Weyl vs. Dirac Semimetals: Stability & Properties

Hello! Welcome to our final lesson in this module on topological matter.

In the previous lesson, we explored the fascinating world of Weyl semimetals. We learned that their bulk band structure features topologically protected nodes called Weyl points, which act as monopoles of Berry curvature. This non-trivial bulk topology mandates the existence of exotic surface states known as Fermi arcs, which we saw beautifully confirmed by ARPES experiments.

Today, we will build directly on that foundation to address our final learning outcome: to contrast the stability and properties of Weyl and Dirac semimetals. While both are "topological semimetals," their underlying symmetries, stability, and physical manifestations are crucially different. We will find that a Dirac semimetal can be understood as a symmetric but unstable "parent" state, from which the robust Weyl semimetal emerges when certain symmetries are broken.

1. Degeneracy and Symmetry: The Fundamental Distinction

The most fundamental difference between Dirac and Weyl semimetals lies in the degeneracy of their band-touching points, or nodes. This difference is dictated by fundamental symmetries.

• A Weyl point is a touching of two non-degenerate bands. It is the simplest possible band crossing in 3D.
• A Dirac point is a touching of four bands (or two doubly-degenerate bands). It can be conceptualized as two Weyl points of opposite chirality residing at the same point in momentum space.

This difference in degeneracy is a direct consequence of the crystal's symmetries. In quantum mechanics, the combination of time-reversal symmetry (T) and inversion symmetry (I) forces every energy band to be doubly degenerate at all points in the Brillouin zone (this is an extension of Kramers' theorem).

Therefore, to get the non-degenerate bands required for a Weyl point, you must break either T or I.

• Weyl Semimetal (WSM): Exists in crystals that lack either inversion symmetry (non-centrosymmetric) or time-reversal symmetry (e.g., magnetic materials).
• Dirac Semimetal (DSM): Can exist in crystals that possess both T and I symmetry. Since T and I together enforce double degeneracy, a special condition is needed to allow two of these degenerate bands to cross without gapping. This protection is typically provided by a crystalline symmetry, such as a rotation axis.

To delve deeper into these symmetry requirements, please read the following sections from the review article "Chemical Principles of Topological Semimetals".

2. Topological Stability and the Dirac-to-Weyl Transition

The symmetry constraints directly impact the stability of the nodal points.

Weyl points are topologically stable. As we learned previously, they are monopoles of Berry curvature with a quantized charge ($\chi =\pm 1$). This topological charge cannot be removed by small perturbations. A Weyl point can only be annihilated if it is brought together in momentum space with another Weyl point of opposite charge.

Dirac points are not, in general, topologically stable. A Dirac point at the critical point between a topological and a trivial insulator is "accidental" and can be gapped by a generic perturbation. However, a Dirac point in a DSM like Na₃Bi or Cd₃As₂ is stabilized by additional crystalline symmetries. If you introduce a perturbation that breaks this protecting crystalline symmetry (while preserving T and I), the Dirac point will be gapped.

This vulnerability of the Dirac point is also its most interesting feature: it is a parent state for the more robust Weyl semimetal. By breaking either T or I in a Dirac semimetal, the four-fold degeneracy is lifted, and the Dirac point splits into a pair of Weyl points.

The following short video clip and lecture notes provide two perspectives on this crucial transition.

3. Contrasting Physical Properties

The differences in symmetry and topology lead to distinct experimental signatures.

Surface States

• Weyl Semimetal: The non-zero topological charge of the bulk Weyl points requires the existence of Fermi arc surface states connecting the projections of the bulk nodes. This is a robust, defining feature.
• Dirac Semimetal: The net topological charge of a Dirac point is zero ($+1+(-1)=0$). Therefore, there is no bulk topological invariant that mandates protected surface states. While some DSMs may host surface states due to other reasons (e.g., crystalline symmetry protection), they do not feature the characteristic Fermi arcs that define a WSM.

Transport: The Chiral Anomaly

One of the most exotic transport phenomena in WSMs is the chiral anomaly. In high-energy physics, this refers to the breaking of classical chiral symmetry at the quantum level. In a WSM, it manifests as the non-conservation of the number of electrons near a single Weyl node in the presence of parallel electric ($\vec{E}$) and magnetic ($\vec{B}$) fields. Electrons are "pumped" from one Weyl node to the other through the bulk.

This effect leads to a unique transport signature: a negative longitudinal magnetoresistance, where the resistance decreases when a magnetic field is applied parallel to the current.

• In a WSM, this is an intrinsic property.
• In a DSM, the chiral anomaly is absent in zero magnetic field. However, applying a magnetic field breaks T-symmetry, splits the Dirac points into Weyl points, and thus induces the chiral anomaly. The observation of negative longitudinal magnetoresistance in materials like Na₃Bi was a key piece of evidence for its identity as a DSM.

Optical Properties

Given your interest in optical properties, it's worth noting a distinction here as well. The linear dispersion in both DSMs and WSMs leads to a characteristic frequency dependence of the optical conductivity ${\sigma }_1(\omega )$. For 3D nodal semimetals, theory predicts a linear dependence, ${\sigma }_1(\omega )\propto \omega$, which is distinct from the frequency-independent conductivity of 2D graphene and the more complex behavior of gapped semiconductors.

For more details on these properties and material examples, you can refer back to the review article LINK, particularly the sections "3D Dirac Semimetals," "Weyl Semimetals," "Transport," and "Optical Properties."

Conclusion

We have seen that while Dirac and Weyl semimetals are close relatives, they are distinct topological phases defined by their symmetries. The Dirac semimetal is a highly symmetric, four-fold degenerate state protected by crystalline symmetries, while the Weyl semimetal is a more robust, two-fold degenerate state that emerges when either time-reversal or inversion symmetry is broken.

Key Takeaways: Summary of Differences

This lesson concludes our journey through the fascinating landscape of topological matter. We have progressed from gapped topological insulators with their conducting edge states to gapless semimetals, culminating in the exotic Weyl semimetal with its bulk monopoles and surface Fermi arcs. These concepts represent some of the most active and exciting frontiers in modern condensed matter physics.