Topological Materials - Appunti

L,

balls surrounding any bad point in p . A connection on a complex line bundle which

α

A(p),

we call is the same as an abelian gauge potential in high-energy physics.

However, parallel-transport cannot be defined globally along a closed loop if the Berry

phase is nonzero. Thus, the Berry connection is not flat, and we can introduce a cur-

0

F L → B

vature = dA, which divided by 2π defines the first Chern class of the line bundle

F

←→

c (L)

1 2π

If we take its flux across the sphere S which contains the p bad point, we obtain the

α α

associated winding number (Chern number) F

Z

w (S) =

α 2π

S

α

The Chern number concept made its first appearance in the electronic structure in 1982,

in the famous TKNN paper about quantum Hall effect.

The Bianchi identity for any abelian gauge field asserts that the exterior differentia-

tion satisfies dF = 0, thus we obtain the Nielsen-Ninomiya theorem from the generalized

Stokes one F

Z Z

dF X X

0= = = w(S ) (6.5)

α

2π 2π

S

α

α α

For a many band system (non-abelian), the whole treatment done till now remains the

F →

same, with the exception that the form TrF is used, and the conclusions reached

don’t change. Therefore, the Nielsen-Ninomiya theorem proves that the Weyl nodes have

to appear as doublets with opposite chiralities, and can be interpreted as positive and

negative monopoles of the Berry connection.

6.3 Weyl Electrons in Condensed Matter [9]

We have seen how chiral Dirac Hamiltonians arise in condensed matter, however, this

has striking consequences only if the band crossing, due to band inversion, is very close

80 CHAPTER 6. WEYL AND DIRAC SEMIMETALS

to the Fermi energy. Ideally we want the Fermi surface to consist only of a finite set of

Weyl points at which two bands cross precisely at . There will have to be an even

F

number of them, with chiralities adding up to zero as stated by the N-N theorem. The

easiest way to have all these points at the same energy is to impose discrete space and/or

time orientation-reversing symmetries, that permute all the Weyl nodes and exchange +

and - points. For TR symmetry, the picture will then look like what is shown in figure

(6.1), with a left-right symmetry that exchanges the four two-fold degenerate Weyl points.

The minimum possible number will then be 8 due to chiral compensation. Moreover, the

condition for these energies to be close to the Fermi one is free. In fact, given a band

Hamiltonian, any nearby band Hamiltonian, having the same symmetries, leads quali-

tatively to a symmetric picture, where at any point p we associate two specular energy

values, one below and one above the Fermi energy. For instance, under inversion symme-

try, taking one electron per unit cell will add up to fill exactly the band till the crossing

middle, defining it as the Fermi energy.

Figure 6.1: Creation of four symmetric Weyl points through an accidental degeneracy

in the 3D momentum space, tuned by an increasing hopping parameter or SOC. The

Nielsen-Ninomiya theorem is satisfied by the appearance of specular nodes at the same

energy, but with opposite chirality. The introduction of inversion symmetry would cause

the nodes to collapse into a single Dirac point at a time-reversal invariant momentum.

These new kind of semimetals (Weyl semimetals) can be defined, respectively, by the

absence of broken parity and/or time-reversal symmetry. In particular, this can be seen

through the exploitation of accidental degeneracies in a four band system, which can be

modelled by a continuum k-space picture with two orbitals plus spin, which Hamilto-

nian contains spin-orbit coupling (SOC) terms and non-invariant Lorentz contributions.

×

Expanding around the Γ point, we consider the most general 4 4 matrix

0

·

H = v τ (σ p) + mτ + bσ + b τ σ (6.6)

F x z z z x

0

·

mI + bσ + b σ v σ p

z x F

= 0

· −mI −

v σ p + bσ b σ

F z x

where the τ ’s are Pauli matrices for the pseudospin (effective spin) orbital degrees of

n

6.3. WEYL ELECTRONS IN CONDENSED MATTER [9] 81

0

freedom, while b and b are two Zeeman splittings. Although idealized as time-inversion

0

breaking fields in this model, b and b represent only analogies to the symmetry breaking

perturbations that may be encountered in lattices. In general they should be considered

as effective internal fields. These broken symmetries cause the fourfold degeneracy of a

0

Dirac accidental crossing (m = b = b = 0) to split into different doubly degenerate nodes

away from the origin, and defined up to a new quantum number, like chirality for broken

0

6 6

parity. Our focus will be on the conditions 0 = m < b = 0 and b = 0, which can define a

gapped or Weyl semimetal phase of the system. Thus, we can write the associated energy

dispersion as r q

2 2

0

2 2 2 2 2

m + b + v p + s 2b v p + m (6.7)

(p) = s

0

ss z

F F

0 ±1

where s and s are the chirality quantum numbers which can take values.

Figure 6.2: (left to right) Energy spectra of (0, p , p ) for the different possible phases

0

ss y x 0

given by the Hamiltonian (6.6): the Dirac spinless Hamiltonian (m = b = b = 0), gapped

0 0

semimetal (m = 1, b = 0.5, b = 0), Weyl semimetal (m = 0.5, b = 1, b = 0), and

0

line-node semimetal (m = b = 0, b = 1). A phase boundary exists between the gapped

and Weyl semimetal phase; the latter obtained only for m/b < 1. The Dirac semimetal

6

dispersion in sensible to even small perturbations (m = 0), hence, the zero energy mode

is not protected and an energy gap opens through band inversion. By applying a broken

parity symmetry, the Dirac cone splits the four-fold degeneracy at the high symmetry

point Γ, creating two two-fold degenerate Weyl nodes at low-symmetric k values in the

BZ. Moreover, the SOC magnitude plays a fundamental role for the transition between

the gapped and gapless Weyl semimetal state, compensating the disorder effect induced

by m.

6.3.1 Robustness of the Phase

From a simple analysis of the Weyl phase, one could speculate that this is a very un-

stable situation, and that any small perturbation, like disorder, would open a gap that

removes the Weyl node. However, in three spacial dimensions there exist a topological

82 CHAPTER 6. WEYL AND DIRAC SEMIMETALS

protection of the phase against small perturbations. This protection is simple to under-

stand. The condition for the points to annihilate in couples requires them to have well

defined coordinates, hence the momentum has to be a good quantum number and trans-

lational invariance has to be unbroken (weak disorder). Under this symmetry, two bands

×

create a two-level system described by the 2 2 Pauli matrices. Because there are only

three of them, they can acquire the three lattice momenta as coefficients, such that, any

other other perturbation, that doesn’t break any symmetry, has to couple with the Pauli

matrices, and can only be reinterpreted as a shift in lattice momenta.

The topological way of understanding this robustness is through the Berry phase

concept. Given the fermion doubling effect of the Nielsen-Ninomiya theorem, this states

that the Weyl nodes are opposite monopoles of the Berry connection with a same energy

behaviour. Therefore, it requires the Chern number of the band system to vanish, and

the topological stability is related exactly to its conservation under smooth modifications

of the Hamiltonian, that is a consequence of the charge conservation symmetry U (1).

6.3.2 Chiral Anomaly

The distance between high-energy physics and condensed matter physics has been de-

creasing in recent years, and nowadays we can see how phenomena such as the chiral

anomaly gets translated in the condensed matter context. µ

µ

The basic idea of chiral anomaly is that the conservation laws ∂ J = 0 and ∂ J = 0

µ µ 5

cannot be simultaneously satisfied. Thus, if we take the former as sacrosanct, the current

of the left and right chiral fermions cannot be individually conserved. The Adler-Bell-

Jackiew anomaly equation µ ∗ 2

∂ J = F F /8π

µ 5

thus leads to breaking of chiral symmetry by the anomaly. In the Weyl semimetal context,

these equations can be rewritten as 2

∂ e

∂ − ± ·

(n + n ) = 0, (n n ) = E B (6.8)

R L R L 2

∂t ∂t h

This means that, by applying a suitable electromagnetic field, we can change the relative

number of chiral charged particles. Without the Nielsen-Ninomiya doubling theorem,

this effect would have meant a non-conservation of charged quasiparticles, indeed, in our

simplest case it only implies a nonzero associated Chern number (broken TRS). Thus,

Weyl nodes will act as sources or sinks of electrons, leading to nodal polarization. Since

they aren’t completely independent, this change in chemical potential per node will give a

conductivity contribution. Indeed, in the continuum limit of the linear energy dispersions,

the chiral flow would mean a transfer of particles between the Weyl points through the

infinite Dirac sea. However, in solid state physics we take in mind the presence of a

periodic electric field, which bounds the linear dispersions. A priori, this highlights the

6.3. WEYL ELECTRONS IN CONDENSED MATTER [9] 83

impossibility to see any transport effect given by the pure chiral anomaly effect. An exit

route from this problem comes from the presence of a relaxation mechanism between the

Landau levels. In fact, this creates a cut off of the periodic Bloch field effect, making the

chiral anomaly observable in longitudinal magnetotransport effects.

A simple way to understand the chiral anomaly effects to transport in 3D is to first

consider the application of the magnetic field in the clean system, which leads to Landau

levels that disperse only along the field direction. Given the Weyl Hamiltonian under

− ·

a magnetic field, H = χv (p eA) σ, if we call the conserved momentum along the

F

·

field p = p B, for example p , we can divide the Hamiltonian into two elements in the

B x

Landau gauge: −

H = χv [(p eB)σ + p σ ] + χv p σ (6.9)

F y y z z F x x

The former term is a 2D node in a field, which has a known quantized energy of the form

p p

2 ∈

2|n|/l , for n and l = The latter, instead, gives a chiral

= v Z }c/eB.

}sgn(n) B

n F B

0-th order Landau level = χv p (polarizing the spin to σ = 1), that disperses linearly

0 F x x

in different directions for the two coupled nodes. This result states that the behaviour of

the particles in this level is exactly one-di