\mathbb{Z}_2$ invariant. Because the three-dimensional strong index is not defined in a gapless Weyl semimetal, these six $\mathbb{Z}_2$ indices are independent (Lau et al., 2017).

For a given plane, the Fu–Kane sewing-matrix expression is

$$(-1)^{\nu}=\prod_{\Gamma_i}\frac{\sqrt{\det[w(\Gamma_i)]}}{\operatorname{Pf}\,w(\Gamma_i)},$$

while in the Wilson-loop formulation one evaluates the winding parity of the Wannier-center phases $\theta_m(k_\perp)$. By bulk–boundary correspondence, a plane with $\nu=1$ requires an odd number of Kramers-pair crossings along the corresponding projected line in the surface Brillouin zone; a plane with $\nu=0$ requires an even number (Lau et al., 2017).

The essential result is that these parity constraints can force the appearance of a closed Fermi pocket centered at a surface TRIM. The pocket originates from a surface Dirac cone pinned to that TRIM and is required when the crossing parity along projected TRI lines cannot be supplied by open arcs alone. The arcs then terminate on, reattach to, or intertwine with the pocket, producing a linked arc–pocket network (Lau et al., 2017).

A central physical picture follows from simultaneous Chern and $4n$0 accounting. A projected Weyl node of charge $4n$1 emits or absorbs one arc; a projection with effective charge $4n$2 must be the endpoint of two arcs. At the same time, a $4n$3 plane contributes one Kramers pair of helical edge modes, which at the surface forms a Dirac cone pinned to a TRIM. At fixed energy, that cone yields a closed Fermi pocket. The linked structure is therefore enforced by parity of helical crossings, not by Weyl-charge counting alone (Lau et al., 2017).

A useful distinction emerges between two connectivity classes. In arcs-only configurations, each projected node connects directly to an opposite-chirality partner. In arcs-plus-pocket configurations, arc reconnection changes which node pairs are joined, and the extra Kramers-pair crossing required by a $4n$4 plane is supplied by the TRIM-centered pocket. This is the canonical meaning of linked Fermi arcs in the contemporary Weyl-semimetal literature (Lau et al., 2017).

3. Surface Lifshitz transitions and material realization in strained LaPtBi

Changing arc connectivity without changing the bulk $4n$5 data produces a surface Lifshitz transition (Lau et al., 2017). Microscopically, arcs bend and intersect as a control parameter varies, reconnect to different endpoints, and create or remove a TRIM-centered pocket so that the odd/even crossing parity along projected TRI lines remains preserved. The six $4n$6 remain unchanged throughout; what changes is the topology of the surface Fermi surface.

The minimal tight-binding model introduced in (Lau et al., 2017) realizes this mechanism on a cubic lattice: $4n$7 Here $4n$8 and $4n$9 are Pauli matrices in spin and orbital space. For suitable parameters the model has four Weyl nodes, with Wilson-loop evaluation giving $k_i=0,\pi$0 and all other $k_i=0,\pi$1 equal to zero. As the inversion-breaking parameter $k_i=0,\pi$2 is tuned, the surface arcs intersect and reconnect; after reconnection, an additional elliptical Fermi pocket appears around the surface TRIM $k_i=0,\pi$3, originating from a surface Dirac cone and required by the $k_i=0,\pi$4 constraint (Lau et al., 2017).

The half-Heusler compound LaPtBi under in-plane compressive strain provides a first-principles material realization. Density-functional calculations show a time-reversal-invariant Weyl phase with eight Weyl nodes on the bulk planes $k_i=0,\pi$5 and $k_i=0,\pi$6. On the (001) surface, the eight nodes project pairwise onto four surface momenta, so each projected point carries effective topological charge $k_i=0,\pi$7 and must emit two arcs. Simultaneously, the TRI planes $k_i=0,\pi$8 and $k_i=0,\pi$9 are fully gapped and have nontrivial $i\in\{x,y,z\}$0 index $i\in\{x,y,z\}$1, established from Wannier-charge-center Wilson loops (Lau et al., 2017).

In LaPtBi, tuning only the Fermi level can drive the surface Lifshitz transition. At the Weyl-node energy, arcs connect projected nodes without additional pockets. As $i\in\{x,y,z\}$2 is raised, the arcs fuse and reconnect, and a closed Fermi pocket appears around $i\in\{x,y,z\}$3, the apex of a surface Dirac cone pinned at $i\in\{x,y,z\}$4. Further raising $i\in\{x,y,z\}$5 causes the pocket to shrink to the Dirac point and then expand again; eventually a second reconnection can restore the original arcs-only topology (Lau et al., 2017). A plausible implication is that chemical-potential control provides a particularly clean experimental route to linked-arc topology because it changes only the surface constant-energy contour, not the bulk topological indices.

4. Boundary-condition mechanisms: intervalley coupling and opposite-surface linkage

A distinct route to linked Fermi arcs arises from boundary-condition engineering. In the analytical continuum model of (Devizorova et al., 2016), the decisive ingredient is intervalley interfacial interaction at a sharp surface. The bulk is described by a two-valley $i\in\{x,y,z\}$6 Hamiltonian with Weyl points at $i\in\{x,y,z\}$7, while the surface at $i\in\{x,y,z\}$8 is encoded by a boundary condition

$i\in\{x,y,z\}$9

where the interface-mixing matrix $\mathbb{Z}_2$0 is constrained by Hermiticity and inversion symmetry and is parameterized by an intravalley phase $\mathbb{Z}_2$1 and an intervalley interaction $\mathbb{Z}_2$2 (Devizorova et al., 2016).

At zero energy, in the absence of intervalley coupling $\mathbb{Z}_2$3, the surface-state contours are “rays” emanating from the Weyl-point projections. Their orientation is controlled by $\mathbb{Z}_2$4. When the $\mathbb{Z}_2$5 rays do not intersect, increasing $\mathbb{Z}_2$6 deforms them and links them into a single open contour, a Fermi arc. The threshold condition is

$\mathbb{Z}_2$7

When the $\mathbb{Z}_2$8 rays cross, any arbitrarily small $\mathbb{Z}_2$9 induces an anticrossing and produces an arc. In this usage, linked Fermi arcs are open contours generated by interfacial intervalley coupling (Devizorova et al., 2016).

The same work introduces a four-valley approximation to capture connectivity involving valleys near opposite Brillouin-zone edges. In that setting, rays can connect “distant” valleys, and additional closed contours may appear that do not intersect Weyl points. These extra contours arise when unclosed branches from different valley pairs recombine. This suggests that arc linkage can be mediated not only by bulk topological indices, but also by multivalley surface hybridization on the scale of the full Brillouin zone (Devizorova et al., 2016).

A different but related linkage mechanism was formulated in a slab construction based on stacking alternating two-dimensional electron and hole Fermi surfaces (Hosur, 2012). The model Hamiltonian is

$\mathbb{Z}_2$0

with alternating couplings $\mathbb{Z}_2$1 or $\mathbb{Z}_2$2. The inequalities $\mathbb{Z}_2$3 on the desired arc set and $\mathbb{Z}_2$4 elsewhere determine which surface hosts the zero modes. In thin slabs, the Fermi arcs from opposite surfaces hybridize, allowing nested backscattering and strong Friedel oscillations in the top-surface LDOS; as thickness increases, bottom-surface amplitudes retreat exponentially and the strong oscillations disappear (Hosur, 2012). Here the relevant linkage is between opposite surfaces rather than within a single surface Brillouin zone.

These boundary-sensitive mechanisms clarify a common misconception. Fermi arcs are often described as if their connectivity were determined uniquely by bulk chirality assignments. The analytical and slab constructions show that boundary conditions, interface coupling, and finite-thickness hybridization can strongly reorganize which surface contours are visible and how they connect, even when the bulk Weyl content is fixed [(Devizorova et al., 2016); (Hosur, 2012)].

5. Interface reconstruction, photonic implementations, and alternative meanings of linkage

At interfaces of twisted Weyl semimetals, surface-arc reconstruction becomes especially pronounced. In the commensurate-twist framework of (Abdulla et al., 2021), two identical Weyl-semimetal slabs are joined with opposite in-plane rotations, creating a superlattice mini Brillouin zone. The interface tunneling conserves momentum modulo superlattice reciprocal vectors and hybridizes multiple folded surface-state copies. The existence of interface-localized states is determined by

$\mathbb{Z}_2$5

where $\mathbb{Z}_2$6 is the tunnel coupling (Abdulla et al., 2021).

As $\mathbb{Z}_2$7 varies, the interface arcs change connectivity, link same-chirality Weyl-point projections, split and merge, and in some parameter windows detach completely from the Weyl-point projections to become closed Fermi loops without endpoints. In these regimes, all interface states are exponentially localized into both slabs and no part of the contour hybridizes with the bulk (Abdulla et al., 2021). This is not the same phenomenon as the TRIM-pocket linkage of (Lau et al., 2017), but it belongs to the same broader class of topology-controlled arc reconstructions.

Photonic Weyl systems have introduced yet another usage. In “Probing and harnessing photonic Fermi arc surface states using light-matter interactions” (García-Elcano et al., 2022), Fermi arcs connect opposite-chirality Weyl points in a photonic slab, and hinge-induced negative refraction routes arc modes across adjacent facets. In that context, “linked” refers both to the momentum-space linkage of arc endpoints and to a robust quantum link between emitters mediated by the surface states. Depending on the ratio between emitter decay time $\mathbb{Z}_2$8 and round-trip time $\mathbb{Z}_2$9, the system realizes either a dissipative chiral quantum channel with concurrence approaching $$(-1)^{\nu}=\prod_{\Gamma_i}\frac{\sqrt{\det[w(\Gamma_i)]}}{\operatorname{Pf}\,w(\Gamma_i)},$$0 or a revival regime with coherent loop-mediated coupling and transient concurrence $$(-1)^{\nu}=\prod_{\Gamma_i}\frac{\sqrt{\det[w(\Gamma_i)]}}{\operatorname{Pf}\,w(\Gamma_i)},$$1 (García-Elcano et al., 2022).

Direct experimental observation of photonic Fermi arcs was reported in chiral hyperbolic metamaterials (Yang et al., 2017). There, the observed arcs connect oppositely charged Weyl-point projections in momentum space and persist across physical edges and three-dimensional steps without backscattering. The paper uses “linked” primarily in the sense of connectivity across distinct bulk iso-frequency surfaces and across distinct sample faces, rather than in the torus-linking or arc–pocket sense (Yang et al., 2017).

These cases underscore that the term has migrated beyond its original electronic-topology usage. This suggests that one should specify the mechanism whenever the phrase is used: intervalley-linked, arc–pocket-linked, interface-reconstructed, opposite-surface-linked, or torus-linked.

6. Global topology, diagnostics, and stricter topological linking

The global-topological perspective of “Global topology of Weyl semimetals and Fermi arcs” (Mathai et al., 2016) recasts Fermi arcs as relative homology classes. For a boundary projection $$(-1)^{\nu}=\prod_{\Gamma_i}\frac{\sqrt{\det[w(\Gamma_i)]}}{\operatorname{Pf}\,w(\Gamma_i)},$$2, bulk invariants in $$(-1)^{\nu}=\prod_{\Gamma_i}\frac{\sqrt{\det[w(\Gamma_i)]}}{\operatorname{Pf}\,w(\Gamma_i)},$$3 map to boundary invariants in $$(-1)^{\nu}=\prod_{\Gamma_i}\frac{\sqrt{\det[w(\Gamma_i)]}}{\operatorname{Pf}\,w(\Gamma_i)},$$4 through a push-forward $$(-1)^{\nu}=\prod_{\Gamma_i}\frac{\sqrt{\det[w(\Gamma_i)]}}{\operatorname{Pf}\,w(\Gamma_i)},$$5. In $$(-1)^{\nu}=\prod_{\Gamma_i}\frac{\sqrt{\det[w(\Gamma_i)]}}{\operatorname{Pf}\,w(\Gamma_i)},$$6, arcs are oriented relative cycles in $$(-1)^{\nu}=\prod_{\Gamma_i}\frac{\sqrt{\det[w(\Gamma_i)]}}{\operatorname{Pf}\,w(\Gamma_i)},$$7, with endpoint data fixed by the projected Weyl charges. The framework makes precise that insulator invariants can “decorate” arcs without changing their endpoint constraints (Mathai et al., 2016).

That work also identifies the strict topological regime in which linked Fermi arcs should literally be understood as linked curves: a surface Brillouin zone of dimension at least three, so that one can define a Gauss linking number between disjoint boundary curves. In that setting, linked bulk Weyl surfaces can project to linked surface arcs protected by ambient isotopy, provided the projected endpoints remain distinct and the generalized charge-neutrality condition is preserved (Mathai et al., 2016). This is conceptually different from the more common three-dimensional-solid case, where the surface Brillouin zone is two-dimensional and ordinary knot/link invariants do not apply.

A concrete modern realization of strict torus-linked arcs is the Floquet-driven Weyl phase generated from intersecting straight nodal lines (Liu et al., 6 Jul 2025). The undriven lattice model

$$(-1)^{\nu}=\prod_{\Gamma_i}\frac{\sqrt{\det[w(\Gamma_i)]}}{\operatorname{Pf}\,w(\Gamma_i)},$$8

hosts straight nodal lines. Under off-resonant circularly polarized light, the Floquet Hamiltonian acquires the form

$$(-1)^{\nu}=\prod_{\Gamma_i}\frac{\sqrt{\det[w(\Gamma_i)]}}{\operatorname{Pf}\,w(\Gamma_i)},$$9

with momentum-dependent coefficients $\theta_m(k_\perp)$0 and $\theta_m(k_\perp)$1 determined by Bessel functions of the drive amplitude (Liu et al., 6 Jul 2025). For representative parameters $\theta_m(k_\perp)$2, the driven phase contains eight pairs of Weyl points, and on a $\theta_m(k_\perp)$3-normal surface each projected net charge $\theta_m(k_\perp)$4 anchors two Fermi arcs (Liu et al., 6 Jul 2025).

The crucial topological feature is that the layer Chern numbers $\theta_m(k_\perp)$5 and $\theta_m(k_\perp)$6 have magnitude $\theta_m(k_\perp)$7 and flip sign across planes containing same-chirality Weyl points. This forces the surface arcs to run across the entire surface Brillouin zone and sew into two closed loops on the compactified torus. Evaluating the Gauss integral gives $\theta_m(k_\perp)$8 for the numerically extracted contours (Liu et al., 6 Jul 2025). In this precise sense, the loops formed by the arc segments are topologically linked.

Experimentally, linked-arc states leave characteristic signatures in momentum-resolved and scattering probes. In the TR-invariant Weyl-semimetal case, the coexistence of arcs and Dirac pockets produces universal quasi-particle interference structure. The joint density of states

$\theta_m(k_\perp)$9

contains, in addition to the pinch point at $\nu=1$0 and crescent-shaped arc–arc features, a disk-like contribution from the Dirac pocket and kidney-shaped features from arc–pocket scattering, whose $\nu=1$1 scale is set by the arc–pocket separation and whose broadening is set by the pocket diameter (Lau et al., 2017). In thin slabs, opposite-surface linkage produces strong Friedel oscillations through nested backscattering, whereas in thick slabs only weaker top-surface oscillations remain (Hosur, 2012). In photonic systems, arc geometry can be imaged by Bloch-mode populations, time-of-flight momentum distributions, and far-field emitter-array form factors (García-Elcano et al., 2022).

These diagnostics also help distinguish linked-arc topologies from superficially similar features. A closed contour around a TRIM is not itself a Fermi arc; in the TR-invariant Weyl case it is a Dirac-cone pocket whose existence is enforced by $\nu=1$2 parity accounting (Lau et al., 2017). Likewise, closed loops at twisted interfaces are interface-localized Fermi loops, not ordinary single-surface arcs (Abdulla et al., 2021). The commonality lies in connectivity reconstruction, not in a single universal contour type.

7. Conceptual scope and related but distinct usages

The literature contains several adjacent concepts that are sometimes folded into discussions of linked Fermi arcs. In topological metals with Chern-charged bulk Fermi-surface sheets, Haldane described Fermi arcs as “Fermi-level plumbing” conduits that attach tangentially to projected bulk Fermi surfaces and transfer quasiparticles between apparently disconnected sheets, with net attachment chirality equal to the total projected Chern number (Haldane, 2014). This is a distinct but related notion of linkage: the arcs form a chiral network between projected bulk Fermi-surface regions rather than between isolated Weyl-node projections.

In Dirac semimetals with multiple Dirac-point pairs, such as the cubic Pd/Pt oxides, symmetry-allowed surface reconnections produce heart-shaped and diamond-shaped arc patterns on the (001) surface (Li et al., 2016). The heart-to-diamond change is a surface Lifshitz transition driven by energy variation, and the multiple Dirac-point projections permit hybridization among different arc families. Because the underlying nodes are Dirac, not Weyl, and because the arcs arise from helicoid/anti-helicoid surface states, this usage is parallel to but not identical with the TR-invariant Weyl-semimetal case (Li et al., 2016).

By contrast, “Fermi and Luttinger arcs: two concepts, realized on one surface” concerns correlated-electron cuprates rather than topological semimetals (Worm et al., 2023). There, a continuous contour satisfying $\nu=1$3 alternates between pole-defined Fermi-arc segments and zero-defined Luttinger-arc segments across the antiferromagnetic zone boundary. This is a Green’s-function linkage between poles and zeros, not a Weyl-node connectivity problem (Worm et al., 2023). Similarly, pseudogap-induced coexistence of Fermi arcs and Fermi pockets in cuprates describes front-side and back-side segments meeting at hot spots, again outside the Weyl-semimetal framework (Zhao et al., 2016).

The stricter encyclopedia usage therefore requires contextual discrimination. In the narrowest and most topologically specific sense, linked Fermi arcs are the arc–pocket networks forced by independent $\nu=1$4 invariants in time-reversal-invariant Weyl semimetals (Lau et al., 2017), or the torus-linked arc loops of Floquet-engineered Weyl phases (Liu et al., 6 Jul 2025). In a broader but still semimetal-specific sense, the term can encompass any surface-arc structure whose connectivity is globally constrained or reconstructed by topology, interface coupling, or boundary conditions [(Devizorova et al., 2016); (Abdulla et al., 2021); (Hosur, 2012)].