Chiral Weyl Semimetals

• Chiral Weyl semimetals are defined by isolated Weyl nodes with integer chirality that act as monopoles of Berry curvature, underpinning exotic transport and optical effects.
• They exhibit topological electromagnetic responses such as the anomalous Hall and chiral magnetic effects, measurable via spectroscopy and magnetotransport experiments.
• Experimental realizations like TaAs enable detailed studies of ultrahigh mobility, nonlinear optical responses, and strain-induced pseudo-gauge field effects in these materials.

A chiral Weyl semimetal (WSM) is a three-dimensional gapless topological phase in which nondegenerate electronic bands intersect at isolated points (“Weyl nodes”) in momentum space. Each Weyl node acts as a source or sink of Berry curvature and possesses a topologically protected integer “chirality.” The nontrivial band topology underpins a suite of exotic transport, magneto-optical, and collective phenomena, centrally governed by the Adler–Bell–Jackiw (ABJ) chiral anomaly—the nonconservation of separate left- and right-chiral fermion charges under applied fields. Chiral Weyl semimetals provide a condensed-matter realization not only of relativistic anomaly physics, but also host additional topological electromagnetic and optical responses with no high-energy analog.

1. Electronic Structure: Weyl Nodes, Chirality, and Symmetry

The low-energy quasiparticles in a WSM are described by the Weyl Hamiltonian: $$H_\chi(\mathbf{k}) = \chi \hbar v_F\,\mathbf{q} \cdot \boldsymbol{\sigma}$$ where $\mathbf{q} = \mathbf{k} - \mathbf{k}_W$ is momentum relative to the node, $v_F$ is the Fermi velocity, $\boldsymbol{\sigma}$ are Pauli matrices, and $\chi = \pm 1$ labels the chirality (“handedness”) of the node. Each fundamental band-touching point is monopolar in Berry curvature: $$\boldsymbol{\Omega}(\mathbf{k}) = \chi\,\frac{\mathbf{q}}{2|\mathbf{q}|^3}$$ with the monopole charge $\chi = \pm 1$ set by the sign of the determinant of the velocity matrix at the node. The Nielsen–Ninomiya theorem enforces that Weyl nodes of opposite chirality appear in pairs, ensuring zero total chirality in a periodic Brillouin zone.

Symmetry classification determines the multiplicity and protection of nodes. While three-dimensional WSMs require only translation symmetry, in two-dimensional systems additional chiral (sublattice) symmetries are necessary. Explicit construction demonstrates stable WSM phases in all five Altland–Zirnbauer chiral classes in 2D, each characterized by a $\mathbb{Z}$ winding number, leading to topologically protected band touchings and perfectly flat edge states, in contrast to the dispersive Fermi arcs of 3D WSMs (Abdulla, 9 Jan 2024).

2. Chiral Anomaly and Anomaly-Induced Responses

The chiral anomaly, central to WSM phenomenology, appears as nonconservation of the axial (chiral) current in parallel electric and magnetic fields: $$\partial_\mu j_5^\mu = \frac{e^2}{2\pi^2\hbar^2}\,\mathbf{E}\cdot\mathbf{B}$$ This has a direct spectral origin: in a magnetic field, the spectrum consists of chiral Landau levels, with the $n=0$ branch unidirectional, leading to spectral flow under the application of $\mathbf{E} \parallel \mathbf{B}$ (Kim et al., 2017).

Topological electromagnetic response in WSMs is governed by an axion term in the effective action: $$S_\theta = \frac{e^2}{4\pi^2}\int d^4x\,\theta(x)\,\mathbf{E}\cdot\mathbf{B},\qquad \theta(\mathbf{r},t) = 2\mathbf{b}\cdot\mathbf{r} - 2b_0 t$$ where $2\mathbf{b}$ and $2b_0$ are the momentum- and energy-separation of Weyl nodes. This term underlies both the anomalous Hall effect (AHE) and the chiral magnetic effect (CME): $$\mathbf{J}_{\mathrm{AHE}} = \frac{e^2}{2\pi^2}\,\mathbf{b}\times\mathbf{E},\qquad \mathbf{J}_{\mathrm{CME}} = \frac{e^2}{2\pi^2}\,b_0\,\mathbf{B}$$ These responses are robust even in the presence of weak symmetry-breaking gaps, reflecting their anomaly origin (Zyuzin et al., 2012).

Additional chiral anomaly mechanisms—distinct from the high-energy axial anomaly—arise from spin current injection. A gradient of spin-chemical potential along a Weyl node’s spin-polarization axis induces an effective chiral pumping, resulting in quantized charge transfer between nodes, a novel “spin anomaly” with consequences for spin-charge conversion in chiral WSMs (Gao, 2021).

3. Experimental Realizations and Probes

A prototypical material realization is TaAs, in which ab initio calculations and quantum oscillation experiments have determined that the Fermi level lies within a narrow window such that each Weyl node (type W1 or W2) is enclosed by an isolated Fermi surface pocket of definite chirality. Angle-dependent Shubnikov–de Haas and de Haas–van Alphen experiments reveal distinct frequencies and effective masses for each pocket, enabling a full mapping of the chiral Fermi surfaces. Clear evidence for Fermi arcs is provided by ARPES, confirming nontrivial surface states (Arnold et al., 2016).

The chiral anomaly manifests most directly via negative longitudinal magnetoresistance when $\mathbf{E}\parallel\mathbf{B}$, but detailed transport signatures require careful alignment and compensation for conventional magnetoresistive effects.

Plasmon spectroscopy provides a robust probe of chiral-anomaly physics: in intrinsic WSMs, the chiral chemical potential induced by $E\cdot B$ splits the nodes, generating a collective plasmon mode at frequency $\omega_0 \propto (E\cdot B)^{1/3}$, which redshifts and then blueshifts as the system is swept through the anomaly-driven Lifshitz transition. These features are observable by EELS, infrared/THz reflectivity, or inelastic X-ray scattering, and provide a direct fingerprint of the chiral anomaly (Zhou et al., 2014).

Ultrafast and optical probes access even richer anomaly-induced responses. Chiral Weyl semimetals lacking mirror symmetry, such as RhSi, demonstrate a quantized circular photogalvanic effect (CPGE), proportional to the total Berry monopole charge of the accessible node(s). When photon energies are below the energy separation of nodes with opposite chirality, a plateau in the CPGE is observed, providing a direct measurement of the integer Berry charge—a topological invariant—through nonlinear optics (Rees et al., 2019).

Circular dichroism measurements can resolve the anomaly-induced valley chemical potential imbalance, yielding a tunable gyrotropic coefficient and associated optical rotation linked directly to $E\cdot B$ pumping and vanishing in Dirac (nonchiral) systems (Hosur et al., 2014).

4. Collective Excitations and Magneto-Optical Phenomena

Chiral states in multi-Weyl semimetals (nodes with higher monopole charge $m$) lead to $m$ anomalous chiral Landau levels in magnetic field, producing a sequence of low-frequency peaks in magneto-optical absorption and in the Faraday rotation, whose number and strength increase with $m$. The tilting of Weyl cones significantly modulates the resonance conditions and spectral features, providing control over Faraday rotation and related effects (Saha et al., 26 Dec 2024).

A WSM with broken time-reversal symmetry acts as a gyrotropic medium; domain walls hosting a sign change in the magnetization support unidirectional (chiral) electromagnetic surface waves—photonic analogs of quantum Hall edge states (Zyuzin et al., 2014). More generally, the electromagnetic response in WSMs can be cast in terms of a frequency-dependent gyration vector proportional to the node separation, giving rise to strong nonreciprocal optical phenomena, including broadband polarization rotation and chiral-selective transmission (“filtration” without magnetic field) (Chtchelkatchev et al., 2020).

Photoinduced anomalous Hall effects arise when circularly polarized light shifts the Weyl nodes in momentum space in a chirality-correlated fashion. The net nodal shift generates an AHE proportional to the light intensity, again set by the node separation and the topological structure of the band crossings (Chan et al., 2015).

5. Strain, Pseudogauge Fields, and Pseudochiral Anomalies

Strain engineering introduces effective axial gauge fields (pseudomagnetic and pseudoelectric fields) that couple to the chiral charge of Weyl fermions with opposite sign at opposite nodes. These “pseudochiral anomalies” lead to distinct physical consequences, such as negative resistance in twisted nanowires, topological hydrodynamic flow (“coaxial cable” current patterns), and hybrid collective modes (plasmon-magnon coupling or magnetoacoustic attenuation) (Pikulin et al., 2016). For time-dependent strains, optical absorption resonances arise at the “pseudocyclotron” frequency, and planar Hall effects emerge under strain-induced planar pseudo-magnetic fields (Heidari et al., 2019).

In addition, spatially varying magnetic fluctuations, as in magnetically doped topological insulators, play the role of chiral gauge fields, enabling manipulation and detection of the chiral anomaly through magnetization textures and spin waves (Liu et al., 2012).

6. Scattering, Transport, and Ultrafast Dynamics

Wave-packet scattering in chiral WSMs is governed by a chirality-protected shift in the scattering amplitude, sharply suppressing large-angle backscattering as the Fermi level approaches the Weyl node. This results in a dramatic enhancement of the transport lifetime over the quantum (level-broadening) lifetime, producing the ultrahigh mobility observed in these materials (Jiang et al., 2016).

Nonlinear Hall responses mediated by the chiral anomaly are enabled in noncentrosymmetric WSMs with node tilting. Chiral-anomaly-induced valley charge pumping, together with Berry curvature anomalous velocities, leads to a Hall signal quadratic in electric field and linear in magnetic field, which can dominate at low doping and is tunable via crystal symmetry and strain (Li et al., 2020).

Asymmetric WSMs—where Weyl nodes have different Fermi velocities or energies—allow the generation of a finite chiral chemical potential from non-chiral pumping, driving a “built-in” chiral magnetic effect even in the absence of parallel electric and magnetic fields, with measurable consequences for magnetotransport (Kharzeev et al., 2016).

Topological edge and surface channels in WSMs are also governed by the projection of Weyl nodes onto the boundary Brillouin zone. In the presence of structural features such as step edges, chiral modes localize algebraically with a $1/r$ tail, forming a discrete ladder of sub-arc surface modes governed by step height and node separation (Takane, 2017).

7. Extensions, Limitations, and Outlook

In strong magnetic fields, the canonical anomaly picture breaks down once the magnetic length becomes comparable to the nodal separation. Opposite-chirality zeroth Landau levels can hybridize, opening a gap and quenching the anomaly-induced conductivity and associated collective modes. The characteristic threshold for this breakdown is set by the material’s nodal geometry and the strength/orientation of the applied field (Kim et al., 2017).

Chiral anomaly physics persists in the presence of disorder, moderate interactions, and even weak symmetry-breaking gaps, provided the gap is small relative to the ultraviolet bandwidth (Zyuzin et al., 2012). Open questions persist regarding the interplay between topology, electron correlations, disorder, and external drives (ultrafast/nonlinear regimes), as well as the design of devices harnessing chiral optical selection rules, nonreciprocal transport, and strain-based responses in chiral Weyl semimetals.