Pseudospin-1 Weyl Semimetals

• Pseudospin-1 Weyl semimetals are topological systems characterized by a threefold band crossing where two linearly dispersing bands and one flat band meet, imparting a net Berry flux of magnitude 2.
• They are realized in diverse platforms such as acoustic 3D phononic crystals, cold-atom setups, and crystalline materials, providing experimental versatility and robust topological signatures.
• Distinctive transport and optical responses, including double Fermi arcs, perfect collimation, and the Imbert-Fedorov shift, validate the unique pseudospin-1 dynamics in these semimetals.

Pseudospin-1 Weyl semimetals, often called spin-1 Weyl semimetals, are topological semimetals in which the low-energy node is a threefold band crossing described by a spin-1 generalization of the Weyl Hamiltonian rather than by the two-component Weyl Hamiltonian of ordinary Weyl semimetals. Their defining local structure is a node at which two linearly dispersing bands and one flat band meet, so the effective internal degree of freedom is a pseudospin-1 representation generated by spin-1 matrices. In the canonical case, the dispersive bands carry Berry-flux monopole charge of magnitude $2$, the flat band is topologically trivial, and the bulk-boundary correspondence yields double Fermi arcs rather than the single arc of a charge-$1$ Weyl node (Deng et al., 2020, Ahmad et al., 2024).

1. Low-energy structure and nomenclature

The standard continuum description is

$H=v\,\mathbf{k}\cdot \mathbf{S},$

with $\mathbf{S}=(S_x,S_y,S_z)$ the spin-1 matrices obeying

$[S_i,S_j]=i\epsilon_{ijk}S_k.$

Within the broader pseudospin-$N/2$ family, the pseudospin-1 case corresponds to $N=2$, so the internal Hilbert space is three-dimensional and $S_z$ has eigenvalues $+1,0,-1$ (Hao et al., 2018). In the chiral form used for 3D multifold semimetals,

$$H(\mathbf{k})=\sum_{\chi=\pm 1} \chi\, \hbar v_F\, \mathbf{k}\cdot \mathbf{S},$$

the spectrum is

$1$0

namely two cones plus a flat band at the node (Ahmad et al., 2024).

This three-band structure is the central distinction from both ordinary Weyl and Dirac semimetals. Ordinary Weyl nodes are twofold and carry $1$1; pseudospin-1 nodes are threefold and carry $1$2 on the dispersive branches. The flat band is not an incidental byproduct: it enters the topology, the surface-state connectivity, and the response theory. In tilted models, the dispersive sector can be modified without tilting the flat band through

$1$3

so the tilt acts only on the dispersive subspace (Ahmad, 9 Jun 2026).

A related usage appears in some three-band cold-atom constructions, where pseudospin-1 Weyl physics is realized through Weyl-type degeneracies between the lower two bands or the upper two bands, with the third band acting as a spectator that changes the topology and robustness. This suggests that “pseudospin-1 Weyl physics” is used in the literature both for genuine threefold spin-1 Weyl nodes and for three-band Weyl systems whose topology is inherited from a spin-1 internal structure (Luo et al., 2019).

2. Topological charges, Berry flux, and internal textures

The topological content of a pseudospin-1 Weyl semimetal exceeds a simple rescaling of Weyl-node charge. In the minimal chiral model, the dispersive bands carry Chern numbers

$1$4

while the flat band is topologically trivial (Ahmad et al., 2024). In generalized triple-component semimetals with

$1$5

the node carries net Chern number

$1$6

so already the $1$7 case has charge magnitude $1$8 (Haidar et al., 8 Jan 2025).

The acoustic spin-1 Weyl realization makes this charge structure explicit. There the three-band charge pattern is reported as $1$9 or $H=v\,\mathbf{k}\cdot \mathbf{S},$0 for the bottom, middle, and top bands, and at the specific high-symmetry points $H=v\,\mathbf{k}\cdot \mathbf{S},$1 and $H=v\,\mathbf{k}\cdot \mathbf{S},$2 the charges are $H=v\,\mathbf{k}\cdot \mathbf{S},$3 and $H=v\,\mathbf{k}\cdot \mathbf{S},$4, respectively. The same work emphasizes that these nodes are “monopoles of Berry flux in momentum space, which are twice those of the WPs with spin-1/2,” and computes the charge from jumps of band Chern numbers across the node, for example

$H=v\,\mathbf{k}\cdot \mathbf{S},$5

for the first band at $H=v\,\mathbf{k}\cdot \mathbf{S},$6 (Deng et al., 2020).

In spin-1 systems, topology is not exhausted by the spin vector alone. A three-component state admits both a spin vector $H=v\,\mathbf{k}\cdot \mathbf{S},$7 and a rank-2 spin tensor

$H=v\,\mathbf{k}\cdot \mathbf{S},$8

In the cold-atom three-band proposal, the nontrivial bands exhibit spin vortices where $H=v\,\mathbf{k}\cdot \mathbf{S},$9, and the Berry phase along a loop may arise entirely from the tensor sector: $\mathbf{S}=(S_x,S_y,S_z)$0 This indicates that the spin-1 node is naturally associated with vector-and-tensor texture rather than with a Bloch-sphere description alone (Luo et al., 2019).

3. Realizations in acoustic, atomic, and crystalline platforms

A direct realization of a spin-1 Weyl semimetal was reported in acoustics using a 3D phononic crystal based on a layer-stacking Lieb lattice. The unit cell contains three non-equivalent acoustic cavities $\mathbf{S}=(S_x,S_y,S_z)$1, $\mathbf{S}=(S_x,S_y,S_z)$2, and $\mathbf{S}=(S_x,S_y,S_z)$3, with tubes acting as acoustic analogs of tight-binding hoppings. Equal in-plane couplings reproduce a Lieb-lattice-like structure, while chiral interlayer couplings generate the effective gauge flux required for Weyl physics. The spin-1 Weyl points arise from an accidental degeneracy between a 1D irreducible representation and a 2D irreducible representation at $\mathbf{S}=(S_x,S_y,S_z)$4 and $\mathbf{S}=(S_x,S_y,S_z)$5, obtained by tuning a cavity-size parameter $\mathbf{S}=(S_x,S_y,S_z)$6. The measured nodes occur at approximately $\mathbf{S}=(S_x,S_y,S_z)$7, and the system is described as a spin-1 Weyl semimetal in a clean frequency window, meaning that the relevant frequency region contains only the spin-1 Weyl points and no nearby unwanted nodal points (Deng et al., 2020).

Cold-atom proposals realize pseudospin-1 Weyl physics in a different manner. In a 1D triple-well superlattice,

$\mathbf{S}=(S_x,S_y,S_z)$8

the three sites in a unit cell are identified as $\mathbf{S}=(S_x,S_y,S_z)$9, $[S_i,S_j]=i\epsilon_{ijk}S_k.$0, and $[S_i,S_j]=i\epsilon_{ijk}S_k.$1. Raman-assisted tunneling generates intra-cell couplings $[S_i,S_j]=i\epsilon_{ijk}S_k.$2 and $[S_i,S_j]=i\epsilon_{ijk}S_k.$3, inter-cell coupling $[S_i,S_j]=i\epsilon_{ijk}S_k.$4, and transverse momentum kicks $[S_i,S_j]=i\epsilon_{ijk}S_k.$5 that act as effective 2D SOC. The system hosts two Weyl points, $[S_i,S_j]=i\epsilon_{ijk}S_k.$6 between the two lower bands and $[S_i,S_j]=i\epsilon_{ijk}S_k.$7 between the two upper bands. Because these nodes belong to different band pairs, they cannot annihilate each other under smooth parameter changes; changing Raman intensities, phases, detunings, incidence angles, or polarizations typically only moves them in momentum space. This is the sense in which the third band makes the Weyl physics robust (Luo et al., 2019).

In crystalline materials, pseudospin-1 fermions are predicted for body-centered cubic symmetry, specifically in space groups $[S_i,S_j]=i\epsilon_{ijk}S_k.$8, $[S_i,S_j]=i\epsilon_{ijk}S_k.$9, and $N/2$0, at the $N/2$1 point of the Brillouin zone (Ahmad et al., 2024). Later transport analyses are formulated explicitly for candidate chiral crystals in space groups $N/2$2 $N/2$3, $N/2$4 $N/2$5, and $N/2$6 $N/2$7 (Ahmad, 9 Jun 2026). Contemporary multifold-semimetal experiments discussed in this context include chiral materials such as CoSi (Haidar et al., 8 Jan 2025).

4. Surface states, boundary geometry, and optical-like transverse responses

The most visible surface consequence of a charge-$N/2$8 pseudospin-1 node is the appearance of double Fermi arcs. In the acoustic realization, two branches of surface arcs occur between the 1st and 2nd bands and also between the 2nd and 3rd bands. These arcs are observed on both the normal (N) surface and the hollow (H) surface, indicating that the phenomenon is a bulk-boundary consequence of the spin-1 Weyl nodes rather than a surface-specific accident (Deng et al., 2020).

Because the surface arcs are open contours, the acoustic system exhibits robust propagation against multiple joints, reflection immunity, and topological negative refraction of acoustic surface arc waves. The reported surface-arc propagation is anticlockwise along the boundary and does not reflect or scatter because of the nonclosed arc geometry. This provides a macroscopic manifestation of the same topology that, in electronic settings, is usually inferred indirectly from spectroscopy or transport (Deng et al., 2020).

A complementary wavepacket-level signature is the Imbert-Fedorov shift. For pseudospin-$N/2$9 semimetals with

$N=2$0

reflection from an interface produces a transverse displacement

$N=2$1

For the pseudospin-1 case this reduces to

$N=2$2

The effect is interpreted as a pseudospin Hall effect of topological fermions, and equivalently as a consequence of conserving

$N=2$3

during reflection (Hao et al., 2018).

5. Interface transport, barrier phenomena, and magnetotransport

Pseudospin-1 semimetals have unusual boundary conditions because integer-spin current conservation does not require continuity of the full spinor. For a spin-$N=2$4 Weyl system with a $N=2$5-component wavefunction, only $N=2$6 components must be continuous across an interface. Thus, for pseudospin-1, current conservation requires continuity of only two out of three components. In the junction analysis, the first and third components are matched while the middle component may be discontinuous. This nonstandard matching yields transport effects absent in pseudospin-$N=2$7 Weyl systems (Nandy et al., 2019).

One consequence is perfect collimation in normal metal-barrier-normal metal junctions. At

$N=2$8

for electron-mediated transport, or

$N=2$9

for hole-mediated transport, all non-normal incident quasiparticles are reflected with unit probability and only the normally incident channel transmits. The same work finds the exact conductance symmetry

$S_z$0

with $S_z$1 for particle transport and $S_z$2 for hole transport (Nandy et al., 2019).

Barrier transport in the linear pseudospin-1 model also supports ordinary Klein tunneling and super-Klein tunneling. For a scalar barrier, the transmission is perfect at normal incidence, and at the special point

$S_z$3

the barrier is perfectly transparent for all incident angles: $S_z$4 At this super-Klein point the Fano factor vanishes,

$S_z$5

Adding a vector potential shifts the transverse momentum, destroys the all-angle transparency, and displaces the high-transmission regions in angle space (Mandal, 2020).

In bulk magnetotransport, the chiral anomaly survives but is modified by the three-band structure, internode scattering, orbital magnetic moment corrections, and charge-conservation constraints. A quasiclassical Boltzmann analysis with $S_z$6 finds that the longitudinal magnetoconductance is quadratic in magnetic field, positive for weak internode scattering, and negative beyond a critical internode scattering strength. The threshold is lower for pseudospin-1 fermions than for pseudospin-$S_z$7 Weyl fermions, and without the orbital magnetic moment the longitudinal magnetoconductance remains positive for all $S_z$8 (Ahmad et al., 2024).

The planar Hall conductance introduces an additional anisotropic diagnostic. For

$S_z$9

the untilted pseudospin-1 system yields

$+1,0,-1$0

is quadratic in $+1,0,-1$1 at weak fields, and changes sign when intervalley scattering exceeds a critical value. Tilt breaks the angular structure in a direction-dependent way: $+1,0,-1$2 and the planar Hall conductance is nonmonotonic in tilt magnitude (Ahmad, 9 Jun 2026). More general weak-field planar-Hall theory for pseudospin-1 triple-component semimetals incorporates Berry curvature, orbital magnetic moment, intrinsic anomalous Hall, Lorentz-force terms, and internode scattering on an equal footing, with the response coefficients acquiring explicit $+1,0,-1$3-dependent structure (Haidar et al., 8 Jan 2025).

6. Flat-band physics, diagnostics, and interpretive cautions

A persistent misconception is that the flat band should directly conduct because it intersects the node. The literature instead treats the flat band as having zero equilibrium group velocity and therefore zero direct equilibrium conductivity. In the linear 3D tunneling model, the flat-band current expectation vanishes,

$+1,0,-1$4

so transport is carried only by the dispersive branches (Mandal, 2020). Closely related pseudospin-1 Dirac-Weyl analyses make the same point in driven settings: the flat band itself has zero conductivity in equilibrium, but it can nonetheless alter non-equilibrium current strongly (Wang et al., 2017).

That non-equilibrium role is twofold. In a sudden electric-field quench of the $+1,0,-1$5D pseudospin-1 Dirac-Weyl system, the weak-field current is interband dominated and is about twice larger than that of the pseudospin-$+1,0,-1$6 system because of the interplay between the flat band and the negative band; in the strong-field regime, the intraband current is $+1,0,-1$7 times larger because the additional flat-band channel increases the effective hole population left in the dispersive sector (Wang et al., 2017). These are not direct 3D semimetal results, but they clarify why the flat band cannot be dismissed as inert.

Detection strategies are platform dependent. In acoustics, the Brillouin-zone structure, surface arcs, and bulk nodes are directly accessible in a macroscopic 3D sample (Deng et al., 2020). In the cold-atom setting, the linear dispersion may be measured with momentum-resolved rf spectroscopy, while momentum-resolved Rabi spectroscopy plus time-of-flight imaging reconstructs Bloch wavefunctions near the Weyl points; the third band acts as a reference, and pseudospin-resolved detection can be implemented with Stern–Gerlach separation (Luo et al., 2019). For the Imbert-Fedorov shift, proposed detection geometries include semimetal slabs between insulating layers, or photonic-crystal analogues in which the shift accumulates over multiple total reflections (Hao et al., 2018).

Interpretively, magnetotransport in pseudospin-1 systems requires more care than in ordinary Weyl semimetals. Recent analyses emphasize that orbital magnetic moment effects and global charge conservation qualitatively alter the low-field response, and that negative magnetoresistance alone may not be a sufficient diagnostic for identifying chiral anomaly in higher-pseudospin systems (Ahmad et al., 2024). A plausible implication is that reliable identification of pseudospin-1 Weyl semimetals will continue to rely on combined evidence: band topology, surface-state connectivity, and transport signatures that are consistent with the three-band spin-1 structure rather than with a two-band Weyl phenomenology alone.