Ideal Weyl Semimetal: Minimal Weyl Node Physics

• Ideal Weyl semimetal is a 3D quantum material defined by a minimal pair of Weyl nodes precisely tuned to the Fermi level, ensuring a clean band structure free from trivial pockets.
• Experimental platforms, ranging from intrinsic crystals to engineered systems, use pressure, magnetic fields, or light to tune node positions and chirality for distinct transport phenomena.
• Realizations in ultracold atoms, metamaterials, and noncentrosymmetric crystals offer versatile means to probe exotic Weyl fermion physics and topological surface Fermi arcs.

An ideal Weyl semimetal is a three-dimensional (3D) quantum material whose electronic structure hosts only the minimal set of isolated Weyl nodes (band crossings with linear dispersion), all positioned exactly at the Fermi level and free from additional trivial Fermi surface pockets. In such systems, low-energy bulk carriers are exclusively Weyl fermions, and the associated topologically protected surface Fermi arcs exhibit maximal clarity, offering an optimal setting for observing exotic transport signatures of Weyl physics such as the chiral anomaly. The concept generalizes existing topological semimetals by demanding “clean” band structure, minimal (often just one or two pairs) Weyl nodes, and a chemical potential ideally tuned to the node energy. Multiple implementations—ranging from noncentrosymmetric crystals to magnetically driven or optically induced phases—realize this archetype in both quantum materials and engineered platforms.

1. Defining Characteristics of the Ideal Weyl Semimetal

An ideal Weyl semimetal (IWSM) is stringently defined by the following criteria:

• Minimal Weyl Node Count and Position: All Weyl nodes lie exactly at the Fermi energy, with no additional electron or hole pockets—analogous to how graphene’s Dirac points reside precisely at the Fermi level in two dimensions (Ruan et al., 2015).
• Spin/Momentum Isolation: Weyl nodes are separated in momentum space, ensuring they are not masked by trivial bands.
• Chirality: Each Weyl point acts as a source or sink of Berry curvature and possesses a quantized chirality (topological charge), commonly ±1.
• Band Structure: The underlying low-energy theory near a Weyl node is described by a linear Hamiltonian,

$$H_{Weyl} = \vec{v} \cdot (\mathbf{k} - \mathbf{k}_W) \cdot \vec{\sigma},$$

where $\vec{v}$ is the velocity tensor, $\mathbf{k}_W$ is the node’s location in momentum space, and $\vec{\sigma}$ are Pauli matrices.

• Absence of Trivial Fermi Surfaces: The Fermi level crosses only the Weyl nodes and not any extraneous, topologically trivial bands. This “clean” electronic structure is crucial for unambiguous observation of topological effects (Ruan et al., 2016, Liu et al., 24 Mar 2024).

The table below outlines key ideal Weyl semimetal features realized in various material platforms:

Feature	Natural Crystals (e.g., XCrTe, EuCd₂As₂)	Engineered Systems (Ultracold Atoms/Metamaterials)
Weyl node count (minimum)	1 pair	1 pair
Fermi level tuning	Magnetic order / pressure / alloying	External field, Raman lattice configuration
Additional pockets?	None in ideal phase	None
Tunable node position	Magnetization/strain	Optical/geometric control

2. Experimental Realizations and Theoretical Models

Material Platforms:

• Intrinsic Crystalline IWSMs: Compounds such as the half-Heusler XCrTe (X = K, Rb) manifest a half-metallic ground state with only one spin channel crossing the Fermi level and exactly a single pair of Weyl points (Liu et al., 24 Mar 2024). EuCd₂As₂, under field or Ba-alloying, offers a single pair of Weyl nodes due to half-metallicity and large exchange splitting (Wang et al., 2019, Soh et al., 2019), while chalcopyrite compounds and strained HgTe-class materials realize higher-multiplicity IWSMs with four pairs of symmetry-related Weyl nodes (Ruan et al., 2015, Ruan et al., 2016).
• Field/Pressure-Controlled Phases: In layered rare-earth pnictides, such as EuCd₂As₂ or MnBi₂₋ₓSbₓTe₄, ideal Weyl states are induced by aligning the magnetic moments via moderate magnetic fields or hydrostatic pressure, driving the system through magnetic phase transitions that open or close gaps at high-symmetry points (Soh et al., 2019, Yu1 et al., 2022, Jiang et al., 2023).
• Non-equilibrium and Engineered Systems:
  • Optically Driven IWSMs: Nonlinear phononic excitation or circularly polarized light (CPL) can induce transient or metastable ideal Weyl states, as in light-driven HgTe (Shin et al., 2023) and FM MnBi₂Te₄ (Fan et al., 8 May 2024). These methods achieve precise tuning of Weyl node number, position, and even type (I/II/III) as a function of driving field.
  • Ultracold Atom Platforms: A 3D optical Raman lattice can simulate the IWSM phase by creating two Weyl nodes in the Brillouin zone. The position, chirality, and surface arcs are verified via “virtual slicing” of reconstructed spin textures and quench dynamics (Lu et al., 2019, Wang et al., 2020, Li et al., 2021).
  • Classical Metamaterials: Zero-index Weyl metamaterials, realized in sonic or photonic crystals, can host a minimal Weyl pair with well-separated momentum positions and exceptional transmission properties (Zangeneh-Nejad et al., 2020).

Theoretical Tools:

• First-principles DFT with GGA/mBJ corrections predicts Weyl node positions, energies, and surface arc structure (Ruan et al., 2015, Ruan et al., 2016).
• Tight-binding/k·p Hamiltonians encode the symmetry protection and minimal model of IWSM phases, e.g.,

$$\mathcal{H}_{Weyl} = v_x k_x \sigma_x + v_y k_y \sigma_y + v_z k_z \sigma_z$$

• Critical symmetry operations (C₂T, inversion breaking, rotoinversions) determine node multiplicity and location (Ruan et al., 2015, Liu et al., 24 Mar 2024).
• Nonlinear phononics models (for light-driven transitions),

$$\begin{aligned} \ddot{Q}_x + \gamma \dot{Q}_x + \Omega^2 Q_x &= -2k_{nl} Q_x Q_{ind} + Z^* E(t) \ \ddot{Q}_{ind} + \gamma \dot{Q}_{ind} + \Omega^2 Q_{ind} &= -k_{nl} Q_x^2 \end{aligned}$$

characterize lattice distortions leading to topological transitions (Shin et al., 2023).

3. Symmetry, Protection, and Tunability

Symmetry Protection:

• Inversion or Time-Reversal Breaking: All ideal Weyl semimetals require breaking of either inversion (e.g., via noncentrosymmetric structure) or time-reversal symmetry (e.g., magnetic order) (Ruan et al., 2015, Liu et al., 24 Mar 2024, Wang et al., 2019).
• Topological Charge and Chern Invariants: The quantized topological charge of each Weyl node,

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

is set by the Berry curvature flux through a surface $S$ enclosing the node.

• Symmetry-Driven Node Multiplicity: The minimal node count is (i) two in time-reversal-breaking or (ii) four in time-reversal-symmetric, inversion-breaking lattices (Ruan et al., 2015, Liu et al., 24 Mar 2024, Chang et al., 2015).
• Tunable Node Location and Chirality: Magnetization or strain direction, light field amplitude, or lattice distortion enable continuous tuning of node positions and the associated Chern vector,

$\vec{v} = \sum_i \chi_i \mathbf{k}_W^i,$

where $\chi_i$ is the chirality (Liu et al., 24 Mar 2024). This directly affects the anomalous Hall response ($\sigma_{xy} = -(e^2/\pi h)q_z$ in the [001] direction, for example).

Tunability:

• Magnetization axis rotation in XCrTe shifts the Weyl nodes across different high-symmetry lines and modulates Hall conductivity (Liu et al., 24 Mar 2024).
• Pressure or chemical alloying induces or stabilizes the IWSM phase (e.g., EuCd₂As₂ under high pressure or Ba substitution) (Yu1 et al., 2022, Wang et al., 2019).
• Light amplitude and polarization in driven MnBi₂Te₄ or HgTe directly control whether the realization is type-I, II, or III IWSM, Fermi arc length, and even the existence of Weyl nodes (Fan et al., 8 May 2024, Shin et al., 2023).

4. Surface States, Fermi Arcs, and Transport

Fermi Arcs:

• IWSMs exhibit minimal, unambiguous Fermi arc surface states that connect the projections of the bulk Weyl nodes with opposite chirality. The absence of trivial pockets ensures that surface ARPES or STM measurements can resolve these arcs without spectral complications (Liu et al., 24 Mar 2024, Lu et al., 2019).
• The arc pattern is topologically linked to the Chern number change across $k_z$ slices (Liu et al., 24 Mar 2024, Ruan et al., 2015):

| Node/Arc Characteristic | Clean Materials (e.g., XCrTe, GdSI) | Complex Materials (e.g., TaAs) | |------------------------|-------------------------------------------|-----------------------------------------------| | Surface arc topology | Simple, single arc | Web of arcs, multiple node pairs | | Fermi level crossing | Only Weyl points | Multiple pockets, possible overlap |

Transport Phenomena:

• Chiral anomaly and negative magnetoresistance: The lack of extra carriers enhances the visibility of anomalous transport, e.g., negative longitudinal magnetoresistance from the Adler–Bell–Jackiw anomaly (Ruan et al., 2015, Soh et al., 2019).
• Anomalous Hall effect: Directly proportional to the Chern vector distance; in IWSMs, anomalous Hall conductivity features quantized or strongly tunable plateaus (Jiang et al., 2023, Liu et al., 24 Mar 2024).
• Berry curvature effects: Clean node separation leads to substantial Berry curvature and thus dominant topological contributions to magnetotransport, Nernst, and optical effects (Soh et al., 2019, Ruan et al., 2015).
• Light-induced phenomena: CPL control enables real-time tuning of Fermi arc length, type-I/II/III node transitions, and manipulation of nonlinear Hall signals (Fan et al., 8 May 2024).

5. Topological Transitions and Tuning Parameters

Phase Evolution:

• Metal–Insulator and Topological Crossover: Modest lattice modification in Ta₃S₂ (by <4%) transitions the system from a type-II Weyl phase to a topological insulator with non-trivial $\mathbb{Z}_2$ index; the presence or annihilation of Weyl nodes is determined by symmetry and tuning parameters (Chang et al., 2015).
• Type-I/II/III Node Tuning: In optically driven systems (e.g., FM MnBi₂Te₄ under CPL), increasing the light intensity evolves a type-II Weyl node (over-tilted cone) through a critical type-III regime (flat dispersion in a specific direction), to a conventional type-I node—accompanied by a corresponding shortening and straightening of the surface Fermi arc (Fan et al., 8 May 2024).
• Pressure and Magnetic Control: Pressure in EuCd₂As₂ sequentially drives transitions from an in-plane antiferromagnetic to in-plane FM, then out-of-plane FM state, only the last providing the IWSM phase with Weyl nodes close to $E_F$ (Yu1 et al., 2022).

6. Implications for Fundamental Physics and Technology

Fundamental Significance:

• Direct platform for Weyl fermion physics: IWSMs provide unambiguous settings for probing chiral anomaly, topological surface transport, and novel collective excitations (e.g., emergent Goldstone modes in cold-atom setups) (Lu et al., 2019, Li et al., 2021).
• Connection to High-Energy Physics: The band theory realization of Weyl fermions enables table-top tests of anomaly-related phenomena, paralleling concepts in quantum field theory.

Technological Potential:

• Spintronics: Nearly 100% spin polarization of the conducting Weyl states in half-metallic IWSMs is ideal for devices exploiting spin transport (Liu et al., 24 Mar 2024).
• Quantum Devices and Sensors: Robust, tunable topological responses (quantized Hall currents, topological magnetoresistance) suit transistor, memory, and quantum logic designs (Jiang et al., 2023, Ruan et al., 2015).
• Dynamically Tunable Materials: Light-driven or field-induced IWSMs constitute a pathway for ultrafast control and reconfiguration of material properties on demand (Shin et al., 2023, Fan et al., 8 May 2024).

7. Future Directions and Open Problems

• Material Discovery: Continued search and synthesis for IWSMs with robust, tunable Weyl nodes at the Fermi level and no trivial pockets remains an active area (e.g., magnetic half-Heuslers, 3D honeycomb lattices, and strain-engineered chalcopyrites) (Liu et al., 24 Mar 2024, Ruan et al., 2015, Nie et al., 2017).
• Ultrafast Topological Switches: Development of devices leveraging light-induced or field-induced topological transitions for rapid, reversible control of transport properties (Fan et al., 8 May 2024, Shin et al., 2023).
• Probing Correlated Weyl Phases: The presence of van Hove singularities and proximate Fermi-level Weyl nodes in some IWSMs (e.g., Ta₃S₂) suggests fertile ground for emergent correlated and superconducting phases (Chang et al., 2015).
• Direct Surface State Imaging: Advanced spectroscopies and microscopy (e.g., ARPES, STM) are essential for resolving the surface Fermi arcs and quantifying the topological invariants without interference from trivial bands (Ruan et al., 2015, Ruan et al., 2016).
• Integration with Quantum Simulation: Atom-optical platforms, with the ability to tune node number, chirality, and Fermi arc topology, offer unparalleled avenues for simulating complex topological phenomena not easily accessible in solid-state materials (Wang et al., 2020, Lu et al., 2019).

In summary, the ideal Weyl semimetal concept encapsulates a “clean” 3D Weyl fermion system—minimal nodes at the chemical potential, absent trivial pockets, and robust surface arcs. Realizable in bulk crystals, engineered metamaterials, and quantum simulation, IWSMs provide a pristine arena for the exploration and exploitation of topological transport and quantum field phenomena in condensed matter systems.