Topological Weyl Semimetal State

Updated 4 March 2026

Topological Weyl semimetals are gapless 3D quantum states where conduction and valence bands cross at isolated Weyl points that act as quantized monopoles of Berry curvature.
They are classified into Type I and Type II based on the tilt of the Weyl cones, leading to either point-like Fermi surfaces or overlapping electron and hole pockets with distinct Fermi arc features.
Their robust topological invariants enable unique phenomena in materials like TaAs and MoTe2, paving the way for applications in low-dissipation electronics and quantum devices.

A topological Weyl semimetal (WSM) is a gapless three-dimensional quantum phase in which conduction and valence bands cross at isolated points in momentum space—“Weyl points”—each acting as a quantized monopole of Berry curvature. These crossings, being protected by topology and crystal symmetry constraints, generate a host of robust phenomena distinct from both conventional metals/semimetals and topological insulators. There exist two distinct classes: Type I, whose low-energy excitations are upright Weyl cones with a point Fermi surface at the node energy, and Type II, in which the cone is strongly over-tilted, resulting in touching electron and hole pockets. The existence of Weyl nodes leads to nontrivial transport (chiral anomaly), magnetic, and surface-state properties, and the associated “Fermi arc” surface bands provide a direct spectroscopic signature of the topological WSM state (Liang et al., 2016, Xu et al., 2015, Yan et al., 2016, Wang et al., 2016, Belopolski et al., 2016).

1. Topological Character and Minimal Weyl Hamiltonian

A WSM is characterized by isolated twofold band crossings in the Brillouin zone (BZ). Near each Weyl node $\mathbf{k}_W$ , the low-energy theory is governed by a two-band Hamiltonian: $H(\mathbf{k}) = \sum_{i=x,y,z} T_i (k_i - k_{W,i}) \, \mathbb{I} + \sum_{i=x,y,z} v_i (k_i - k_{W,i}) \, \sigma_i$ where $v_i$ are the Fermi velocities (possibly anisotropic) and $T_i$ encode tilt. The eigenenergies are

$E_{\pm}(\mathbf{k}) = \mathbf{T} \cdot (\mathbf{k} - \mathbf{k}_W) \pm \sqrt{\sum_i (v_i (k_i - k_{W,i}))^2}$

For $|\mathbf{T}| < \min_i |v_i|$ , the node is Type I: Fermi surface reduces to a point at the Weyl energy.
For $|\mathbf{T}| > \min_i |v_i|$ , the node is Type II: the cone is over-tilted, so a constant-energy plane through the node contains touching electron and hole pockets (Liang et al., 2016, Wang et al., 2016).

Each Weyl node is a source ( $C=+1$ ) or sink ( $C=-1$ ) of the Berry curvature, with the Chern number

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

where $\boldsymbol{\Omega}(\mathbf{k})$ is the Berry curvature and $\mathcal{S}$ encloses the node. The stability of Weyl nodes is topologically protected: the net chirality in the BZ must vanish; isolated nodes can only be gapped by annihilation with partners of opposite sign.

2. Symmetry Realization, Materials, and Band Topology

Weyl points require the breaking of either inversion ( $\mathcal{P}$ ) or time-reversal ( $\mathcal{T}$ ) symmetry. In the archetypal TaAs family (TaAs, TaP, NbAs, NbP), $\mathcal{P}$ is broken, and $\mathcal{T}$ maintained; in magnetic WSMs such as Co $_3$ Sn $_2$ S $_2$ , $\mathcal{T}$ is broken and $\mathcal{P}$ preserved. The count and placement of Weyl nodes depend on crystal space group, SOC, and lattice parameters (Xu et al., 2015, Liu et al., 2021, Yang et al., 2015, Belopolski et al., 2016).

Canonical WSM materials and the classes they represent:

Compound	Symmetry breaking	Weyl type	Characteristic nodes (per BZ)
TaAs, NbAs, TaP	Inversion	I	24
WTe $_2$ , MoTe $_2$ , Mo $_x$ W $_{1-x}$ Te $_2$	Inversion	II	8
Co $_3$ Sn $_2$ S $_2$	Time-reversal (magnetic)	I	12

In type-II WSMs such as Td-MoTe $_2$ and WTe $_2$ , theoretical calculations and ARPES demonstrate strongly tilted Weyl cones and Fermi surfaces composed of electron and hole pockets touching at energies up to ∼40–90 meV above $E_F$ (Liang et al., 2016, Wang et al., 2016, Belopolski et al., 2016).

3. Fermi-Arc Surface States and Experimental Signatures

Bulk-boundary correspondence dictates that a 2D surface BZ enclosing the projection of two bulk Weyl nodes of opposite chirality hosts open surface states (Fermi arcs) joining these points. The Chern number on individual $k_z$ -slices jumps by ±1 when $k_z$ passes through a Weyl node (Xu et al., 2015, Liang et al., 2016, Xu et al., 2015, Belopolski et al., 2016, Benito-Matías et al., 2018).

ARPES and STM experiments consistently resolve:

Singular, non-closed surface-state branches (arcs) terminating on the projections of bulk Weyl nodes.
In type-II WSMs, the Fermi arc merges into bulk electron and hole pockets, a distinctive hallmark of the Lifshitz transition unique to over-tilted cones.

Explicit example: In Td–MoTe $_2$ , only one Fermi-arc branch (SS) connects identified electron (E2) and hole (H4) pockets, reflecting the minimal nodal configuration allowed by symmetry and in agreement with DFT calculations (Liang et al., 2016).

STM-QPI and surface JDOS studies further discriminate oscillatory vs. simple exponential decay of surface states, reflecting the interplay between bulk mass parameters and Fermi velocity (Benito-Matías et al., 2018, Zhang et al., 2017).

4. Topological Invariants and Lifshitz Transitions

Each Weyl node’s Chern number is defined via Berry curvature flux integration. As system parameters (chemical potential, lattice constants, correlations) tune the positions or annihilate pairs of Weyl nodes (e.g., via temperature, strain, or field), the system undergoes topological Lifshitz transitions signaled by changes in node count or type and in surface-arc connectivity (Liang et al., 2016, Liu et al., 2021, Xu et al., 2017).

In magnetic WSMs, the transition from a Weyl to a trivial state (e.g., Co $_3$ Sn $_2$ S $_2$ across $T_C$ ) entails recombination and annihilation of nodes and the disappearance of Fermi arcs. This is captured by ARPES as a collapse of the band splitting and surface spectral weight (Liu et al., 2021).

5. Transport Properties and Anomalous Responses

The chiral anomaly in WSMs produces negative longitudinal magnetoresistance (NLMR) due to charge pumping between nodes of opposite chirality under parallel $\mathbf{E}$ and $\mathbf{B}$ . In type-I systems the effect is maximal along the node separation axis. In type-II WSMs the response is highly angle dependent; the existence of extended electron and hole Fermi surfaces admits additional scattering channels, modifying both the magnitude and direction dependence of NLMR (Liang et al., 2016, Zhao et al., 2017, Belopolski et al., 2016).

For a pair of nodes separated by $2k_0$ , the anomalous Hall conductivity is quantized: $\sigma_{yz} = \frac{e^2}{\pi h} k_0$ For tilted systems (type-II and hybrid Weyl semimetals), this quantization is partially lost as only the “class-1” Fermi arc segment contributes a nontrivial chiral channel, with the residual Hall response proportional to the arc length (Zhao et al., 2017).

6. Tunability, Material Platforms, and Applications

Type II Weyl states are exceptionally sensitive to lattice constants, pressure, strain, alloying, and electronic correlations. Small changes (∼1%) in lattice parameters of MoTe $_2$ can switch between phases with different node count and type (2 or 4 node pairs, type I or II). Thin film exfoliation and alloying (W/Mo ratio) enable “topological switching”—direct tuning of topological features and device-relevant phenomena (Liang et al., 2016, Belopolski et al., 2016).

The observation of nontrivial surface arcs, together with the strongly angle- and field-dependent chiral anomaly, opens routes to low-dissipation electronics, spintronic devices, topological switches, and potentially to the realization of topological superconductors in systems exhibiting both Weyl nodes and superconductivity (as in MoTe $_2$ under pressure) (Jiang et al., 2016, Liang et al., 2016, Belopolski et al., 2016).

7. Minimal Models and Theoretical Extensions

Minimal lattice models (both time-reversal-breaking and inversion-breaking) precisely capture the tunable transition between type-I and type-II WSM phases, the emergence and annihilation of Weyl nodes, surface-arc reconfiguration, and even the appearance of non-topological “track states” that emerge at the type-I/II boundary (McCormick et al., 2016). Optical lattice implementations allow for the engineering and control of Weyl nodes, including regimes with mixed-type (type-1.5) nodes (Kong et al., 2016, Jiang, 2011).

The general phenomenology of bulk monopole charges, Fermi arcs, chiral-anomaly-driven response, and topological phase transitions establishes the topological Weyl semimetal as a paradigmatic quantum phase mediating between trivial metals and quantum Hall/axion insulator states.

References:

(Liang et al., 2016, Xu et al., 2015, Yan et al., 2016, Liu et al., 2021, Wang et al., 2016, Belopolski et al., 2016, McCormick et al., 2016, Benito-Matías et al., 2018, Zhao et al., 2017, Xu et al., 2017)