Single-Weyl Hamiltonians

• Single-Weyl Hamiltonians are minimal two-band 3+1D models featuring linear band-touching points with unit topological charge, essential for describing chiral quasiparticles.
• They employ a robust canonical formulation with clear Berry curvature, Chern number, and momentum-space monopole structures that validate quantum transport and topological phenomena.
• Real-world realizations in crystalline, magnetic, and artificial systems highlight the effects of anisotropy, tilt, and symmetry constraints in engineering and observing Weyl nodes.

A single-Weyl Hamiltonian is a minimal two-band model in 3+1 dimensions that describes a linear band-touching (Weyl) point in momentum space with unit topological (monopole) charge. This Hamiltonian serves as the prototypical theory of chiral quasiparticles in condensed matter, quantum transport, and topological phases, and underlies much of the theory of Weyl semimetals, chiral anomaly, and momentum-space monopole physics. In its canonical form, the single-Weyl Hamiltonian exhibits a simple, robust relationship between its spectral, Berry, and topological attributes, and provides the foundation for more complex models involving symmetry breaking, anisotropy, lattice regularization, and physical realizations in solid-state and artificial systems.

1. Canonical Formulation and Eigenstructure

The minimal single-Weyl Hamiltonian in continuum momentum space $(\mathbf{p}\in\mathbb{R}^3)$ is given by

$$H_W(\mathbf{p}) = \boldsymbol{\sigma}\cdot\mathbf{p} = p_1\sigma_1 + p_2\sigma_2 + p_3\sigma_3,$$

where $\{\sigma_i\}$ are the Pauli matrices. The spectrum is linear, $E_\pm(\mathbf{p}) = \pm |\mathbf{p}|$, with degenerate crossing at $\mathbf{p}=0$. The eigenstates $|+(\mathbf{p})\rangle$ and $|-(\mathbf{p})\rangle$ are obtained by the unitary diagonalization $$U(\mathbf{p})\,H_W(\mathbf{p})\,U(\mathbf{p})^\dagger = \mathrm{diag}(|\mathbf{p}|,-|\mathbf{p}|)$$, and the Hamiltonian can be written via projectors as $$H_W(\mathbf{p}) = |\mathbf{p}|[P_+(\mathbf{p}) - P_-(\mathbf{p})]$$ (Elbistan, 2014).

2. Topological Invariants: Winding, Monopole, and Chern Number

The topological classification of single-Weyl Hamiltonians is encoded in multiple equivalent quantities:

• Winding Number: The 3+1D winding number of the Euclidean Green’s function

$$C_3 = \frac{1}{24\pi^2}\int d\omega\,d^3p\,\epsilon^{\mu\nu\rho\sigma}\,\mathrm{Tr}\left[(G\partial_\mu G^{-1})(G\partial_\nu G^{-1})(G\partial_\rho G^{-1})(G\partial_\sigma G^{-1})\right]$$

evaluates to $C_3=1$ for the single-Weyl node (Elbistan, 2014).

• Momentum-space Monopole: The winding can be recast as the divergence of the vector field $b_3^a(\mathbf{p}) = p^a/(2|\mathbf{p}|^3)$, i.e., the Dirac monopole field, such that $$\nabla_\mathbf{p}\cdot\mathbf{b}_3 = 4\pi\delta^3(\mathbf{p})$$, directly relating the topological charge to the physical monopole at the Weyl point.
• Berry Connection and Curvature: For the positive energy band, the Berry connection is $$A_a(\mathbf{p}) = i\langle+(\mathbf{p})|\nabla_{p_a}|+(\mathbf{p})\rangle$$, yielding Berry curvature $$F_{ab}(\mathbf{p}) = \epsilon_{abc}\,p^c/(2|\mathbf{p}|^3)$$, which also integrates to unit flux through any enclosing $S^2$.
• First Chern Number: The surface integral $\frac{1}{2\pi}\int_{S^2}F$, where $F$ is the Berry curvature, evaluates to $C_1=1$. All these structures are manifestations of the same unit monopole charge (Elbistan, 2014).

3. Anisotropy and Tilt in Real Materials

Real-world single-Weyl nodes generically display cone anisotropy and tilt—features encapsulated in the generalized low-energy Hamiltonian

$$H(\mathbf{q}) = \sum_{i,j}v_{ij}q_j\sigma_i + \sum_{i} t_i q_i I_2,$$

where $v_{ij}$ is the Fermi-velocity tensor (capturing anisotropy), $t_i$ is the tilt-velocity, and $I_2$ is the $2\times2$ identity (Grassano et al., 2019). The spectrum is

$$E_\pm(\mathbf{q}) = T(\mathbf{q}) \pm U(\mathbf{q}),\quad T(\mathbf{q}) = \sum_it_iq_i, \quad U(\mathbf{q}) = \sqrt{\sum_{i,j}(v_{ij}q_j)^2}.$$

If $|T(\hat{\mathbf{q}})|<U(\hat{\mathbf{q}})$ for all directions, one has a type-I node (pointlike Fermi surface); otherwise, for some direction with $|T|>U$, a type-II node arises (open Fermi surface). Ab initio fits to TaAs, TaP, NbAs, NbP yield a hierarchy of velocity and tilt parameters, with W2 nodes exhibiting stronger anisotropy and tilt but all remaining strictly type-I (Grassano et al., 2019).

The Berry curvature, robust against tilt, generalizes to

$$\mathbf{\Omega}(\mathbf{q}) = \frac{1}{2}\,\mathrm{sgn}(\det \hat{v}_F)\frac{v_x v_y v_z\,\mathbf{q}}{(v_x^2q_x^2 + v_y^2q_y^2 + v_z^2q_z^2)^{3/2}},$$

with unit monopole charge. Deviations from the ideal “hedgehog” spin texture reflect spin-momentum locking breakdown in the presence of strong anisotropy (Grassano et al., 2019).

4. Lattice Realizations and Single-Weyl Regularization

In lattice models, single-Weyl Hamiltonians arise as nodes of minimally doubled lattice fermion constructions. The Karsten–Wilczek Hamiltonian, for instance, has two Weyl nodes which can be reduced to a single node via a species-splitting mass term in a Nambu–BdG basis: $$h^{\mathrm{single}}(\mathbf{p}) = \frac{1}{2}\Bigl[\sigma_1\sin p_1 + \sigma_3\sin p_2 + \tau_3\sigma_2\sin p_3 + \sigma_2r(2-\cos p_1 - \cos p_2) + \tau_1\sigma_2(1-\cos p_3)\Bigr],$$ with only one node at $\mathbf{p}=0$ when $|r|>1$ (Misumi, 27 Dec 2025). The protecting symmetry is a momentum-dependent, non-onsite “axial” symmetry $S_\chi(\mathbf{p})$, analogous to a Ginsparg–Wilson relation, rather than a conventional onsite chiral symmetry.

A one-parameter deformation $$\delta h_\mu(\mathbf{p}) = \mu\,\sigma_2\,S_\chi(\mathbf{p})$$ preserves all symmetries, but for $|\mu|$ exceeding a critical threshold, additional nodes emerge and the single-node regime is lost. Radiative corrections in interacting lattice field theory render moderate parameter tuning essential to maintain single-Weyl behavior, analogous to mass renormalization in standard Wilson fermion constructions (Misumi, 27 Dec 2025).

5. Mechanisms for Isolated and Paired Weyl Nodes

Single-Weyl nodes frequently appear in pairs of opposite chirality due to the Nielsen–Ninomiya no-go theorem; however, exceptions exist:

• Annihilation by Magnetic/Structural Tuning: In EuCd$_2$As$_2$, a Dirac point splits into two Weyl node pairs under ferromagnetic order; with sufficiently large exchange splitting, one pair moves to the zone center and the nodes annihilate, leaving a single pair of Weyl nodes (Wang et al., 2019). The effective $4\times4$ $k\cdot p$ model projects onto a single-Weyl structure for the surviving node, with velocities and topological charge fixed by the band parameters.
• Circumventing No-Go via Nodal Walls: In certain noncentrosymmetric, time-reversal symmetric crystals (e.g., SG 19 with strong SOC), a lattice can realize a single Weyl node at the $\Gamma$ point, surrounded by symmetry-protected nodal walls on the Brillouin zone faces. The low-energy $k\cdot p$ theory remains linear, and Berry curvature calculation gives unit monopole charge, but the absence of an everywhere-gapped 2D Brillouin plane precludes the existence of surface Fermi arcs (Yu et al., 2019).

6. Physical Realizations and Artificial Systems

Single-Weyl Hamiltonians are realized in a range of platforms:

• Crystalline Weyl Semimetals: Compounds such as TaAs, TaP, NbAs, and NbP display multiple Weyl nodes (W1, W2) of the single-Weyl type over their Brillouin zone; each is characterized by the linearized form but with strong real-material renormalizations of velocities and tilt (Grassano et al., 2019).
• Magnetic Weyl Semimetals: Systems such as ferromagnetic EuCd$_2$As$_2$ and its Ba-alloyed derivatives produce a single pair of Weyl nodes by manipulating exchange splittings and structural distortions, directly engineering the effective $k\cdot p$ parameters (Wang et al., 2019).
• Weyl Josephson Circuits: Artificial superconducting circuits can simulate the single-Weyl Hamiltonian by mapping circuit parameters (gate charges, phase biases) onto effective momentum coordinates, with full control over topological transitions and monopole charge testing enabled by the circuit's tunability (Fatemi et al., 2020).

A table summarizes select realizations:

System/Class	Hamiltonian Structure	Topological Charge	Notable Features
Ideal continuum Weyl	$\sigma\cdot\mathbf{p}$	$+1$/$-1$	Linear, isotropic, fully local monopole
TaAs-family (DFT)	$v_{ij}\,q_j\sigma_i +t_iq_iI$	$\pm 1$	Anisotropic, tilted cones, multiple nodes
Single-node lattice (KW+mass)	See above ($h^{\rm single}$)	$+1$/$-1$	Only one node; requires fine-tuned symmetry
EuCd$_2$As$_2$ (FM phase)	Projected $4\times4$ $k\cdot p$	$\pm 1$	Single pair by Dirac annihilation
Weyl Josephson Circuit	$d_i(\phi)\sigma_i$	$\pm 1$	Tunable, artificial parameter space
T-invariant lattice (SG19+SOC)	$\rightarrow$ eff. $\sigma\cdot\mathbf{q}$	$\pm 1$	Isolated Weyl at $\Gamma$, nodal walls

7. Surface States, Edge Modes, and Symmetry Constraints

The existence of Fermi arc surface states characteristically distinguishes Weyl semimetals with isolated single-Weyl nodes. However, the presence of nodal walls or lack of a gapped 2D plane in the Brillouin zone obviates the standard bulk-boundary correspondence, as in the T-invariant nodal-wall systems where no Fermi arcs appear (Yu et al., 2019). Symmetry constraints, including mirror, screw, and time-reversal operations, determine both the protection and the annihilation conditions for Weyl nodes. Non-onsite symmetries, as in lattice-regularized single-Weyl constructions, require momentum-dependent generators, reflecting the subtle interplay between topology and lattice realization (Misumi, 27 Dec 2025).

In all settings, the monopole charge remains quantized and robust: the spatial configuration, dispersion details, or symmetry class may strongly modify observable physics (spin texture, transport, density of states), but the topological invariant persists as the unifying feature of the single-Weyl Hamiltonian.