Weyl Semimetal and Fermi Velocity

Question

In a Weyl semimetal, the Nielson-Ninomiya theorem enforces the fact that number of positive and negative chirality Weyl points must be equal.

Is there any restriction on the form of the Weyl points? That is to say, given a pair of Weyl points, can the positive chirality point have a different Fermi velocity than the negative chirality point (in the absence of symmetries relating the two)?

If it is true that the Fermi velocities are allowed to be different, are there any physical consequences of this difference?

Answer 1

In its first version, Nielsen-Ninomiya paper was related to the question of whether one can map high-energy physics onto a lattice and in particular neutrino(massless fermions). But their work was later extended to what you call theorem.

Regarding your question, in its original form for neutrinos, there was no question of Fermi velocity since any massless fermion will move at a velocity c. Now in condensed matter, which I am more familiar with, the Nielsen-Ninomiya does not care about the Fermi velocity of the different cones but only about their chirality.

A way to see this, using more recent arguments coming from topological insulators physics is that each weyl nodes of chirality X is a monopole of Berry curvature 1/2X, since any finite system has 0 chern number, due to its edges, you must have a compensation between the monopoles.

Note however, that these are not "all" possible forms. While this allow you to have tilted cones, things can change when you begin considering so called "multi-Weyl" semimetals in which some Weyl cone may carry a chirality +/-2 or even +/-3 in which case the topological argument does not garantee you to have equal number of weyl of each chirality but only an overall 0-chirality when summing all nodes

If you are interested a great review to understand this can be found in Weyl and "Dirac semimetals in three-dimensional solids" by N. P. Armitage, E. J. Mele, and Ashvin Vishwanath

For observable consequences, I do not know exactly. Most topological features global to "all" such Weyl semimetals do only care about their chirality and not their precise shapes. But for sure you may observe consequences of this when considering specific observables whose symmetry is explicitly broken by these unequal shapes (time reversal+inversion).

Answer 2

The existence of Weyl fermions appears to be dependent on the repeating structure of the crystal lattice, specifically translation symmetry. Based on paragraph three of this paper, it looks like the Weyl points will even accommodate defects in Weyl semimetals.

Equal positive and negative chirality Weyl points are imposed on a Weyl semimetal as a way to ensure that it does not spontaneously accumulate chiral charge from Fermi arcs. Anything you do to slow the Fermi velocity of one chirality will prompt other translationally symmetrical 'cells' to interact with the disrupted pair in order to regain charge/momentum balance.

According to the talk (at 13:00) given here by Carlo Beenakker, individual Weyl cones can appear to have a unidirectional current. Charge is conserved by the interaction of left and right-handed cones across the bulk. In the situation you discuss, perhaps there are an even number of left handed and right-handed cones in the bulk, but they are not distributed evenly – this creates a ‘Chiral Chemical Potential Difference’.

The idea is that the uneven charge allows uneven current to appear over short distances. Changes in current produce a measurable magnetic field termed a ‘Chiral Magnetic Effect’, which is discussed (at 17:00). Experimental evidence for CME from CCPD does not appear to exist. However, computer simulations and indirect measurements suggest CCPD theory is valid.

Given that Weyl Fermions of both chirality are massless, will travel at the same speed through electric gradient or changing magnetic field. The speaker proposes a experiment which can produce evidence of CME (at 23:10) which needs a seed magnetic field to influence direction that left and right-handed fermi arcs bend.

It does not appear possible to preferentially accelerate a chiral current without using a CCPD, which is experimentally unconfirmed. The proposed magnetic field seeding will affect Weyl fermions equally because they have the same (zero) mass.

Contrast this with electrons and holes, where mobility is different for each ‘particles’.

Saved from discussion below:

Consider the following simplified example: "We have 1 'coulomb' of right-chirality weyl fermions and 2 'coulombs' left-chirality weyl fermions available to a subgroup of atoms in a weyl semimetal, which is chirality balanced in bulk.

Chiral Chemical Potential Difference with cross-section.

A: The 'chiralstatic?' repulsion produced by their respective concentration gradients causes a current of 1 'ampere' of right-chiral fermions to travel (faster) across a $10mm^2$ cross section of the weyl semimetal as 1 'ampere' of left-chiral fermions travel (slower) in the opposite direction for a short period of time. B: This happens until the chiralities are still imbalanced (.50 'coulombs right', .50 'coulombs left') but have non-equal concentration gradients such that all subsequent flow has $I_{NET}$=.25 'ampere' left-chiral fermions."

This example simplifies the current flow from what would obviously be a changing number to a constant number, but there are still two distinct periods. A has equal current, so by $Magfield=\frac{\mu_o∗I}{2∗pi∗d}$ the current carrying cross section has no magnetic field. B has a non-equal current, producing a temporary magnetic field. B would be an observable physical consequence, but it does originate from the faster speed of the right-chiral current. It originates from the net current of .25 left-chiral 'amperes' during stage B.

Stage A has a net magnetic field of zero because the dense left-chiral current appears magnetically opposite to the fewer contracted reference frame right-chiral fermions that are going faster.