Tunable Dirac-Weyl Semimetal Phase

• Tunable Dirac-Weyl semimetal phase is a quantum state with symmetry-protected Dirac and Weyl nodes whose locations, dispersion, and topology can be externally controlled.
• External parameters such as strain, magnetic fields, optical pumping, and alloying enable precise tuning of phase transitions and electronic properties.
• These materials offer a versatile platform for exploring novel transport phenomena, including chiral anomaly, Hall conductivity, and the interplay with superconductivity.

A tunable Dirac-Weyl semimetal phase refers to a quantum state of matter in which symmetry-protected Dirac or Weyl band touchings exist in the bulk electronic structure, and where the number, location, topology, or dispersion of these nodes can be externally controlled, typically by electric, magnetic, structural, or optical means. Tunability fundamentally distinguishes these phases from their static, symmetry-fixed counterparts and enables in situ manipulation of band topology, Fermi velocities, surface states, and intrinsic responses such as chiral anomaly and Hall conductivity.

1. Definition and Structural Realization

Dirac-Weyl semimetals lie at the intersection of topological and symmetry-protected band theory, hosting low-energy quasiparticles with relativistic-like dispersion, including fourfold-degenerate Dirac fermions and their twofold-degenerate Weyl cousins. Dirac nodes require the coexistence of time-reversal and inversion symmetry, along with discrete crystal point-group elements (e.g., $C_n$ rotation), whereas Weyl nodes emerge when either time-reversal or inversion is broken, pairing Kramers partners into chiral monopoles of Berry flux.

Tunable Dirac-Weyl semimetal phases have been realized in a variety of crystalline, engineered, and driven systems. In two-dimensional materials, the hydrogenated borophane $\beta_{12}$-B$_5$H$_3$ presents a paradigm: it possesses a Pmm2 symmetry and mirror-protected Dirac points that are responsive to uniaxial strain. The band crossings are protected by mirror reflections $m_x$ and $m_y$, with atomic-scale hydrogenation eliminating phonon soft modes and stabilizing the 2D Dirac framework. Lattice parameters and atomic sites are precisely determined, and electronic/phononic structure calculations employ high-cutoff plane-wave DFT and fine-mesh Brillouin zone integration for quantitative accuracy (Zhong et al., 2024).

2. Generic Hamiltonian and Symmetry Analysis

The low-energy physics of tunable Dirac-Weyl systems is governed by multi-component $k\cdot p$ Hamiltonians of the form

$$H(\mathbf{k}) = v_x k_x \sigma_x + v_y k_y \sigma_y + m(\epsilon)\,\sigma_z + w_x k_x \sigma_0$$

where $v_{x,y}$ are direction-dependent Fermi velocities, $\sigma_{i}$ are Pauli matrices, $m(\epsilon) \simeq \gamma(\epsilon_c - \epsilon)$ is a strain- or field-tuned mass, and $w_x$ a tilt parameter. The mass controls the transition between topological phases: for $m<0$ there are two Dirac nodes, for $m=0$ a quadratic merging, and for $m>0$ a single Dirac node surviving along the mirror line.

Crucially, in tunable Dirac-Weyl platforms, the protection and annihilation/creation of nodes is dictated by commensurate crystalline symmetries. Mirror lines with $\pm 1$ eigenvalue assignment forbid gapping along high-symmetry directions, and their preservation or breaking under strain, field, or chemical doping governs phase transitions.

3. External Control Parameters and Phase Transitions

Tunability is achieved by varying external parameters that alter symmetry or band structure:

• Strain (mechanical/electronic): In $\beta_{12}$-B$_5$H$_3$, uniaxial tensile strain along $a$-axis tunes the band inversion amplitude $\Delta(\epsilon)$ linearly, annihilating two symmetry-related Dirac cones at a critical strain $\epsilon_c=3.8\%$. For $\epsilon_a<\epsilon_c$, three Dirac cones exist (two Type-I, one Type-II); for $\epsilon_a>\epsilon_c$, only the mirror-protected Dirac point along T–Y survives (Zhong et al., 2024).
• Magnetic Field: In 3D Dirac semimetals, applying a Zeeman field along the symmetry axis splits Dirac points into two or more Weyl nodes, with node separation controllable via field orientation and magnitude. If the magnetic field is tilted to break residual mirror symmetry, further topological transitions—for example, from nodal ring to gapped Weyl modes—can be induced (Zhong et al., 9 Dec 2025, Li et al., 2021, Zheng et al., 2016).
• Optical Pumping (Floquet Engineering): Periodically driven systems under circularly polarized light realize Floquet-induced Weyl nodes whose location and type (I/II) can be arbitrarily controlled by drive amplitude, frequency, and polarization (Bucciantini et al., 2016, Chan et al., 2016, Hübener et al., 2016). This platform enables photoinduced Dirac–Weyl phase transitions on ultrafast (fs–ps) timescales.
• Alloying and Symmetry Reduction: Alloy engineering (e.g., MgTa$_{2-x}$Nb$_x$N$_3$) breaks global symmetries in a tunable way, continuous in $x$. Varying $x$ moves the system through a progression: Dirac → triple-point → Weyl, with corresponding changes in nodal degeneracy and surface arc topology (Huang et al., 2018).

Table: External Control Knobs for Dirac-Weyl Semimetal Tunability

Control Parameter	System/Prototype	Effect on Phase
Uniaxial strain	$\beta_{12}$-B$_5$H$_3$	Multiplicity and type of Dirac nodes
Magnetic field	3D Dirac semimetals (Na$_3$Bi, Cd$_3$As$_2$)	Dirac→Weyl splitting, node motion
Circular light (CPL)	3D Dirac (Na$_3$Bi, Cd$_3$As$_2$)	Floquet Weyl cones, Type–I/II, node separation
Alloy/doping	MgTa$_{2-x}$Nb$_x$N$_3$	Dirac→Triple-point→Weyl transition
Altermagnetic exchange	2D Dirac lattice + d-wave AM	Dirac-Weyl hybrid, node rotation

4. Topological and Spectral Signatures

Tunable Dirac-Weyl phases exhibit characteristic features in their bulk, surface, and edge spectra:

• Anisotropic Fermi velocities and tilting: Fermi velocity tensor and tilt parameter $\tau$ are strain- or field-dependent, with values for $\beta_{12}$-B$_5$H$_3$ ranging from ($v_x$, $v_y$) = (0.80, 0.30)—(0.68, 0.41) ×$10^6$ m/s as strain increases. The tilt parameter changes from $\tau=0.15$ (Type-I) down to $\tau=0.08$, with a large tilt $\tau>1$ for Type-II nodes at zero strain (Zhong et al., 2024).
• Node Merging and Nodal-line Evolution: The coalescence and annihilation of Dirac or Weyl points produce massive Dirac phases or topological gaps, as illustrated by the field-driven transition in ZrTe$_5$ from 3D Weyl semimetal to 2D massive Dirac by exceeding $\Delta_Z > W_b$ (Zeeman energy exceeds interlayer bandwidth) (Zheng et al., 2016).
• Surface and Hinge States: The connectivity of Fermi arcs or edge/higher-order modes maps directly to the bulk node configurations and their symmetry protection. In phase-engineered topolectrical circuits, the Fermi arc/higher-order hinge arcs can be programmed by circuit parameters, with phase boundaries in the (coupling, resistance) space (Rafi-Ul-Islam et al., 2023).
• Berry Curvature and Dynamical Responses: The Berry curvature monopole charge, Chern number jumps on $k$-fixed slices, and Hall conductivity are mathematically tied to node location and tunable via external parameters. In Floquet-driven and phonon-pumped platforms, nonlinear responses such as switchable Berry curvature dipole moments enable ultrafast control of Weyltronics (Aryal et al., 2021, Mishra et al., 27 Jan 2025).

5. Correlated and Coexistent Orders

Tunable Dirac-Weyl platforms enable the realization and interplay of multiple emergent orders:

• Superconductivity: In $\beta_{12}$-B$_5$H$_3$, a robust phonon-mediated superconductivity (Eliashberg $T_c = 32.4$ K, enhanced to 42 K by strain along $b$) is realized in a Dirac semimetal background. Such dual tunability provides a novel platform for studying the interplay of Dirac Fermions and superconductivity, including potential for Majorana physics and correlated topological quantum phases (Zhong et al., 2024).
• Spin-charge Conversion and Magnetotransport: Magnetically tunable Dirac-Weyl systems support substantial spin Hall (and charge Hall) responses that depend quantitatively on the symmetry, strength, and orientation of applied fields. The efficiency of spin-to-charge conversion ($\sim$15–20%) in class-I Dirac semimetals can be optimized via careful symmetry and field management (Li et al., 2021).
• Chiral Anomaly and Topological Transport: The emergence of Weyl nodes under fields or strain gives rise to anomalous Hall effect, negative magnetoresistance, and nonlinear Hall effects with field- and angle-dependent signatures, essential for probing and exploiting chiral anomaly transport phenomena (Zhong et al., 9 Dec 2025).

6. Material Realizations and Experimental Feasibility

A diversity of materials and meta-materials realize tunable Dirac-Weyl phases:

• Elemental and alloyed semimetals: Bulk Na$_3$Bi, Cd$_3$As$_2$, and MgTa$_{2-x}$Nb$_x$N$_3$ offer platforms where Dirac and Weyl nodes can be tuned by magnetic field, alloying, pressure, and symmetry breaking (Narayan et al., 2014, Huang et al., 2018).
• Low-dimensional and designer systems: 2D borophanes, topological circuit/metamaterial networks, heterointerfaces, and cold-atom optical lattices demonstrate high-flexibility control over Dirac–Weyl transitions via strain, synthetic gauge fields, or engineered hopping and resistance elements (Zhong et al., 2024, Liu et al., 24 Jan 2026, Rafi-Ul-Islam et al., 2023, Jiang, 2011).
• Driven phases: Ultrafast light-induced driving (Floquet engineering) and optically pumped phonon modes enable dynamically tunable Dirac–Weyl topologies on sub-picosecond timescales, with the possibility of field-programmable switches, helicity-tunable THz emission via photon drag, and Berry-dipole Hall devices (Chan et al., 2016, Bucciantini et al., 2016, Aryal et al., 2021, Mishra et al., 27 Jan 2025).

7. Comparative Perspective and Research Directions

Tunable Dirac–Weyl phases provide an unprecedented platform for engineering, controlling, and investigating fundamental topological phenomena:

• They enable controlled quantum phase transitions, including multi-node annihilation, integration of line nodes, higher-order Weyl points, and surface/higher-order topological states.
• Dual tunability (Dirac node number/type and $T_c$ for superconductivity) in certain compositions (e.g., $\beta_{12}$-B$_5$H$_3$) establishes novel regimes for exploring the competition and coexistence of topological and ordered phases (Zhong et al., 2024).
• They facilitate the exploration of correlations between topological band inversions, anomalous/quantized transport, and symmetry-protected degeneracy breaking, connecting the design of quantum materials with device-oriented function (reconfigurable spintronics, programmable THz sources, etc.) (Li et al., 2021, Zhong et al., 9 Dec 2025, Mishra et al., 27 Jan 2025).

No general consensus exists on a universal mechanism for all possible Dirac–Weyl transitions—material- and symmetry-specific details (e.g., kinetic birefringence, interaction terms, or proximity effects) play a decisive role. However, the unifying theme is external control over topological band connectivity and node topology, rendering these systems a central focus for contemporary topological quantum materials research.