Spin-1 Weyl Hamiltonian in 3D Systems

• Spin-1 Weyl Hamiltonian is a three-band low-energy model characterized by a triply degenerate node with one flat band and two linearly dispersive bands.
• It is realized on a 3D Lieb lattice where chiral interlayer hopping induces anisotropic dispersion and a topological charge of ±2, distinguishing it from ordinary Weyl points.
• The nontrivial Berry flux leads to robust surface arc states, offering insights into directional surface transport in chiral phononic crystal systems.

A spin-1 Weyl Hamiltonian is a three-band low-energy Hamiltonian describing a triply degenerate point in three-dimensional momentum space at which two bands exhibit cone-like linear dispersion and a third band is flat at the touching energy. In the phononic realization reported in "Spin-1 Weyl Point and Surface Arc State in a Chiral Phononic Crystal" (Shi et al., 2020), this structure emerges from a three-band tight-binding model of a 3D Lieb lattice with chiral interlayer coupling, and the resulting node carries topological charge $\pm 2$ rather than the unit charge of an ordinary Weyl point.

1. Spectral definition and band topology

In the formulation used in (Shi et al., 2020), a spin-1 Weyl point is formed by three bands touching at a single point in 3D momentum space. Two of these bands show cone-like dispersion, while the third band is flat. The defining low-energy structure is a three-component Hamiltonian linear in momentum, of the generic form

$H(\mathbf{k})=\sum_{j=1}^3 v_j k_j S_j$

up to an overall energy shift, where the $S_j$ are spin-1 matrices acting on a three-component basis.

The key spectral consequence is that the spin-1 matrices have eigenvalues $+1$, $0$, and $-1$ along any direction. Accordingly, near the node the spectrum consists of one flat band at the touching energy and two dispersive bands with opposite linear slopes. In the specific model of (Shi et al., 2020), the effective vector is

$$\mathbf{d}(\mathbf{k})=\bigl(-t_{n1}k_x,\,-t_{n1}k_y,\,-4t_c k_z\bigr),$$

which yields the energies

$$E(\mathbf{k})=\varepsilon,\quad \varepsilon \pm \sqrt{t_{n1}^2(k_x^2+k_y^2)+(4t_c)^2k_z^2}.$$

This gives an anisotropic cone structure in the dispersive branches and leaves the middle band at the touching energy.

The same work emphasizes a topological distinction from an ordinary Weyl point: the triply degenerate node carries topological charge $\pm 2$ (Shi et al., 2020). Within the paper’s description, the spin-1 Weyl point is a monopole of Berry flux whose total Chern number is twice that of an ordinary Weyl point. The two dispersive bands carry Berry curvature corresponding to helicities $+1$ and $H(\mathbf{k})=\sum_{j=1}^3 v_j k_j S_j$0, while the flat band carries zero.

2. Three-band tight-binding realization on a 3D Lieb lattice

The full construction begins from a three-band tight-binding Hamiltonian on a 3D Lieb lattice with chiral interlayer coupling (Shi et al., 2020). In momentum space, the Hamiltonian is written as

$H(\mathbf{k})=\sum_{j=1}^3 v_j k_j S_j$1

with

$H(\mathbf{k})=\sum_{j=1}^3 v_j k_j S_j$2

$H(\mathbf{k})=\sum_{j=1}^3 v_j k_j S_j$3

$H(\mathbf{k})=\sum_{j=1}^3 v_j k_j S_j$4

The basis corresponds to the three sublattice sites in the Lieb unit cell: the $H(\mathbf{k})=\sum_{j=1}^3 v_j k_j S_j$5 site, the $H(\mathbf{k})=\sum_{j=1}^3 v_j k_j S_j$6 site, and the $H(\mathbf{k})=\sum_{j=1}^3 v_j k_j S_j$7 site. The parameters have the following meanings.

Parameter	Meaning
$H(\mathbf{k})=\sum_{j=1}^3 v_j k_j S_j$8	onsite energy on the $H(\mathbf{k})=\sum_{j=1}^3 v_j k_j S_j$9 sublattice
$S_j$0	onsite energy on the $S_j$1 and $S_j$2 sublattices
$S_j$3	nearest-neighbor in-plane hopping between $S_j$4 and $S_j$5
$S_j$6	in-plane next-nearest-neighbor hopping between $S_j$7 and $S_j$8
$S_j$9	long-range hopping between $+1$0 sites
$+1$1	long-range hopping between $+1$2 sites
$+1$3	chiral interlayer hopping

The paper identifies the chiral interlayer hopping $+1$4 as the term responsible for the synthetic gauge flux along $+1$5 and stresses that this coupling makes the system genuinely 3D (Shi et al., 2020). In this model, the $+1$6 dependence enters through the complex combination $+1$7, with $+1$8 containing $+1$9 and $0$0 containing $0$1. This is the mechanism by which the layered Lieb-lattice structure acquires Weyl-like splitting in the out-of-plane direction.

3. Low-energy reduction to the spin-1 Weyl Hamiltonian

For the essential physics, the model is simplified by retaining only the in-plane nearest-neighbor hopping $0$2 and the chiral interlayer hopping $0$3, with $0$4 (Shi et al., 2020). The Hamiltonian is then expanded around the $0$5 point,

$0$6

which produces the linearized form

$0$7

Here, $0$8 denotes the small deviation from the $0$9 point, $-1$0 is the $-1$1 identity matrix, and the three spin-1 matrices are

$-1$2

The term $-1$3 shifts all three bands uniformly and does not affect topology. The remaining terms are linear in $-1$4, $-1$5, and $-1$6, so the Hamiltonian has exactly the form expected for a spin-1 node. The resulting anisotropy is explicit: the in-plane slopes are set by $-1$7, whereas the out-of-plane slope is set by $-1$8.

Within the paper’s framework, this low-energy expansion is the explicit spin-1 Weyl Hamiltonian (Shi et al., 2020). It acts on a three-component basis and reproduces the characteristic triply degenerate spectrum with one flat band and two linearly dispersive bands. The paper further notes that near the $-1$9 point the first and third bands are linear, while the second band is nearly flat. This distinction is useful because it separates the idealized linearized Hamiltonian, in which the middle band is flat at the touching energy, from the full-band description, in which the same band is described as nearly flat near the node.

4. From stacked Lieb layers to a 3D triply degenerate node

The physical origin of the spin-1 Weyl form is tied to the layered Lieb-lattice geometry and its chiral interlayer coupling. Each 2D layer is effectively a Lieb lattice, which already supports a flat band together with a Dirac-like crossing (Shi et al., 2020). When these layers are stacked in three dimensions and connected by the chiral interlayer hopping $$\mathbf{d}(\mathbf{k})=\bigl(-t_{n1}k_x,\,-t_{n1}k_y,\,-4t_c k_z\bigr),$$0, the degeneracy becomes fully three-dimensional.

The paper’s mapping can be summarized as follows. A 2D Lieb flat-band structure provides the coexistence of a flat band and Dirac-like crossing. The added chiral interlayer hopping introduces $$\mathbf{d}(\mathbf{k})=\bigl(-t_{n1}k_x,\,-t_{n1}k_y,\,-4t_c k_z\bigr),$$1-dependent complex hopping through the $$\mathbf{d}(\mathbf{k})=\bigl(-t_{n1}k_x,\,-t_{n1}k_y,\,-4t_c k_z\bigr),$$2 and $$\mathbf{d}(\mathbf{k})=\bigl(-t_{n1}k_x,\,-t_{n1}k_y,\,-4t_c k_z\bigr),$$3 terms in $$\mathbf{d}(\mathbf{k})=\bigl(-t_{n1}k_x,\,-t_{n1}k_y,\,-4t_c k_z\bigr),$$4 and $$\mathbf{d}(\mathbf{k})=\bigl(-t_{n1}k_x,\,-t_{n1}k_y,\,-4t_c k_z\bigr),$$5. This breaks the simple 2D degeneracy into a full 3D point node and makes the crossing linear in all three momentum directions near the Weyl point (Shi et al., 2020).

At the $$\mathbf{d}(\mathbf{k})=\bigl(-t_{n1}k_x,\,-t_{n1}k_y,\,-4t_c k_z\bigr),$$6 point, the low-energy expansion reduces to the spin-1 Weyl operator

$$\mathbf{d}(\mathbf{k})=\bigl(-t_{n1}k_x,\,-t_{n1}k_y,\,-4t_c k_z\bigr),$$7

up to anisotropic velocities and sign conventions. This establishes the spin-1 Weyl Hamiltonian not as an abstract three-band model but as the effective theory of a concrete 3D tight-binding construction. A plausible implication is that the essential ingredients are not the full set of longer-range hopping terms, but the combination of Lieb-lattice flat-band physics and a chiral $$\mathbf{d}(\mathbf{k})=\bigl(-t_{n1}k_x,\,-t_{n1}k_y,\,-4t_c k_z\bigr),$$8-dependent interlayer coupling.

5. Topological charge, Berry flux, and Wilson-loop characterization

The linearized Hamiltonian around the $$\mathbf{d}(\mathbf{k})=\bigl(-t_{n1}k_x,\,-t_{n1}k_y,\,-4t_c k_z\bigr),$$9 point describes a spin-1 Weyl point of topological charge $$E(\mathbf{k})=\varepsilon,\quad \varepsilon \pm \sqrt{t_{n1}^2(k_x^2+k_y^2)+(4t_c)^2k_z^2}.$$0, and the opposite charge $$E(\mathbf{k})=\varepsilon,\quad \varepsilon \pm \sqrt{t_{n1}^2(k_x^2+k_y^2)+(4t_c)^2k_z^2}.$$1 appears at the $$E(\mathbf{k})=\varepsilon,\quad \varepsilon \pm \sqrt{t_{n1}^2(k_x^2+k_y^2)+(4t_c)^2k_z^2}.$$2 point (Shi et al., 2020). The topological interpretation given in the paper is that the node is a monopole of Berry flux whose total Chern number is twice that of an ordinary Weyl point.

The work confirms this numerically using the Wilson-loop method. For the three bands around the $$E(\mathbf{k})=\varepsilon,\quad \varepsilon \pm \sqrt{t_{n1}^2(k_x^2+k_y^2)+(4t_c)^2k_z^2}.$$3 point, the Wannier centers shift by $$E(\mathbf{k})=\varepsilon,\quad \varepsilon \pm \sqrt{t_{n1}^2(k_x^2+k_y^2)+(4t_c)^2k_z^2}.$$4, $$E(\mathbf{k})=\varepsilon,\quad \varepsilon \pm \sqrt{t_{n1}^2(k_x^2+k_y^2)+(4t_c)^2k_z^2}.$$5, and $$E(\mathbf{k})=\varepsilon,\quad \varepsilon \pm \sqrt{t_{n1}^2(k_x^2+k_y^2)+(4t_c)^2k_z^2}.$$6, implying charge $$E(\mathbf{k})=\varepsilon,\quad \varepsilon \pm \sqrt{t_{n1}^2(k_x^2+k_y^2)+(4t_c)^2k_z^2}.$$7. Around the $$E(\mathbf{k})=\varepsilon,\quad \varepsilon \pm \sqrt{t_{n1}^2(k_x^2+k_y^2)+(4t_c)^2k_z^2}.$$8 point, the opposite shifts give charge $$E(\mathbf{k})=\varepsilon,\quad \varepsilon \pm \sqrt{t_{n1}^2(k_x^2+k_y^2)+(4t_c)^2k_z^2}.$$9 (Shi et al., 2020). The flat band contributes zero in this description, while the two dispersive bands encode the nontrivial Berry-flux structure.

This characterization is significant because the charge $\pm 2$0 is not an incidental property of a particular parametrization; it is a topological invariant of the node. In the language used by the paper, the spin-1 Weyl point is therefore distinguished both by its three-band spectrum and by its doubled topological charge relative to the ordinary two-band Weyl case.

6. Surface arcs, gapped slices, and directional surface transport

The bulk topology implies nontrivial surface states. At fixed $\pm 2$1, the bulk degeneracy is lifted and two nontrivial gaps appear, each with Chern number $\pm 2$2 (Shi et al., 2020). By bulk-edge correspondence, this guarantees surface arc states.

In the supercell spectrum, these surface states appear as bands connecting the projected Weyl nodes in the surface Brillouin zone. The paper describes them as nearly straight acoustic Fermi arcs. These arcs connect the $\pm 2$3 and $\pm 2$4 Weyl points and support robust one-way propagation around corners and defects without reflection (Shi et al., 2020).

The straight-type character of the surface arcs is a notable feature of this realization. This suggests strongly directional surface transport, consistent with the computational demonstration of robust propagation in the designed chiral phononic crystal. Within the scope of the reported results, the spin-1 Weyl Hamiltonian is therefore not only a bulk low-energy description but also the generator of a bulk-edge structure manifested as topologically protected surface transport.

7. Role in the chiral phononic-crystal realization

The paper presents the spin-1 Weyl Hamiltonian in the context of a chiral phononic crystal that carries spin-1 Weyl points and straight-type acoustic Fermi arcs (Shi et al., 2020). In that setting, the three-band tight-binding model is not merely illustrative; it serves as the conceptual template for the phononic implementation.

The compact low-energy description is

$\pm 2$5

derived from a 3-band Lieb-lattice tight-binding model with chiral interlayer coupling. In the paper’s own framework, this Hamiltonian produces one flat band, two linear cone bands, a triply degenerate node, topological charge $\pm 2$6, and surface Fermi arcs connecting opposite-charge nodes (Shi et al., 2020).

The broader significance assigned in the work is that such a platform provides a way to manipulate acoustic waves in 3D structures and to explore energy transport properties in 3D spin-1 Weyl systems. Within the limits of the reported evidence, the spin-1 Weyl Hamiltonian is thus both a topological band-theory object and the effective operator underlying a specific chiral phononic realization.