Skip to content

Weyl Basics

Basics of Weyl Physics

Kramer's Theorem

For every energy eigenstate of a time-reversal symmetric system with half-integer total spin, there is at least one more eigenstate with the same energy. That is, every energy level is at least doubly degenerate if it has half-integer spin. Kramers Theorem

Time Reversal Symmetry

for spin- \(\frac12\) particles \(T^2 = \pm 1\)

Weyl Fermions

When the mass term in the Dirac equation vanishes, two independent solutions appear. They are chiral fermions with opposite handedness, called Weyl fermions. Although they have not been discovered as a type of fundamental particle within the Standard Model, quasiparticles with these properties were discovered about five years ago in condensed matter systems called Weyl semimetals.

The chirality of the Weyl fermion comes from the fact that the direction of its pseudospin is always locked to the direction of the translational motion and is either parallel (positive chirality) or antiparallel (negative chirality) to it. The chiral electronic structure leads to particular charge dynamics: the chiral anomaly and chiral magnetic effect.

Weyl Semimetals

  • Weyl semimetals have some amount of protection from interaction or disorder.
    • This could be because disorder is average and on average keeps the symmetry
    • Band structure of semimetal has very low density of states on the Fermi level

Helicity

source

Every matter particle (electrons, quarks, etc.) is spinning, i.e. each matter particle carries some intrinsic angular momentum. This spin is an inherently quantum mechanical property of fundamental particles. (red: indicates spin and gray: indicates the direction). The particle shows right-handedness. This is called helicity of the particle. (specifically right helicity)

spin.gif Right Helicity

left-helicity.png Left Helicity

Mass is a something that tells us whether helicity is an “intrinsic” property of the particle. If a particle is massless, then its helicity has a fixed value in all reference frames. On the other hand, if a particle has any mass, then helicity is not an intrinsic property since different observers (in valid reference frames) can measure different values for the helicity (left- or right-helicity). So even though helicity is something which is easy to visualize, it is not a “fundamental” property of most particles.

Chirality

Just as we say that a particle can have left- or right-handed helicity, we also say that a particle can have left- or right-handed chirality. As we said above, for massless particles the chirality and helicity are the same. A massless left-chiral particle also has left-helicity.

However, a massive particle has a specific chirality. A massive left-chiral particle may have either left- or right-helicity, depending on your reference frame relative to the particle(imagine overtaking a vehicle). In all reference frames, the particle will still be left-chiral, no matter what helicity it is.

It is an inherently quantum mechanical sense in which a particle is left- or right-handed. For now, let us focus on fermions, which are “spin one-half.” Recall that this means that if you rotate an electron by 360 degrees, you don’t get the same quantum mechanical state: you get the same state up to a minus sign! This minus sign is related to quantum interference. A fermion’s chirality tells you how it gets to this minus sign in terms of a complex number.

complexrotation.png

What happens when you rotate a left- vs right-chiral fermion 360 degree about its direction of motion. Both particles pick up a -1, but the left-chiral fermion goes one way around the complex plane, while the right-chiral fermion goes the other way. The circle on the right represents the complex phase of the particle’s quantum state; as we rotate a particle, the value of the phase moves along the circle. Rotating the particle 360 degrees only brings you halfway around the circle in a direction that depends on the chirality of the fermion.

Physically, it means that the phase of the wavefunction changes. Rotating a fermion shifts its quantum wavefunction in a way that depends on the fermion chirality.

Particles with different chiralities are really different particles(see below image) If we have a particle with left-handed helicity, then we know that there should also be a version of the particle with right-handed helicity. On the other hand, a particle with left-handed chirality needn’t have a right-chiral partner.

(electron and positron have same charge but opposite chirality.)

electron_positron.png

Chiral Chemical Potential (CCP)

Chiral Magnetic Effect (CME)

  • It is the phenomenon of electric charge separation along the external magnetic field that is induced by the chirality imbalance. The CME is a macroscopic quantum effect — it is a manifestation of the chiral anomaly creating a collective motion in Dirac sea.
  • CME refers to the appearance of additional charge current along the magnetic field when the two Weyl nodes have different chemical potential from each other.

MF is a crucial part in CME, as it breaks the rotational invariance and creates a preferred orientation for the spins of the fermions.

Chiral Anomaly

It is one of the measurable properties of materials with Weyl nodes in their band structure.
It refers to the charge pumping effect from one Weyl point to the other when electric and magnetic fields are applied parallel to each other.