Universiteit van AmsterdamInstitute of Physics

Master Thesis

Weyl semimetals

Author:

Bogdan GALILO

Supervisor:

Dr. Ari TURNER

July 23, 2013

Abstract

Weyl semimetals are novel topological phases of three-dimensional materials,

which are characterized by the existence of a set of linear-dispersive band-touching

points, called Weyl nodes. They are considered to be protected against small per-

turbations and disorder.

In the current work the problem of the protection of Weyl semimetals is addressed

on a simple lattice model realizing two Weyl nodes. To test the protection we add an

external periodic in position space potential. The energy spectrum is investigated

analytically using second order Brillouin-Wigner perturbation theory in regime of

weak potential. A transition from a gapless to a gapped bulk phase is observed and

a phase diagram is provided. The energy spectrum is also investigated numerically

using exact diagonalization of the Hamiltonian. It turns out that even though Weyl

semimetals are expected to be robust against small perturbation, we find that one

can open a gap by a right adjustment of the wavenumber of the external potential.

Also, in the search for observable features of Weyl semimetals the magnetic neutron

scattering problem on a free Weyl fermion is considered. The differential cross-

section is computed in Born approximation and the angular and magnetic aspects of

the scattering process are discussed. Some features of the linear dispersion of Weyl

fermions like threshold neutron scattering momentum and the jump in the maximal

neutron scattering angle are discussed in the context of experimental realization.

– 1 –

Contents

I. Introduction 3

II. Weyl semimetals in periodic potential 10

A. Lattice model 10

B. Regime of weak potential 13

C. Phase diagram and discussions 19

III. Neutron Scattering on a Weyl fermion 22

A. Differential cross-section in Born approximation 22

B. Polarization and scattering angle 29

C. Conclusions and Discussions 32

IV. Conclusions and Discussions 34

A. Lattice model of a Weyl semimetal 37

References 39

– 2 –

– 3 –

I. INTRODUCTION

One of the greatest achievement in science of the 20th century is the one in the under-

standing the fundamental interactions and the elementary particles. Thanks to the con-

tribution of many scientists nowadays we have elaborated theories of special and general

relativity, of quantum mechanics, quantum field theory etc., which made possible not only

the formulation of the Standard Model of the interaction of elementary particles but even

looking beyond it. However, even though our world is made out of elementary particles it is

quite rare for them to be “alone”. Rather particles like to gather into complex systems which

far exceeds in variety the elementary particles itself. Moreover the collective behaviour is so

rich that many phenomena can be described in terms of new particles (called quasi-particles)

which are different from the actual constituent elementary particles. For instance, Cooper

pairs responsible for the BCS superconductivity differ from electrons in many aspects such as

statistics and charge. However, one could think of re-gaining the properties of the elementary

particle physics in collective phenomena. With this respect, recently great attention was de-

voted to realization of Dirac and Majorana fermions as low-energy excitations in condensed

matter systems. These types of excitations are interesting due to its special properties and

the possible applications. For instance, in systems such as graphene the low energy physics

is described by Dirac fermions and Lorentz symmetry locally holds. Remarkable properties

such as high charge mobility and non-zero quantum conductivity at low temperatures have

been observed for graphene. Unlike Dirac fermions, Majorana fermions, which carry zero

charge and are antiparticles of themselves, don’t describe any elementary particle from the

set of standard model particles (however, the nature of neutrinos is not completely settled

out yet). However, driven by the interesting property of quantum information protection,

scientific community makes great efforts in realization of these states in nature. In that

context one can extend this idea and look for realizations of other elementary excitations in

many-body physics which historically were introduced to describe fundamental interactions

but were left out of the game. The closest to Dirac fermions are perhaps the Weyl fermions,

introduced by Hermann Weyl one year later than Dirac introduced his famous equation.

Dirac in attempt to linearize the Schrodinger equation, which is quadratic in derivatives,

wrote his equation in 1928 [1] which turned out to successfully describe free electrons and

positrons. A Dirac fermion is the solution of the Dirac equation HΨ = EΨ given the Dirac

– 4 –

Hamiltonian is

H = pxαx + pyαy + pzαz +mβ,

where px, py and pz are the momentum components, m is the mass of the Dirac fermion

and αx, αy, αz and β are anticommuting objects αi, αj = δij, αi, β = 0 and β2 = 1.

-1.0 -0.5 0.5 1.0ÈpÈ

-1.0

-0.5

0.5

1.0

E

FIG. 1: Energy levels of a Dirac Hamil-

tonian for different values of masses (gap)

m. The dashed line is for m = 0. When

m 6= 0 there is a gap ∆ = 2m between

positive and negative energy bands.

If mass m is not zero the minimal representation

for α-s and β is four dimensional square matrices

and hence the Dirac fermion wavefunction has four

components. The fourth, anticommuting with α-s,

matrix β makes possible having a mass term

E = ±√p2x + p2

y + p2z +m2.

If the above Hamiltonian was realized in solids,

one would think of having an energy band struc-

ture E± and the mass term would play the role of

a gap (Fig. 1).

Dirac fermions can be realized in graphene, a

two dimensional layer of one carbon atom thick-

ness. The graphene lattice consists of two sublattices (Fig. 2). If there is a symmetry in

the interchange of sublattices then the band structure of the Brillouin zone contains two

isolated points with locally linear dispersion relation given by the Dirac Hamiltonian:

Hgraphene = vFpxσx + vFpyσy +mσz,

Egraphene = ±√v2Fp

2x + v2

Fp2y +m2,

where vF is the effective speed of electronic states and the Pauli matrices act in the space of

the two sublattices, so called pseudospin space (’spin up’ corresponds to the component on

one sublattice and ’spin down’ to that on the other). Since graphene is a two dimensional

lattice it has two dynamical degrees of freedom which correspond to two momenta px and py

on the lattice plane. It can be shown that the two momenta couple linearly to σx and σy in

– 5 –

FIG. 2: Illustration of the graphene lattice (left) and its band structure (right) [3]. the red and blue

nodes belong to the two different triangular graphene sublattices which give rise to pseudospin. In

the band structure pictured on the right one identifies six isolated points where the bands touch,

however only two of them are seating in the same Brillouin zone. Close to these points the energy

spectrum is linear and the Hamiltonian is 2D Dirac Hamiltonian.

an ideal symmetric in the sublattice interchange graphene (in this case the highlighted mass

term m in the above Hamiltonian vanishes and one gets Dirac points as shown in Fig. 2).

However, since there is one more Pauli matrix σz, which anticommutes with the other two,

a gap can open, i.e. a mass term can be generated, by small perturbations, due to disorder,

external magnetic field perpendicular to the lattice plane or some other perturbation which

couples to σz.

Hermann Weyl noticed in 1929 [2] that if the particles are massless then the relativistic

fermions can be described by a two component spinor in contrast to four component Dirac

spinor. Nowadays the corresponding fermions are called Weyl fermions and they obey the

Weyl equation HΨ = EΨ generated by the Weyl Hamiltonian

H = pxσx + pyσy + pzσz,

E = ±√p2x + p2

y + p2z = ±|p|,

where unlike the 3 + 1 dimensional Dirac equation the minimal matrix representation of

σx, σy and σz is two dimensional matrices. This does not leave any room for a mass term

since all three σ-matrices are already coupled to all three momenta p, and there is no

other anticommuting 2 × 2 matrix available. In terms of energy bands, this means that

one cannot open a gap and the fact that the bands touch in some points is stable. In that

sense, if realized in materials, 3D Weyl fermions are expected to be more protected to small

perturbations than the Dirac fermions in graphene. Such a 3D phase of matter that would

– 6 –

have the low energy physics determined by the gapless Weyl-like fermionic excitations is

conventionally called Weyl semimetal.

One important quantity characterizing massless particles is helicity. Helicity is deter-

mined as the projection of the spin vector s on the direction of the momentum of the

particle

χ =p · s|p||s|

.

Given that for a spin 1/2 fermion the spin operator is s = ~σ/2, the helicity operator

for a Weyl fermion is χ = p · σ/|p|. Because the helicity operator χ commutes with the

Weyl Hamiltonian H = vFp ·σ, helicity is a conserved quantity and hence a good quantum

number. The eigenvalues of the helicity operator are ±1, where +1 stands for a fermion with

the spin vector aligned along the momentum, −1 in the opposite direction. Considering the

energy of a Weyl fermion is E = ±|vFp|, the Weyl Hamiltonian is essentially the helicity

operator times the energy

H = vFp · σ = sgn(vF )|E|χ.

This leads to a picture where a Weyl node looks like a hedgehog or a magnetic monopole

in momentm space, with the spin vectors oriented toward the Weyl node or in the opposite

direction (Fig. 3).

FIG. 3: Weyl nodes of opposite helicity. The arrows indicate the direction of the spin vector σ,

which can be parallel or antiparallel to the momentum vector p.

Recently the protection of gapless modes in solid state physics became an important

– 7 –

question. In this perspective it is suitable to introduce Weyl semimetals in the context of

topological phases.

Topological phases of matter were discovered in ’80s with the discovery of the Quantum

Hall Insulators. Contrary to the usual phases of matter topological phases are characterized

by the topological order, which does not follow the Landau symmetry-breaking theory. It

turned out that physical systems such as spin liquids and systems with quantum Hall effect

may exhibit topological order.

An example of topological phases are topological insulators. Contrary to normal insu-

lators, topological insulators are conductive at the surface of the material. Using band

structure terminology this means that there is a gap inside the material and zero-energy,

i.e. gapless, modes on its surface. The existence of such surface states is determined by

the nontrivial topology of the bulk electronic wave functions. The surface states are quite

robust to small disorder provided particular symmetries are preserved. A natural question

FIG. 4: Linear dispersive energy spectrum at the Weyl node (Left) and Weyl node in the Brillouin

zone (Right).

to ask within this framework is whether there are topological gapless phases with gapless

edge states. Weyl semimetals are believed to be a three dimensional realization of such a

phase. They are characterized by the presence of a set of linear-dispersive band-touching

points, called Weyl nodes (Fig. 4). At this points the Hamiltonian expanded in series in

momentum coordinates has the form of the Weyl Hamiltonian or an equivalent one. And

hence the excitations at the Weyl nodes are given by the Weyl fermions. Contrary to topo-

logical insulators, the topological protection of Weyl semimetals is provided not by the bulk

band gap, but by the separation of gapless 3D Weyl nodes in momentum space. Also the

Fermi surface of a surface state of a Weyl semimetal represents an open line called Fermi

– 8 –

Arc, which connects surface projection of Weyl nodes. This is different from Hall insulators

which have a line along the surface of the Brillouin zone (Fig. 5).

NI Weyl Semimetal HI

FIG. 5: Weyl semimetals can be regarded as phases between normal insulators (NI) and Hall

Insulators (HI)

Weyl fermions in 3D systems can be found in band touching points of the valence and

conduction bands with linear dispersion relation. Existence of such points in substances was

suggested in 1970 [4]. Generically the possible low-energy physics at a Weyl node in real

materials can be described by the Hamiltonian

H = (v0 · p)1 +3∑j=1

(vj · p)σj,

where vi are velocity vectors. The chirality of the Weyl-like excitation is determined by

sgn(v1 ·v2×v3). Weyl nodes are predicted to arise in nonstoichiometric silver chalcogenides

(Ag2+δSe and Ag2+δTe) [5], pyrochlore Iridates [6] A2Ir2O7 (A stands for Yttrium or a Lan-

thanide element), heterostructures of normal and topological insulators [8] and in HgCr2Se4

ferromagnetic compounds [9].

It was mentioned that Weyl fermionic systems are robust to small perturbations, but

how much protected they are? All real materials contain some impurities, so how does it

influence on Weyl semimetallic properties? Is there a way to open a gap in Weyl semimetals

and can we induce Weyl nodes in gapped materials? In Section II we address the question

of how robust are Weyl nodes in presence of an external perturbation. We study this on a

simple lattice model of a Weyl semimetal realizing two Weyl nodes. For a perturbation we

apply an external periodic potential. We investigate the model numerically and analytically

using perturbation theory in regime of weak potential. A phase transition from a Weyl

semimetallic phase to an insulating phase is observed. This is discussed in details and a

prescription on how to open a gap in the model is suggested.

In Section III the possibility of identifying Weyl fermions in magnetic neutron scattering

– 9 –

at zero temperature is investigated. Motivated by constant helicity of Weyl fermions and

the idea of regarding Weyl nodes as magnetic monopoles in momentum space, we compute

the differential cross-section for the semimetallic phase (Fermi energy at the Weyl point).

The current work is concluded with a discussion of results and an outlook for further

research is given in the last section.

– 10 –

II. WEYL SEMIMETALS IN PERIODIC POTENTIAL

In the introduction it was pointed out that Weyl semimetals might be robust to small

disorder/perturbation due to the fact that one cannot generate a mass term for a single

Weyl state. This leads to the idea that the annihilation of Weyl states of opposite chirality

is the only way to gap out a Weyl semimetallic phase. In this section we consider a simple

lattice model realizing two Weyl nodes of opposite chirality. To test the protection we add

an external periodic in position space potential. The potential makes fermionic states from

different Weyl nodes “interact” and “annihilate”. The question of whether a gap can be

opened in such a system is addressed and a prescription of how to open a gap is suggested.

In regime of weak potential we compute the energy spectrum and find the condition of

opening a gap. Then we show numerical results of direct diagonalization of the Hamiltonian

and compare them with the results obtained by perturbation theory. We conclude the section

with some discussions.

A. Lattice model

We use the following half-filled 3D two-band Hamiltonian in order to model Weyl semimet-

als [10]

H0 = δk,k′ [2tx(cos kx − cos k0)σx +m(2− cos ky − cos kz)σx

+ 2ty sin kyσy + 2tz sin kzσz] (1)

where m is some adjustable real parameter, tx, ty and tz are the nearest-neighbour hopping

parameters, kx, ky and kz are the momentum components, and σx, σy and σz are the Pauli

matrices.

For m > 2tx(cos k0 + 1) one has two Weyl nodes in the Brillouin Zone at k =

(±k0, 0, 0) (see Fig.6 and the appendix A).

At zero temperature in a half filled system the fermions occupy all negative energy bands,

the Fermi energy is zero and crosses the Weyl nodes. The low energy physics is determined

by the excitations around the Weyl nodes k = (±k0, 0, 0) +p, where |p| k0. One expands

the Hamiltonian (1) in p to get the Hamiltonian HW± describing low energy modes around

– 11 –

the Weyl nodes (±k0, 0) correspondingly

HW± = ∓2tx sin k0pxσx + 2typyσy + 2tzpzσz.

The Hamiltonians HW± describe two Weyl fermions of opposite chirality.

FIG. 6: Two Weyl nodes. Here ky = kz = 0, k0 = π/4, tx = ty = tz = 1 and m = 3

It is considered that small perturbations do not to destroy Weyl semimetal phases. How-

ever real material always have some impurities and one thinks whether it effects the proper-

ties of Weyl semimetals. In order to drastically change this properties one has to change the

topology of the bulk. This is possible by opening a gap, which can be done by creating some

interaction between the Weyl fermions of opposite chirality in order to annihilate them.

We consider an external periodic potential V (x) to generate the interaction between the

fermionic states at the Weyl nodes

H = H0 + hV, V (x) = cos(ωx),

where h is the strength of the potential and ω is its wavenumber.

The external potential V breaks the periodicity along x-direction of the total Hamiltonian.

Nevertheless, if ω/2π = p/q is a rational number the periodicity is recovered and one can

introduce momentum component in x direction. In incommensurate case, i.e. when ω/2π

is irrational number, one cannot define a momentum quantum number in x-direction. Also,

at strong enough potential comparable to 2tx one may expect fractal-like behaviour in the

energy spectrum dependence of the wavenumber ω. This can be heuristically argued by

considering the problem on a one dimensional slice at ky = kz = 0. The Hamiltonian then

– 12 –

can be diagonalized in spin space separately from the position coordinate:

H±(x′, x) = (h cos(ωx)∓ 2tx cos k0) δx′,x ± tx (δx′,x−1 + δx′,x+1) .

The Hamiltonians H±are the well-known Harper Hamiltonians [13] or almost Mathieu oper-

ators. Its spectrum exhibits fractal structure at h = 2tx resembling two overlaped Hofstadter

butterflies shifted each by 2tx cos k0 along the energy axes in opposite directions (Fig. 7).

One however cannot completely trust this picture since for a full understanding one has to

FIG. 7: The two overlapped Hofstadter butterflies in ky = kz = 0 plane at h = 2. Here tx = ty =

tz = 1 and m = 3.

consider the all three directions, which is complicated for numerical study in three dimen-

sions.

We, however, will focus on the regime of weak external potential.

FIG. 8: Interaction of Weyl nodes through

n-order scattering.

In this regime one thinks of the potential V as of

a way to induce momentum exchange k → k ± ω

between the fermionic states. Technically this is

reflected in the presence of Kronecker delta terms

in the interaction potential in momentum space

Vk′k =1

2(δk′,k−ω + δk′,k+ω) .

We treat the above potential perturbatively. Hence

one thinks of having a scattering problem between

the two Weyl nodes. If the distance between the Weyl nodes matches an integer number

– 13 –

of the potential wavenumbers, i.e. 2k0 = n ω with n = 1, 2, 3, ..., then the Weyl nodes can

”communicate” with each other by n-order scattering processes (Fig. 8).

B. Regime of weak potential

Each scattering process is of order of h. Thus for small h the main contribution to

the ”interaction” between the Weyl states comes from the single and double scattering.

FIG. 9: Interaction of Weyl nodes through

double-scattering

It turns out the first order scattering, i.e. when

ω = 2k0, only changes the positions of the Weyl

nodes. Double scattering process, however, can

open a gap. In regime when ω is close to k0 up

to an order of h2, i.e. ω − k0 ∼ h2 1, one

distinguishes few phases.

We use Brillouin-Wigner perturbation the-

ory [11, 12] to investigate second order corrections

to the energy bands of our model of Weyl semimet-

als. This method avoids the need for special treatment of degeneracies. First, we construct

an effective Hamiltonian by projection on the states |k〉 and |k + 2ω〉:

Heff(E) = PH0P +h

2PV P +

h2

4PV Q [E −H0]−1QV P + ..,

where P is the projection operator on the states |k〉 and |k + 2ω〉, and Q = 1 − P is the

projection operator on the rest states.

Then

PH0P =

< k|H0|k > < k|H0|k + 2ω >

< k + 2ω|H0|k > < k + 2ω|H0|k + 2ω >

=

H0(k) 0

0 H0(k + 2ω)

,

PV P =

< k|V |k > < k|V |k + 2ω >

< k + 2ω|V |k > < k + 2ω|V |k + 2ω >

= 0,

(QV P )σ,sq,k = δσ,s

< q 6= k, k + 2ω|V |k >

< q 6= k, k + 2ω|V |k + 2ω >

= δσ,s

Vq,k

Vq,k+2ω

,

– 14 –

(PV Q)s,σk,q = δs,σ(Vk,q Vk+2ω,q

),

and the inverse operator is

[E −H0]−1 = [E −H0 · σ]−1 =E + H0 · σE2 −H0

2 =E +H0

E2 −H20

.

The second order correction to the unperturbed part of the effective Hamiltonian is given

by

(PV Q [E −H0]−1QV P )s,s′

k

=∑

q 6=k,k+2ω

(δk−ω,q + δk+ω,q) (δq,k−ω + δq,k+ω) (δk−ω,q + δk+ω,q) (δq,k+ω + δq,k+3ω)

(δk+ω,q + δk+3ω,q) (δq,k−ω + δq,k+ω) (δk+ω,q + δk+3ω,q) (δq,k+ω + δq,k+3ω)

×[E −H0]−1s,s

′

q =

[E −H0]−1s,s′

k−ω + [E −H0]−1s,s′

k+ω [E −H0]−1s,s′

k+ω

[E −H0]−1s,s′

k+ω [E −H0]−1s,s′

k+ω + [E −H0]−1s,s′

k+3ω

.

In order to find out whether a gap opens or not one investigates the energy spectrum of

the effective Hamiltonian Heff . Unfortunately analytical diagonalization of Heff is difficult

not only to compute, but also to investigate. Instead, we take use of the small parameter h

and assuming the energy bands of Heff are smooth functions of the momentum k and the

wavenumber ω, we look at the states infinitesimally close to the Weyl nodes. Technically

one expands around the Weyl points in the Brillouin zone: kx = −k0 + px, ky = py , kz = pz

and w = (1 + δa)k0, where px, py, pz and δak0 are small, of same order as h2. To further

simplify the calculations we make cyclic rotation in Pauli space: σ ≡ σx, σy, σz → σ ≡

σz, σx, σy. This can be accomplished by unitary transformation in spin space which does

not depend on momenta, and thus does not mix the states. The momenta py and pz are then

naturally combined in one complex variable typy + itzpz = ueiγ, where u =√t2yp

2y + t2zp

2z and

γ = arg(tzpz + itypy).

Then up to second order in h2 one has

Heff = H(0)eff +

h2

4H

(2)eff ,

where

– 15 –

H(0)eff = 2

pxtx sin k0 ueiγ 0 0

ue−iγ −pxtx sin k0 0 0

0 0 −tx sin k0 (2δak0 + px) ueiγ

0 0 ue−iγ tx sin k0 (2δak0 + px)

,

H(2)eff =

1

2tx

− 2 cos k0

cos k0−cos 2k00 − 1

1−cos k00

0 2 cos k0cos k0−cos 2k0

0 11−cos k0

− 11−cos k0

0 − 2 cos k0cos k0−cos 2k0

0

0 11−cos k0

0 2 cos k0cos k0−cos 2k0

.

It turns out that TrH3eff = TrHeff = 0. This implies that there are only two linearly

independent eigenvalues, which can be computed using TrH2eff and TrH4

eff :

E(1,±) = ±1

2

√TrH2

eff −√

4TrH4eff − (TrH2

eff )2,

E(2,±) = ±1

2

√TrH2

eff +√

4TrH4eff − (TrH2

eff )2.

To find possible zero energy points it is useful to consider the determinant of the effective

Hamiltonian, which is the product of the eigenenergies

det Heff = E(1,−)E(1,+)E(2,−)E(2,+) =1

8

(2TrH4

eff − (TrH2eff )

2).

At points where the gap is closed, like at Weyl nodes, the determinant vanishes. Hence a gap

is opened when the determinant is non-zero everywhere in px, py and pz. One notices that the

determinant det Heff is a polynomial in px, py, pz and δa. Hence its zeroes are determined by

the coefficients of these polynomials. We check all the possibilities of existence of zero-energy

points by making some clever steps.

So we are looking for the points px and u at which det Heff vanishes. But the det Heff is

a fourth order polynomial in px and u. Its analytical evaluation on zeroes is complicate and

messy. One however notices that the determinant turns out to be second order polynomial

– 16 –

in δa

det Heff = c2δa2 + c1δa+ c0,

where the coefficients c0, c1 and c2 are themselves polynomials of px and u. The determinant

of the Hamiltonian vanishes at zero energy points. Unless c2 = 0 the existence of such points

is assured by c21 − 4c2c1 ≥ 0.

Next steps show that u has to vanish in order to have zero energy band touching points.

The coefficient c2 vanishes at k0 = nπ (these are special points where the two Weyl points

are initially merged and this is not the case we consider) or at

u2 = −(h2 cos k0 − 4px(sin k0 + sin 2k0 − sin 3k0)t2x)2

64(cos k0 − cos 2k0)2t2x,

which implies that u = 0 and px =h2 cos k0

4(sin k0 + sin 2k0 − sin 3k0)t2x. At this point det Heff =

− h8

216 sin8 k02t4x6= 0. Hence there are no zero-energy points where c2 vanishes.

On the other hand

c21 − 4c2c1 = −

u2k20 cot2 k0

2

64t2x sin4 k02

(1 + 2 cos k0)4F2(px, u) ≤ 0, (2)

where F(px, u) is a second order polynomial in px

F(px, u) = d2p2x + d1px + d0,

with the coefficients

d2 = 212 sin6 k0

2

(2 cos

k0

2+ cos

3k0

2

)2

t4x,

d1 = −24h2(sin k0 + sin 3k0 − sin 4k0)t2x,

d0 = h4(5 + 4 cos k0 + 4 cos 2k0) + 210u2 sin4 k0

2(1 + 2 cos k0)2t2x.

Therefore the condition of existence of the zero energy points reduces to

F(px, u) = 0. (3)

– 17 –

To satisfy the above equality on can first look at cases when u = 0 or d2 = 0 and hence

k0 ∈

0,±2π3,±π,±4π

3

.

At k0 ∈±2π

3,±4π

3

one has d2 = d1 = 0 and d0 = h2 6= 0. Similarly at k0 = ±π the

coefficients d2 = d1 = 0 and d0 = 5h4 + 1024u2t2x, which does not have real valued solution

for u. Thus there are no zero-energy points at k0 ∈±2π

3,±π,±4π

3

where d2 = 0.

The only condition left that guarantees the existence of real solution of equation (3) is

d21 − 4d2d0 ≥ 0. But

d21 − 4d2d0 = −214t4x cos2 k0

2(1 + 2 cos k0)4 sin6 k0

2

(h4 + 210u2t2x sin4 k0

2

)is non-negative. This implies that d2

1 − 4d2d0 has to vanish, which holds for k0 = ±2π/3

(was discussed earlier; does not bring zero-energy solutions) for h2 6= 0.

The only option left to satisfy (2) is to set u =√

t2yk2y + t2zk

2z = 0, which implies that

δa =h4(1 + 4 cos k0) + 210p2

xt4x(1 + 2 cos k0)2 sin2 k0 sin4 k0

2

27k0t2x sin k0 sin2 k02

(1 + 2 cos k0)(h2 cos k0 − 24pxt2x sin k0 sin2 k0

2(1 + 2 cos k0)

) . (4)

This is the relation between δa, px, k0 and h2 for which the energy bands get zero values

(the gap might close).

Our aim now is to find values px for which det Heff vanishes. For that purpose we regard

the equation (4) as equation for px:

b2p2x + b1px + b0 = 0,

where the corresponding coefficients are

b2 = 210t4x sin2 k0 sin4 k0

2(1 + 2 cos k0)2,

b1 = 211δak0t4x sin2 k0 sin4 k0

2(1 + 2 cos k0)2,

b0 = h4(1 + 4 cos k0)− 27h2δak0t2x sin k0 sin2 k0

2(1 + cos k0 + cos 2k0).

– 18 –

If the gap closes there must exist px for which it happens. This requires b21 − 4b2b0 ≥ 0, i.e.

b21 − 4b2b0 = 212t4x sin2 k0 sin4 k0

2(1 + 2 cos k0)2

(−h2 + δak0 25t2x sin k0 sin2 k0

2(1 + 2 cos k0)

)(h2(1 + 4 cos k0) + δak0 25t2x sin k0 sin2 k0

2(1 + 2 cos k0)

)≥ 0.

If the above condition is satisfied then the energy bands vanish at

px,± = −δak0 ±

√(δak0 −

h2

C(k0)

)(δak0 +

h2(1 + 4 cos k0)

C(k0)

), (5)

where C(k0) = 32t2x sin k0 sin2 k02

(1 + 2 cos k0).

When the wavenumber ω matches the half of distance between Weyl points, i.e. ω = k0

and δa = 0, the energy vanishes at

px = ± h2√−1− 4 cos k0

32√t4x sin2 k0 sin4 k0

2(1 + 2 cos k0)2

,

which is real only for cos k0 ≤ −1/4. The point k0 = arccos(−1/4) is thus a transition point

were the gap might open/close (Fig. 10).

0.6 0.7 0.8 0.9 1.0k0

-1.0

-0.5

0.5

1.0

px @h2D

FIG. 10: Values of px, starting from the threshold value of arccos(−1/4), for which the gap might

close when ω = k0. Here tx = 1.

– 19 –

The effective Hamiltonian Heff has four eigenvalues

E(1,±) = ±√u2 + 4t2x

(p2x sin2 k0 + 2A

)− 8√t2xA

2,

E(2,±) = ±√u2 + 4t2x

(p2x sin2 k0 + 2A

)+ 8√t2xA

2,

where A = δak0 (sin k0 + px cos k0) (px sin k0 + δak0 (sin k0 + px cos k0)).

C. Phase diagram and discussions

In regime of weak periodic potential the system undergoes a topological phase transition,

leading a Weyl semimetal phase to a insulator phase (see Fig. 11). The insulator phase

(Phase I or white region on Fig. 11) is characterized by presence of a gap (Fig. 12). In the

FIG. 11: Topological phase diagram. Phase I (white color) corresponds to insulator phase (a

gap is opened). Phase II (orange) is a Weyl semimetal phase (the dispersion relation where the

energy bands touch zero is locally linear). Red line is a transition line between insulator and Weyl

semimetal phases. In this regime the gap closes at points with quadratic dispersion in px-direction

and linear in two other directions. Phase II? is a regime in which δak0 ≥ h2 and we don’t have

faith in the perturbation method we used. There is also no trust in the grey dark line where both

momenta deviations from both Weyl points px,± exceed h2. Here tx is set to 1.

Weyl semimetal phase (Phase II or orange region on Fig. 11) the energy bands touch zero

in two Weyl nodes (the dispersion relation is locally linear) (see Fig. 14). There is also a

transition regime (red line on Fig. 11) in which the gap closes at (−(1 + δa)k0, 0, 0) and

((1 + δa)k0 + px, 0, 0). However these are not Weyl points since the local dispersion relation

– 20 –

is locally quadratic in px-direction, but linear in the two other directions. (Fig. 13).

-3 -2 -1 0 1 2 3

-400

-200

0

200

400

px@h2D

E@h

2D

-1.0 -0.5 0.0 0.5 1.0

-40

-20

0

20

40

px@h2D

E@h

2D

FIG. 12: Energy spectrum in insulator phase (white region on Fig. 11) vs momenta deviation

px = kx − k0 for k0 = 0.1π, δa = 0.1 (Left) and for k0 = 0.95π and δa = −0.15 (Right). A gap is

open and the minima of positive energy bands are locally stable. Here tx is set to 1.

-1.0 -0.5 0.0 0.5 1.0

-200

-100

0

100

200

px@h2D

E@h

2D

-1.0 -0.5 0.0 0.5 1.0

-200

-100

0

100

200

px@h2D

E@h

2D

FIG. 13: Energy spectrum on transition line (red line on Fig. 11) vs momenta deviation px = kx−k0

for k0 = arccos(−1/4), δa = 0 (Left) and for k0 = 0.3π and δa = 0.0914018 (Right). The gap

is closed but the dispersion is locally quadratic in px. The minima of positive energy bands are

locally stable. This region is a transition regime from a gap phase to a gapless phase. Here tx is

set to 1.

One can also investigate the problem numerically diagonalizing the Hamiltonian and

analyzing its spectrum. On Fig. 15 the energy spectrum for k0 = 0.1π computed by nu-

merical means in the ky = kz = 0 plane is shown. The ”gap” seen in the neighbourhood

of ω = 2k0 = 0.2π actually closes in some other plane ky, kz 6= 0. In this regime the Weyl

fermions interact mainly through single-scattering processes, which only change the position

of the Weyl nodes. The gap opened in neighbourhood of ω = k0 = 0.1π is due to double-

scattering processes between the Weyl fermions of opposite chirality. It is in accordance

with perturbation theory (Fig. 11 and Fig. 12).

– 21 –

-1.0 -0.5 0.0 0.5 1.0-300

-200

-100

0

100

200

300

px@h2D

E1

@h2

D

-1.0 -0.5 0.0 0.5 1.0

-200

-100

0

100

200

u@h2D

E1

@h2

D

FIG. 14: Energy spectrum in the gapless phase (region II or orange region on Fig. 11). Left: Energy

spectrum vs momenta deviation in x-direction px = kx − k0 (Here ky = kz = 0). Energy bands

touch at two point with locally linear dispersion relation. Right: Energy spectrum vs momenta

deviation in Oy and Oz directions represented by u =√t2yk

2y + t2zk

2z . The dispersion relation at

zero-energy band touching points is also linear. This implies that the zero-energy band-touching

points are actually Weyl nodes. Here tx is set to 1, k0 is 0.4π and δa = 0.2.

FIG. 15: Energy spectrum for h = 0.1 and k0 = 0.1π. Left: energy spectrum sliced at ky = kz = 0.

The gap opens when ω is close to k0 = 0.1π (Phase I). The gap seen at 0.2π is not real: the Weyl

nodes moved to other slice ky 6= 0 and kz 6= 0 due to single scattering process. Right: the energy

vs ky at ω = k0 and kz = 0. The gap of order h2 is present (the zoomed picture at the upper

corner).

– 22 –

III. NEUTRON SCATTERING ON A WEYL FERMION

Many features of the molecular and atomic structure can be captured using scattering

processes between particles. In solid-state physics a quite powerful experimental technique

used for measuring crystal and magnetic structure is neutron scattering. Neutrons are

neutral spin 1/2 particles. Due to the absence of electrical charge neutrons are able to

penetrate deeply into solids. Despite being neutral neutrons carry magnetic moment µ.

The vector potential at some position r produced by the neutrons magnetic moment µ at

rn in SI units is

Ar−rn =µ0

4π

µ× (r− rn)

|r− rn|3(6)

This makes possible to use neutrons for testing the electron spin-orbital and magnetic inter-

actions in many materials. For instance paramagnetic phases can efficiently be distinguished

in magnetic neutron scattering with polarized beams by measuring the spin-flip cross-section

with the polarization first parallel and then perpendicular to the momentum transfer [15].

In solid-state physics physicists usually use thermal neutrons to run their experiments on

neutron scattering. This is mostly due to the fact that thermal neutrons have energy (around

0.025eV) and wavelength (λ ≈ 1.6A) comparable to the ones of many excitations in solids.

Schematically a magnetic neutron scattering experiment can be pictured as follows. An

incident polarized neutron beam interacts with a sample and the neutrons scattered out

from the sample are capture by a detector. In magnetic neutron scattering besides neutrons

energy, momentum and the scattering angle, the polarization of the scattered neutrons can

be measured (Fig. 17).

We consider the inelastic scattering process of polarized neutrons on a Weyl fermion.

A. Differential cross-section in Born approximation

Consider a neutron with momentum ki, spin vector si and energy Ein scattering on a

Weyl fermion with the initial momentum pi and energy EiW . The interaction Hint between

the neutron and Weyl fermion changes the Weyl fermions momentum to pf and energy to

– 23 –

FIG. 16: Schematic representation of a neutron scattering process.

EfW , and the scattered neutron to a state with momentum pf , spin vector sf and energy

Efn . The differential cross-section of this scattering process in Born approximation is given

by [14]

d2σ

dΩdEfn

=kf

ki

( mn

2π~2

)2 ∑pi,pf

PiW∣∣∣⟨kf , sf ; pf , E

fW+|Hint|pi, E

iW− ; ki, si

⟩∣∣∣2 δ (Efn + Ef

W+− Ei

n − EiW−

)

where ki =√

k2i and kf =

√k2

f , mn is the neutrons mass and PiW is the probability distri-

bution of the initial Weyl fermionic satates.

In order to find the interaction Hamiltonian of a Weyl fermion with the neutrons magnetic

moment we first recall the Hamiltonian of a free Weyl fermion

H0(r) = −ivF~∇r · σ, (7)

where vF is the effective velocity of the Weyl fermion and σ ≡ (σx, σy, σz) are the Pauli

matrices.

The free Weyl Hamiltonian (7) has two eigenstates ΨW− and ΨW+ corresponding to

negative and positive energy excitations E± = ±vF |p|

ΨW+(p) =expip · r√

VWuw+(p), ΨW−(p) =

expip · r√VW

uw−(p), (8)

where VW is the normalization coefficient and the spinors uw+(p) and uw−(p) depend only

– 24 –

on the momentum orientation p = (sin θ cosφ, sin θ sinφ, cos θ):

uw+(p) =

cos θ/2

eiφ sin θ/2

, uw−(p) =

e−iφ sin θ/2

− cos θ/2

.

In presence of an external vector potential Ar−rn created by a neutron at rn moving

towards an electron at r one has to make a substitution

∇r → ∇r − ie

~Ar−rn , (9)

where e is the electron charge.

After substituting (9) into (7) one identifies the interaction Hamiltonian

Hint(r; rn) = H(r; rn)−H0(r) = −vF e (Ar−rn · σ) . (10)

Before and after the interaction neutrons are free non-interacting particles and its wave

functions are just plane waves

ψin =1√Vneikir, ψfn =

1√Vneikfr,

where Vn is the normalization coefficient.

The initial and final Weyl states ΨiW and Ψf

W are eigenstates of the free Weyl Hamiltonian

(7)

ΨiW =

1√VW

eikiruiw, ΨfW =

1√VW

eikfrufw.

Making use of (6) one integrates over the neutrons and electronic position to get the

– 25 –

scattering amplitude

⟨ΨfW ;ψfn |Hint|ψin; Ψi

W

⟩= VnVW

∫d3rnd

3r(uf∗w ψ

f∗n Hintψ

inu

iw

)= −evFµ0

4π

∫d3rnd

3r e−i(kf−ki)rne−i(pf−pi)r

[µfi × (r− rn)

|r− rn|3

]·(uf∗w σu

iw

)= ievFµ0

([µfi × q

q2

]·(uf∗w σu

iw

))∫d3 rei(pi+q−pf)r

= (2π)3ievFµ0

([µfi × q

q2

]·(uf∗w σu

iw

))δ (pi + q− pf) , (11)

where µfi = 〈sf | µ |si〉 and q = ki − kf is the transfered momentum.

At zero temperature all negative-energy Weyl states are occupied and all positive-energy

states are empty. Hence, due to the Pauli principle the available negative-energy Weyl states

can scatter only to positive-energy Weyl states, which are empty, i.e. only the scattering

process∣∣ψin; Ψi

W−⟩→∣∣∣ψfn; Ψf

W+

⟩is allowed (Fig. 17).

FIG. 17: Schematic illustration of a neutron scattering process on a Weyl node at zero temperature.

One can expect some restrictions on possible values of incident

neutron momenta due to linear dispersion of the Weyl fermions.

FIG. 18: Schematic illustration of

momentum conservation.

This would then allow one to test the linearity of the dis-

persion. Let us look at the possible restrictions that might

come from the momentum and energy conservation. Due

to momentum conservation pf = pi + q vectors pf , pi and

q form a triangle as shown in Fig. 18. According to the

triangle inequality law the length of any side of a triangle

cannot be greater then the sum of the lengths of the other

– 26 –

two sides. Hence one has q ≤ pi + pf . On the other hand

the energy conservation reads

~vFpf + ~vFpi ≡ EfW+− Ei

W− = ∆E ≡ ~2k2i

2mn

− ~2k2f

2mn

.

Consequently one has

∆E

~vF≥ q. (12)

In terms of the incident and scattered neutron momenta ki and kf one has q =√

(ki − kf)2 ≥

ki−kf , where we took into account that ki ≥ kf , i.e. the neutrons lose their energy in order to

excite Weyl fermionic states from the Fermi sea. The inequality (12) gives then a threshold

value of the incident neutron momenta kc at which neutrons start to scatter on Weyl fermions

kf ≥ kc ≡mnvF~

. (13)

The lowest bound ki = kc is reached when kf = ki.

The scattering amplitude (according to (11)) and the differential cross-section for a∣∣ψin; ΨiW−⟩→∣∣∣ψfn; Ψf

W+

⟩scattering process are

⟨ΨfW+;ψfn

∣∣∣Hint

∣∣ψin; ΨiW−⟩

= (2π)3ievFµ0

([µfi × q

q2

]·(uf∗w+

σuiw−))

δ (pi + q− pf) ,

d2σ

dΩdEfn

=kf

ki

(2emnvF

2π~2

)2+∞∫0

dpi

(2π)3p2

i

∫ π

0

dθi sin θi

∫ 2π

0

dφi

∫d3pf

(2π)3|(2π)3δ(pf − pi − q)|2

×∣∣∣∣[µfi × q

q2

]·(uf∗w+

σuiw−)∣∣∣∣2 δ(~2k2

f

2mn

+ vF~pf −~2k2

i

2mn

+ vF~pi

).

It is convenient to introduce spherical coordinates and fix the coordinate system in such

a way that the axis Oz is along the momentum transfer q. In this coordinate system q =

(0, 0, q), pi = pi(sin θi cosφi, sin θi sinφi, cos θi) and pf = pf(sin θf cosφf , sin θf sinφf , cos θf ).

One then notices that since q = pf − pi = (0, 0, q) the polar angle of the Weyl fermion

doesn’t change, i.e. φf = φi.

– 27 –

It is straightforward then to compute the spin and magnetic moment contribution to the

scattering amplitude[µ× q

q2

]·(uf∗w+

σuiw−)

= − iq

(µ+ cos

θf2

cosθi2

+ µ−e−2iφi sin

θf2

sinθi2

), (14)

where we used the relation between neutrons magnetic moment and its spin

µ = −1.91e~

2mn

σ

and have introduced the spin-flip operators µ± = µ1 ∓ iµ2 = −1.91e

mn

(s1 ∓ is2) =

−1.91e~

2mn

σ±.

To find the contribution to the scattering cross-section we first integrate over the polar

angle φi. The only term in the differential cross-section depending on φ is

∫ 2π

0

dφi

∣∣∣∣[µ× q

q2

]·(uf∗w+

σuiw−)∣∣∣∣2 =

2π

q2

(|µ+|2 cos2 θf

2cos2 θi

2+ |µ−|2 sin2 θf

2sin2 θi

2

).(15)

Now it is convenient to introduce new variable η = cos θi. Then∫ π

0dθi sin θi =

∫ 1

−1dη and

cos θf =pzfpf

=pzi + qz√(pi + q)

=ηpi + q√

p2i + 2ηqpi + q2

.

The term (15) depends on η and pi as

∫ 2π

0

dφi

∣∣∣∣[µ× q

q2

]·(uf∗w+

σuiw−)∣∣∣∣2 =

2π

4q2

(|µ+|2 + (|µ−|2

)(1 +

η2pi + ηq√p2

i + 2ηqpi + q2

)

+2π

4q2

(|µ+|2 − (|µ−|2

)(η +

ηpi + q√p2

i + 2ηqpi + q2

)

Making use of the conservation of energy

δ (~vFpf + ~vFpi −∆E) =1

~vF∆E2/~2v2

F + 2ηq∆E/~vF + q2

2 (∆E/~vF + ηq)2 δ

(pi −

∆E2/~2v2F − q2

2(∆E/~vF + ηq)

),

where ∆E = ~2 (k2i − k2

f ) /2mn > 0 is the energy transfer, we integrate over pi and we are

– 28 –

left with the integral over η

d2σ

dΩdEfn

=kf

ki

(eµ0mnvF~2

)2 δ3(0)

4π

1

~vF

I+

(|µfi+ |2 + |µfi− |2

)+ I−

(|µfi+ |2 − |µ

fi− |2)

,

where

I+ =1

q2

(∆E2

~2v2F

− q2

)21∫

−1

dη(1 + η2) (∆E2/~2v2

F + q2) + 4ηq∆E/~vF(∆E/~vF + ηq)4 =

8

3

(∆E2

~2v2F q

2− 1

),

I− =1

q2

(∆E2

~2v2F

− q2

)21∫

−1

dη2 (η∆E/~vF + q)

(∆E/~vF + ηq)3 = 0.

Finally the differential cross-section for the magnetic neutron scattering on a Weyl fermion

is

d2σ

dΩdE=

kf

ki

(eµ0mnvF~2

)2 δ3(0)

4π

8/3

~vF q2

(∆E2

~2v2F

− q2

)(|µfi+ |2 + |µfi− |2

)= α2

EM

v2F

c2

kf

ki

8πV

3(−1.913)2 1

~vF

(∆E2

~2v2F q

2− 1

)(|σfi+ |2 + |σfi− |2

), (16)

where V is the volume of the sample and we have introduce the fine structure constant

αEM =e2µ0c

4π~≈ 1

137.

The momentum transfer q can be expressed in terms of neutron momenta ki and kf , and

in terms of scattering angle θ = (ki · kf)/(kikf):

q =√k2

i − 2kikf cos θ + k2f .

Then the differential cross-section is

d2σ

dΩdE= α2

EM

v2F

c2

8π (−1.913)2

3

δ3(0)

~vFγ

((~ki

2mnvF

)2(1− γ2)2

1 + γ2 − 2γ cos θ− 1

)×(|σfi+ |2 + |σfi− |2

), (17)

where the dimensionless quantity γ =kf

ki

< 1 has been introduced.

– 29 –

B. Polarization and scattering angle

Since the polarization factorizes from the angular dependence of the differential cross-

section formula (17), qualitatively the angular behaviour of the cross-section does not depend

on the polarization of the neutron beam. The cross-section has a maximum at θ = 0 and

monotonically decreases. However the cross-section amplitude depends on the polarization of

the initial and final neutron states. One identifies two important cases: when the polarization

vector P is parallel to the momentum transfer (P‖q) and when the polarization vector is

situated on the perpendicular plane to the momentum transfer (P⊥q).

Polarization along momentum transfer (P‖q)

Suppose we prepare the incident neutron beam polarized in Oz direction, i.e. in direction

of the momentum transfer q. Then the neutrons magnetic state is an eigenvector of the σz

matrix

σzu± = ±u±,

where

u+ ≡ |↑〉 =1√2

1

0

, u− ≡ |↓〉 =1√2

0

1

.

0 Π4 Π2 3Π4 Π

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Θ

d2Σ

dWdE

P°q

Spin-Flip

non-Spin-Flip

FIG. 19: Angular dependence of the differential cross-section for polarization P being along transfer

momentum q. Here ki = 4kf = 3kc. The diff cross-section is in units of α2EM

(vFc

)2 8π3 (−1.913)2 V k

2c

~vF .

– 30 –

In this basis the operators σ± flip the neutrons spin

σ±u± = u∓.

Thus when the neutrons beam is polarized along the momentum transfer (P‖q) the dif-

ferential cross-section of non-spin-flip process is zero and only the spin-flip process gives

contribution (Fig. 19).

Polarization perpendicular to the momentum transfer (P⊥q)

In case of the orthogonal neutron polarization to the momentum transfer (P⊥q) one has

(σx cosα + σy sinα)u± = ±u±,

where α is the polar coordinate in the polarization plane and the eigenvectors u± are

uα+ ≡ |↑〉 =1√2

1

eiα

, uα− ≡ |↓〉 =1√2

1

−eiα

.

Then the action of σ± on these states is

σ+uα+ = σ−u

α− =

1√2

0

1

, σ−uα+ = −σ+u

α− =

eiα√2

1

0

.

Hence for a final neutron state uf = (uf1 , uf2) one has

|µfi+ |2 + |µfi− |2 ∼ |uf1 |2 + |uf2 |2.

This means that the spin-flip and the non-spin flip have the same probability and give same

contribution to the differential cross-section.

– 31 –

0 Π4 Π2 3Π4 Π

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Θ

d2Σ

dWdE

P¦q

Spin-Flip

non-Spin-Flip

FIG. 20: Angular dependence of the differential cross-section for polarization P being orthog-

onal to the transfer momentum q. Here ki = 4kf = 3kc. The diff cross-section is in units of

α2EM

(vFc

)2 8π3 (−1.913)2 V k

2c

~vF .

Maximal scattering angle

It has been already mentioned that due to linear dispersion of Weyl fermions one expects

a critical value of the incident neutrons momenta at which the neutrons start scattering.

However there is one more characteristic parameter of linearity that might be observed in

experiments. The neutrons scatter up to a certain angle θmax:

θmax =

arccos

(4 + 7k2

i /k2c − 2k4

i /k4c

3√

12− 3k2i /k

2c ki/kc

), kc ≤ ki ≤ 2kc

π , ki > 2kc

(18)

Qualitativly, one describes the scattering angle dependence as follows (see Fig. 21). The

neutrons start scattering on Weyl fermions once their momentum ki exceeds the value of kc.

The maximal scattering angle θmax of the outgoing neutrons increases monotonically with

the increase of ki till ki reaches 2kc when θmax reaches π/2. Once ki exceeds 2kc neutrons

start to scattering at any angle. The step-like increase in the maximal scattering angle from

π/2 to π at ki = 2kc can probably be observed in experiments.

– 32 –

0 kc 3 kc2 kc

0

Π4

Π2

3Π4

Π

0

Π4

Π2

3Π4

Π

ki

Θm

ax

FIG. 21: Maximal neutron scattering angle θmax in the semimetallic phase as a function of the

momenta of the incident neutrons ki. The neutrons start to scatter once their momenta reaches

the threshold value of kc = mnvF /~. It smoothly increases till reaches the value of π/2 at ki = 2kc,

where it becomes π after ki exceeds 2kc. The orange plots at 1.5kc, 2kc and 3kc illustrates the

angle dependance of the differential cross-section dσdΩ with the dσ

dΩ being along the horizontal axis

oriented to the left and the angle θ on the vertical axis. The neutrons scatter up to the maximal

angle θmax. The most likely scattering angle however is θ = 0.

0.117 Π Π2 3Π40.0

0.5

1.0

1.5

Θ

dΣ

dW

P°q

non-Spin-Flip

0.117 Π Π2 3Π40.0

0.5

1.0

1.5

Θ

dΣ

dW

P°q

ki=3kc

ki=2kc

ki=1.5kc

Spin-Flip :

Π0.117 Π Π2 3Π4

10-9

10-7

10-5

0.001

0.1

Θ

dΣ

dW

P°q

ki=3kc

ki=2kc

ki=1.5kc

Π

FIG. 22: Differential cross-section for the polarization vector along the momentum transfer at

different incident neutrons momenta ki. The plot on the right is in logarithmic scale. It is clearly

seen that the differential cross-section vanishes ’abruptly’ at certain angles for ki ≤ kc.

C. Conclusions and Discussions

Probably this computation will help with the experimental detection of Weyl semimetals

in magnetic neutron scattering. We found the neutrons prefer not to deviate from the

initial direction (the differential cross-section is peaked at small angles θ). Also, when the

polarization vector of the incident beam is parallel to the momentum transfer the neutrons

flip the spins after being scattered on Weyl fermions. However, when the polarization vector

is sitting in the orthogonal plane to the momentum transfer, the neutrons are equally likely

– 33 –

to flip and not flip its spins. This kind of angular dependence of the differential cross-section

reminds elastic magnetic neutron scattering on paramagnets. In that sense it doesn’t feature

anything specific for Weyl semimetals. Despite that, the presence of the critical value for

momenta of the incident neutrons and as well as of the “jump” in the neutrons maximal

scattering angles may be a sign of linear dispersion in neutron scattering experiments. Let

us thus check whether neutrons in a standart experiment would have enough energy to

scatter off the Weyl fermions. Usually neutron scattering experiments in solid-state physics

are carried out by thermal neutrons, which have an energy of about 0.025eV. The neutrons

with the critical momenta kc = mnvF/~ carry kinetic energy of Ec =~2k2

c

2mn

= mnv2F/2.

Hence one estimates the effective velocity for the electronic states that could be “seen” by

the thermal neutrons

vF =

√2Ec

mn

≈ 10−5c = 103m/s, (19)

where c is the speed of light and the neutron mass is 939MeV/c2.

This velocity is rather low. For instance the effective velocity of the electronic states in

graphene is around 106m/s or 10−2c. If we assume the velocity of the electronic states in

Weyl nodes is the same order as in graphene, then the neutrons critical energy capable to

“feel” Weyl fermions has to be of order of 105eV = 0.1MeV. That energy is outside the

energy range of thermal and corresponds to resonance close to intermediate neutron energy

range. This makes troublesome the experimental detection of the features of Weyl fermions

unveiled in the current section.

One would think trying to excite the Weyl fermionic states with neutral particles lighter

than the neutrons. From the set of neutral spin 1/2 particles lighter than the neutrons are

probably only neutrinos. This leaves us no much room for experimental verification of the

discussed features for Weyl fermions in real materials in the way I described here. But may

be at different way of scattering the Weyl fermions would work.

– 34 –

IV. CONCLUSIONS AND DISCUSSIONS

The aim of the current work is to provide a reasonable insight in the protection of Weyl

semimetallic phases from small perturbations on a simple example of a lattice model realizing

interaction between the Weyl states and to find some hint on possible experimental detection

of Weyl fermions in neutron magnetic scattering.

The question of robustness of the topological properties of Weyl semimetals still attracts

a lot of attention in the scientific community [16, 17]. For small perturbations the Weyl

semimetals have protected properties. But what happens to real materials, which have im-

purities? In the lattice model considered in the current work a gap opening in the Weyl

semimetallic phase was modeled by addressing the problem in an external periodic poten-

tial. It was shown that the Weyl phase is still robust to linear order in perturbation over the

potential. Computations in second order perturbation showed the possibility of opening a

gap in the system, and thus driving the system from a Weyl phase to a bulk gapped phase.

The problem was solved analytically in the regime of weak potential, and numerically by

direct diagonalization of the Hamiltonian. The numerical results are in agreement with the

analytical ones. The work concluded with a phase diagram. For the addressed problem a

prescription to open a gap in the bulk would be to match the distance between Weyl nodes

to be an integer multiple of the wavenumber of the external potential. In contradiction with

this half-intuitive statement it was shown that when the wavenumber ω matches the distance

between Weyl nodes the gap doesn’t open. The Weyl nodes just move around without dis-

appearing. However second order perturbation showed that when the wavenumber matches

half of the distance between Weyl nodes a gap opens up, and it does that even when the

wavenumber slightly deviates from the given prescription.

A next step in understanding the phase transition within this framework would be in-

vestigation of the surface states. This might bring some insight on the character of the

insulating phase obtained after the Weyl semimetallic phase was gapped out. The question

would be what kind of gapped phase is that, is it a topological insulating phase or a normal

insulating phase? What happened to the Weyl semimetallic surface states, the Fermi Arcs?

The author tried to address these questions using numerical direct diagonalization of the

– 35 –

model Hamiltonian but didn’t achieved enough accuracy to resolve the potential gapless

states. The issue was technical and related to the difficulties with the direct diagonalization

in three dimensions. Probably, the question of the presence of gapless surface states would

be addressable using some other numerical methods.

The problem of magnetic neutron scattering on Weyl fermions has been addressed in

Section III. The differential cross-section for a free Weyl fermion in semimetallic phase has

been computed. The angular dependence suggests neutrons prefer not changing its initial

direction after being scattered by the Weyl fermions. The magnetic character resembles the

elastic magnetic neutron scattering on paramagnets. If the polarization vector is aligned

along the momentum transfer, then the neutrons flip the spins. If the neutron polarization

vector is perpendicular to the momentum transfer, then the flipping of neutrons spin is as

likely as not flipping it. Due to linear dispersion of the Weyl fermions, there is a threshold

value kc of momentum of the incident neutrons at which they start scattering on Weyl

fermions. It was estimated that in order to observe kc in neutron scattering experiments,

the effective velocities of the electronic states around the Weyl nodes have to be of order

of 103m/s, which is by three orders of magnitude lower than the velocity of the Dirac

fermions in graphene. Provided the velocity of the Weyl states is of the same order as in

graphene one expects a threshold value for the incident neutron momenta to be of order

of 0.1MeV. This corresponds to resonant close to intermediate neutrons and is out of the

range of the conventional thermal neutrons used in solid-state physics. Another feature

arising from the linearity of the dispersion relation of the Weyl fermions is the “jump” in

the maximal scattering angle once the energy of the incident neutrons is increased. We

showed that when the momenta of the incident neutrons equals twice the threshold value kc,

the maximal neutron scattering angle θmax makes a jump from π/2 to π. It was suggested

in Section III that the threshold angular values can be resolve in logarithmic plots of the

angular differential cross-section.

Despite the above results, it is still unclear what happens to the Weyl system when

it is doped, i.e. the chemical potential is not zero and the Fermi energy is not at the

Weyl point anymore. An appropriate step for improving the given results in the current

framework would be computing the differential cross-section for doped systems at finite

temperature. Also, in real materials there are contributions to the cross-section coming from

other scattering channels. How the picture would change if one takes the lattice structure

– 36 –

into consideration? Will the effects of the linearity of the Weyl fermionic dispersion remain

or will they change?

– 37 –

APPENDIX A: LATTICE MODEL OF A WEYL SEMIMETAL

We have used the following half-filled 3D two-band model given by the Hamiltonian [10]

H0(k) = 2tx(cos kx − cos k0)σx +m(2− cos ky − cos kz)

×σx + 2ty sin kyσy + 2tz sin kzσz, (A1)

where σx, σy and σz are Pauli matrices, kx, ky and kz are the components of the momentum

vector, tx, ty, tz and m are strength constants.

The zero energy points of (A1) are discussed bellow. The presence of two Weyl nodes is

shown and the range of m used in Section II is discussed.

To find the zero-energy points one first diagonalizes the Hamiltonian (A1) and finds the

the energy levels

Ek = ±√am2 + bm+ c,

where

a = (2− cos ky − cos kz)2

b = 4tx(cos kx − cos k0)(2− cos ky − cos kz)

c = 4[t2x(cos kx − cos k0)2 + t2y sin2 ky + t2z sin2 kz

].

There are two Weyl points sitting in (kx, ky, kz) = (±k0, 0, 0). At these points the en-

ergy levels Ek are ±2tx(cos kx − cos k0) and linearly vanish as ±2tx sin k0(kx ± k0) once kx

approaches k0. Except Weyl points, however, there can exist other points in k-space where

the energy also vanishes. One may tune m in order to get rid of these points. Since we have

b2 − 4ac = −16(2− cos ky − cos kz)2(t2y sin2 ky + t2z sin2 kz) ≤ 0, ∀ ky, kz ∈ BZ,

the energy Ek will vanish for some values of m where (2 − cos ky − cos kz)2(t2y sin2 ky +

t2z sin2 kz) = 0 is satisfied, i.e. at ky = kz = 2nπ, ∀n ∈ Z. At ky = kz = 0 energy eigenvalues

Ek vanish for m = −∞. However at kyy = kz = ±π energy eigenvalues Ek vanish for

m = −tx(cos kx − cos k0) ∈ [tx(cos k0 − 1), tx(cos k0 + 1)], i.e at cos kx = cos k0 − mtx

.

– 38 –

Hence, in order to prevent vanishing energy levels except in Weyl points m has to be

outside the range [tx(cos k0 − 1), tx(cos k0 + 1)].

– 39 –

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