Lorentz-Violating Weyl Fermions

• Lorentz-violating Weyl fermions are massless spin-½ excitations whose effective dynamics incorporate background tensors that explicitly break Lorentz symmetry.
• They exhibit modified dispersion relations with overtilted Weyl cones and anisotropic electromagnetic responses derived from effective Hamiltonians in both high-energy and condensed-matter frameworks.
• Experimental realizations in type-II Weyl semimetals like LaAlGe and MoTe₂ validate their unique topological properties and tunable transport phenomena.

Lorentz-violating Weyl fermions are massless spin-½ excitations whose low-energy dynamics are governed by effective Hamiltonians or Lagrangians containing background tensors that explicitly break Lorentz invariance. These systems arise in both high-energy extensions (notably the Standard-Model Extension, SME) and in condensed matter, especially in type-II Weyl semimetals such as LaAlGe, MoTe₂, and MoₓW₁₋ₓTe₂, where symmetry-breaking terms (Zeeman splitting, inversion-breaking spin–orbit coupling, or tilt) generate shifts and overtilting of Weyl cones. Fundamentally, the Lorentz-violating terms produce dispersions and topological responses inaccessible in conventional Weyl or Dirac systems, and their observation in crystalline solids offers a laboratory analog of Lorentz violation in field theory.

1. Theoretical Framework and Effective Hamiltonians

The minimal description of Lorentz-violating Weyl fermions is provided by the SME, with the key CPT-odd operator $b_\mu \bar\psi \gamma_5 \gamma^\mu \psi$ acting on the Dirac spinor $\psi$. The massless limit yields the Lagrangian

$$\mathcal{L}_{\rm Weyl} = \bar\psi\left(i\gamma^\mu\partial_\mu - b_\mu\gamma_5\gamma^\mu\right)\psi,$$

in which $b_\mu$ is a fixed background vector encoding Lorentz violation (Kostelecky et al., 23 Dec 2024). Upon projection onto Weyl chiralities and restoring the Fermi velocity $v_F$, the condensed-matter momentum-space Hamiltonian reads

$$H = \sum_{\chi=\pm} \Psi_\chi^\dagger\left[ \chi v_F\, \boldsymbol{\sigma} \cdot (\mathbf{p} - \chi \mathbf{b}) - \chi b_0 \right] \Psi_\chi,$$

where $\chi$ labels chirality, $\boldsymbol{\sigma}$ are Pauli matrices, and $\mathbf{b}$ (spatial component of $b_\mu$) and $b_0$ shift momenta and energies, respectively (Kostelecky et al., 2021, Kostelecky et al., 23 Dec 2024, Grushin, 2012).

In type-II semimetallic realizations, the most general two-band Hamiltonian near a Weyl node includes a tilt vector $\mathbf{t}$: $$H(\mathbf{k}) = \mathbf{t} \cdot \mathbf{k} \, \mathbb{1}_{2\times 2} + \sum_{i=1}^3 v_i k_i \sigma_i,$$ with $|t_i| > v_i$ indicating overtilt and intrinsic Lorentz violation (Xu et al., 2016, Xu et al., 2016, Belopolski et al., 2016).

2. Spectral Properties and Dispersion Relations

The presence of $b_\mu$ or tilt modifies the Weyl cones:

• For $b_\mu$, the dispersion relation is

$$E_\chi^{\pm}(\mathbf{p}) = -\chi b_0 \pm |\mathbf{p} + \chi \mathbf{b}|,$$

shifting energy by $b_0$ and momentum by $\chi\mathbf{b}$ for chirality $\chi$ (Kostelecky et al., 23 Dec 2024).

• For the tilt, the spectrum becomes

$$E_\pm(\mathbf{k}) = \mathbf{t}\cdot \mathbf{k} \pm \sqrt{(v_x k_x)^2 + (v_y k_y)^2 + (v_z k_z)^2},$$

with overtilted regimes ($|\mathbf{t}| > \min(v_x,v_y,v_z)$) exhibiting electron and hole pockets that touch at the Weyl node, precluding restoration by Lorentz boosts (Xu et al., 2016).

In the condensed-matter context, such overtilted dispersions have been confirmed in LaAlGe and MoTe₂ using ARPES techniques (Xu et al., 2016, Xu et al., 2016, Belopolski et al., 2016).

3. Topological Invariants and Geometric Phases

Lorentz-violating Weyl fermions retain nontrivial Berry curvature and Chern numbers:

• Each shifted or split Weyl node acts as a monopole in momentum space with curvature

$$\Omega_i^{(1,2)}(\mathbf{k}) = \mp \frac{k_i}{2|k|^3},$$

corresponding to first Chern numbers $C_1 = \pm 1$ for spheres enclosing each node (Kostelecky et al., 1 Dec 2025, Kostelecky et al., 2021).

• In scenarios with large Lorentz violation (e.g., $|b^2| \gtrsim m^2$), occupied negative-energy branches bestow the vacuum with a quantized geometric phase $\pm\pi$ (Kostelecky et al., 1 Dec 2025).
• Alternative momentum-space topological invariants, such as four-momentum winding numbers, confirm quantization and - in the presence of Lorentz violation - can label distinct ground states (Kostelecky et al., 1 Dec 2025).

4. Modified Electromagnetic Response and Quantum Anomalies

Integrating out the Lorentz-violating fermions yields macroscopic electromagnetic actions with anomalous terms:

• A radiatively generated Chern–Simons term appears,

$$\Delta L_{\rm CS} = \frac{1}{2} k_\mu \epsilon^{\mu\nu\rho\sigma} A_\nu F_{\rho\sigma},$$

where $k_\mu$ depends on $b_\mu$ and the microscopic model (Grushin, 2012, Kostelecky et al., 2021). In condensed-matter models, lattice regularization fixes the coefficient uniquely, lifting the ambiguity present in quantum field theory (Grushin, 2012).

• The axial anomaly retains its canonical form,

$$\partial_\mu J_5^\mu = \frac{e^2}{16\pi^2} \epsilon^{\mu\nu\rho\sigma} F_{\mu\nu} F_{\rho\sigma},$$

and is insensitive to weak Lorentz-violating tilt (Gómez et al., 2021).

• Transport coefficients, such as the anomalous Hall conductivity,

$$\sigma^{ij} = \frac{1}{2\pi^2} \epsilon^{ijl} b_l \sqrt{1 - m^2/|b|^2},$$

and magnetoelectric responses, are directly calculable in terms of SME coefficients (Kostelecky et al., 2021).

In type-II Weyl semimetals, the overtilted cones introduce strong directionality and open Fermi surfaces; the chiral anomaly and optical responses become anisotropic and are tunable via doping or chemical substitution (Xu et al., 2016, Xu et al., 2016, Belopolski et al., 2016).

5. Experimental Realizations and Observations

Type-II Lorentz-violating Weyl fermion states have been experimentally observed in LaAlGe, MoTe₂, and MoₓW₁₋ₓTe₂:

• In LaAlGe, ARPES and DFT identify sixteen type-II Weyl nodes near the Fermi level with strong tilt, robust against structural perturbations and readily accessible for transport and optical studies (Xu et al., 2016).
• In MoTe₂, four type-II nodes in the $k_z=0$ plane produce electron and hole pockets, with distinctive Fermi arcs confirmed by ARPES (Xu et al., 2016).
• In Mo₀.₂₅W₀.₇₅Te₂, pump-probe ARPES directly observes Fermi arcs above the Fermi level and confirms overtilted dispersions, establishing tunable platforms for Lorentz-violating phenomena (Belopolski et al., 2016).

Optical birefringence in these materials provides quantitative measurement of Lorentz-violating parameters $b_\mu$, with laboratory bounds orders of magnitude less stringent than those from astrophysical searches for fundamental Lorentz violation (Grushin, 2012).

6. Vacuum Structure, Concordance Problem, and Thermodynamic Resolution

Large Lorentz-violating backgrounds ($|b_\mu| \sim m$ or in highly boosted frames) raise the concordance problem: naive vacuum instability due to the mixing of positive and negative energy branches. In condensed matter, as shown for Weyl semimetals, the physical ground state is thermodynamically set by the lattice bath, phonons, and chemical potential, with all negative-energy states occupied, yielding a stable vacuum even under large Lorentz violation (Kostelecky et al., 23 Dec 2024). Boosting the observer frame changes occupation patterns only in coordinate representation, not thermodynamic stability. This analogy extends to fundamental physics via identification of a cosmological bath (e.g., the CMB frame).

Furthermore, when the vacuum fills an entire topologically nontrivial branch, the many-body ground state acquires a quantized geometric phase, establishing a topological vacuum (Kostelecky et al., 1 Dec 2025).

7. Bulk-Boundary Correspondence and Further Topological Structures

Lorentz-violating Weyl and Dirac systems exhibit robust bulk-boundary correspondence:

• Bulk topological invariants (Berry monopole charge, winding numbers) predict drumhead surface bands or Fermi arcs (Kostelecky et al., 2021, Xu et al., 2016, Belopolski et al., 2016).
• Surface-sensitive ARPES in LaAlGe and MoTe₂ directly maps such features; connectivity and number of Fermi surface crossings confirm the topological character of observed arcs (Xu et al., 2016, Xu et al., 2016).
• Fine control over symmetry parameters in crystal or lattice models enables continuous tuning across transitions between type-I, type-II, and nodal line regimes (Xu et al., 2016).

Table: Key Lorentz-Violating Weyl Semimetal Materials

Material	Type	Nodes at EF?	Experimental Probe
LaAlGe	Type-II	Yes (W2)	ARPES, DFT
MoTe₂	Type-II	No (offset)	ARPES
MoₓW₁₋ₓTe₂ (x=0.25)	Type-II	Yes (W₁)	Pump-probe ARPES

The position of nodes relative to the Fermi level and accessibility by spectroscopy or transport vary, with LaAlGe providing the cleanest stoichiometric realization.

• Consequences of a condensed matter realization of Lorentz violating QED in Weyl semi-metals (Grushin, 2012)
• Lorentz violation in Dirac and Weyl semimetals (Kostelecky et al., 2021)
• Physical interpretation of large Lorentz violation via Weyl semimetals (Kostelecky et al., 23 Dec 2024)
• Discovery of Lorentz-violating Weyl fermion semimetal state in LaAlGe materials (Xu et al., 2016)
• Discovery of Weyl semimetal state violating Lorentz invariance in MoTe2 (Xu et al., 2016)
• Discovery of a new type of topological Weyl fermion semimetal state in Mo$_x$W$_{1-x}$Te$_2$ (Belopolski et al., 2016)
• The Axial Anomaly in Lorentz Violating Theories: Towards the Electromagnetic Response of Weakly Tilted Weyl Semimetals (Gómez et al., 2021)
• Lorentz violation and momentum-space geometric phases (Kostelecky et al., 1 Dec 2025)