Understanding chiral crossings
with local and global symmetry constraints
Moritz M. Hirschmann
Max Planck Institute for Solid State Research, Germany
November 10, 2022

Topological band crossings
Two-fold screw rotation C̃2zφb

Implications of band crossings• Top. invariants
Chern number, chirality• Surface states• Effects on transport properties■1

Symmetries relate Weyl points
Symmetry D on chirality νk• Rotations and time reversal: νk = νDk• Mirror and Inversion: νk = −νDk
Yet, time reversal ⇏ four Weyl points.
three Weyl points

two Weyl points8Energy

6420-2-4Γ

ZPX

SG 80 (I41 ) spinless

M Z1

SG 119 (I 4̄m2) spinful

Hirschmann, M. M., Leonhardt, A., Kilic, B., Fabini, D. H., Schnyder, A. P. (2021). Physical Review Materials, 5(5), 054202.
■2

Outline
1. Local constraint
Relation between rotation eigenvalues and the chirality

2. Global constraint
Enforced band exchange by (non)symmorphic symmetry

3. Applications of global and global constraints
Double Weyl points in TaAs
Quasi-symmetry-protected quadruple Weyl points
Fourfold crossing with Chern number ν = 5
Topological nodal planes
Chiral nodal lines

Local constraint I
Consider Weyl points:• rotation symmetry Cn• eigenvalues Cn |Eb ⟩ = λb |Eb ⟩ for band index b

Weylλ-λ+−• crossing between rotation eigenvalues λ+b and λb

Goal: Determine the Chern number ν using−±±∆φb = φ+b − φb with φb = argλb .
1

Chern number ν = Integral of Berry curvature

2

Rotation symmetry Cn simplifies the integration.

φ+1∂SWdsCn2∂SWdsSWdnφ-12∂SW = ∂SW+ ∂SW■3Local constraint IICnφ+1
∂SW

ds2
∂SW

ds

Apply Stokes theorem

4Symmetry-related paths partially cancel

5

At the north and south pole the phase Φ(k)reduces to φ±

ν=SWdnφ-12∂SW = ∂SW+ ∂SW

3n2πn=2πZdn · Ω(k)SWZ!

Zds · A(k) +1∂SWZn=
ds · ∇Φ(k)22π ∂SWn=∆φ
2πds · A(k)mod n2∂SW

mod n
mod nsummarizes tables given in:Fang et al. PRL 108, 266802 (2011), Tsirkin et al., PRB 96, 045102 (2017)■4Outline
1. Local constraint
Relation between rotation eigenvalues and the chirality

2. Global constraint
Enforced band exchange by (non)symmorphic symmetry

3. Applications of global and global constraints
Double Weyl points in TaAs
Quasi-symmetry-protected quadruple Weyl points
Fourfold crossing with Chern number ν = 5
Topological nodal planes
Chiral nodal lines

Global constraint I~
C2xxn(Cnz (x, y, mn )) = ± T (0, 0, m) = ± exp(imkz )
2πp + mkz
1/n=⇒ λCn = (±1)
exp in≡ exp(iϕ(kz ))with band crossings at {kc }φb (kz ) = ϕ(kz ) +X

∆φb,ckc ≤kz

BZ periodicity φb (−π) = φb (π)

X⇒∆φb,cb + 2πcbP21

Γ

Z

P31

Γ2ΓA

m
=0nP61

P41

Γ2

Γ

Z

mod 2π

Γ2ΓA

Γ2

■5

Global constraint II
If all cb are two-band crossings:

b=1b=2m
mod 2πnc1XXm
mod 2π∆φ2,c2 −∆φ1,c1 = −2πnc2c1Xm⇔∆φ2,c2 = −2 · 2π
mod 2πncX∆φ1,c1 = −2π2..
.
any b..
.
Xcb∆φb,cb = −b · 2πmn

mod 2π.

■6Outline
1. Local constraint
Relation between rotation eigenvalues and the chirality

2. Global constraint
Enforced band exchange by (non)symmorphic symmetry

3. Applications of global and global constraints
Double Weyl points in TaAs
Quasi-symmetry-protected quadruple Weyl points
Fourfold crossing with Chern number ν = 5
Topological nodal planes
Chiral nodal lines

Application to band structures Ikz

Chern number ν of a band crossingnν = ∆φ 2π
mod nAL'L

H'M'Eigenvalues and Weyl points for SG 144 (P 31 )(simplest arrangement)

P'ML''ΔHΓ

PK'M''kyKkxEEEE

E–1+1E

EE–1

E+1+1+1–1–1±3∓3+1+1–1+1–1–1+1–1HKHAΓA

H'K'H'kz = -πkz = 0kz = πkz = -πkz = 0kz = πkz = -πkz = 0kz = πTRS
Representation labels :

p=0

p=1

p=2

Y.H. Chan, B. Kilic, M.M.H., C.K. Chiu, L.M. Schoop, D.G. Joshi, A.P. Schnyder, (2019). Physical Review Materials, 3(12), 124204.
■7Application to band structures II
1.00

number ofWeyl points

0.75

E - EF (eV)

0.50

23614
50.25
0.00
0.25

eigenvalue
+i

- 1

0.50

00.75

50

1.00

yC
˜2

Γ

Y1+1- iY

For C2y (0, 12 , 12 ) the global constraint is:X∆φb,cb = b · π

mod 2πcb

every second band exhibits an odd number of Weyl points
M.A. Wilde, M. Dodenhöft, A. Niedermayr, A. Bauer, M.M.H., K. Alpin, A.P. Schnyder, C. Pfleiderer (2021). Nature, 594(7863), 374-379.
■8Double Weyl points in TaO2
Oxygen deficient NbO2 and (proposed) TaO2
• SG 80 I41• C4z T creates Kramers pairs at P
• At P the only symmetry is C2z• ∆φ = 0, ν = 0 mod 2 : double Weyl point- 0.1
right surface

Energy (eV)

- 0.2

- 0.3

ν=-2X

N/P

Γ/M

N

ν=2-0.4-0.5

ΓMPΓ■9Quadruple Weyl
Twofold quadruple Weyl points ν = ±4 in cubic point groups
T. Zhang, R. Takahashi, C. Fang, S. Murakami, Physical Review B, 102, 125148 (2020)

H(k) = d0 + d1 (kx2 + ky2 + kz2 ) σ00a1 (kx2 + ky2 e−i2π/3 + kz2 e−i4π/3 )+ a0 kx ky kz σz +H.C.
0for a0 , a1 ∈ R ⇒ H(k) belongs to point group 432.
Symmetry eigenvalues:
λC4 do not exchange: ν = 0 mod 4
λC3 exchange:
ν = ±1 mod 3
Lowest possible Chern number: ν = ±4(Quadruple Weyl)■ 10

Quasi-symmetric quadruple Weyl
What happens in a cubic point group without fourfold rotation?
One finds the same Hamiltonian:H(k) = d0 + d1 (kx2 + ky2 + kz2 ) σ00
a1 (kx2 + ky2 e−i2π/3 + kz2 e−i4π/3 )+ a0 kx ky kz σz +H.C.
0

now with a1 ∈ C.
But this is not enough to break the C4 symmetry, needs k n with n > 3→ C4 is a quasi-symmetry of the low-energy model.

U (C4z ) =0e−i(arg(a1 )+2π/3)!
ei(arg(a1 )+2π/3),0

U (C4z )† H(kx , ky , kz )U (C4z ) = H(ky , −ky , kz ).
As before: λU (C4z ) :ν = 0 mod 4 → ν = ±4
■ 11

Crossing with chirality ν = 522=12=15220+ 1

1-1
2=5

2=50

0

1, 4, . . .}• C2 : ν = 1 mod 2→ ν ∈ {. . . , −5, −3, −1, 1, 3, 5 . . .}→ Lowest possible chiralities ν = 1, −520.75-5

Cubic point group PG 23• C3 : ν = 1 mod 3→ ν ∈ {. . . , −5, −2,BaAsPt

0.25
0.00EEf [eV]0.50

0.250.50

RX|M

0.75
{3, -5, 5, -3}{-2, 2}

M{21,-21}XR{3,-3}■ 12

Nodal planes
Symmetries:~C2x• two-fold screw rotation C̃2xx
k1~k2k2Kramers theorem applies to θC̃2x because

k1TRS• (θC̃2x )2 = +T(1,0,0) = eikxkx

G~
θ C2x k1 = k1 mod G• time reversal θ = iσy K~
C2x=at kx =π−1

• invariant planes: kx = 0 and kx = π(θC̃2x )2 = −1
⇒ two-fold degenerate plane kx = π

M.A. Wilde, M. Dodenhöft, A. Niedermayr, A. Bauer, M.M.H., K. Alpin, A.P. Schnyder, C. Pfleiderer (2021). Nature, 594(7863), 374-379.
■ 13

Trio of topological nodal planes
Necessary symmetries: Yu, Z. M., et al. (2019). Physical Review B, 100(4), 041118.
• θC̃2x , θC̃2y , θC̃2z=⇒ three intersecting nodal planes• θ2 = −1, single Kramers-Weyl point at Γ with νWeyl = ±1
• Multiplicity of any other Weyl point > 1.
=⇒ nodal planes compensate the Weyl point at Γ with νNP = ∓1.
dall noicagpoloplane

Properties of nodal plane trios:toΩ

Weyl point• No surface states• Vanishing anomalous Halleffect■ 14

Topological nodal plane duoSymmetries:• θC̃2x , θC̃2z ⇒ two nodal planes• Time-reversal symmetry θ is broken.
• Two-fold screw rotation C̃2yC̃2y representation:• C̃2y -invariant lines: rotation axes
• (C̃2y )2 = T(0,1,0) (iσy )2 = −eiky⇒ symmetry eigenvalues ±ieiky /2=⇒ uncompensated, odd number of Weyl points=⇒ The nodal plane must be topological.
M.A. Wilde, M. Dodenhöft, A. Niedermayr, A. Bauer, M.M.H., K. Alpin, A.P. Schnyder, C. Pfleiderer (2021). Nature, 594(7863), 374-379.
■ 15

Topological nodal plane duo
Fermi arc:
single Weyl → Nodal plane

M.A. Wilde, M. Dodenhöft, A. Niedermayr, A. Bauer, M.M.H., K. Alpin, A.P. Schnyder, C. Pfleiderer (2021). Nature, 594(7863), 374-379.
■ 16

Space groups with nodal planes
possible topological nodal plane

enforced topological nodal plane

Paramagnets

MnSi~Ferromagnets

Nd5Si4CoNb3S6~Antiferromagnets

M.A. Wilde, M. Dodenhöft, A. Niedermayr, A. Bauer, M.M.H., K. Alpin, A.P. Schnyder, C. Pfleiderer (2021). Nature, 594(7863), 374-379.
■ 17

Nodal plane duo in SG 94
SG 94 P 42 21 2 with spin-orbit coupling• Two nodal planes at kx = ky = π• Along Z-Γ-Z: hour-glass crossing• rotation 42 : n = 4, m = 2Consider the lowest band b = 1 on the fourfold rotation axis:
global constraint:X∆φ1,c1 = −mc12πnmod 2π

Total chirality νZ-Γ-Z on the fourfold rotation axis:
add local constraint: νZ-Γ-Z ≡Xc1

νc1 =X nc12π∆φ1,c1 = −m

mod 4

νZ-Γ-Z = 2 mod 4 cannot be compensated by generic Weyl points⇒ The nodal plane duo exhibits a charge of νnodal plane = −νZ-Γ-Z ̸= 0.

■ 18

Application: Chiral nodal lines I

Question: Are chiral nodal lines possible?
• Line protected by mirror symmetry→ν=0• Can there be a stable line with ν ̸= nZ
enclosing a Cn rotation axis?
AgF3 - González-Hernández, R., et al PRM 4, 124203 (2020)■ 19

Application: Chiral nodal lines II
Surface including line:nνline + νaxis = ∆φ 2π
mod n

Surface excluding line:nνaxis = ∆φ 2π
mod n⇒ νline = 0 mod n, thus it cannot be protected by rotation symmetry.
Model of a chiral nodal line:
W (k, ϵ) = k · σ + ϵσ0A(k) = kz σ1
W (k, ϵ)A(k)H(k) =A(k)† W (k, −ϵ)
2 0 101 0 2 01Uline = √ 10−2050

10−2[H(k), Uline ] = 0■ 20

Summary• Local constraintn
mod n
ν = ∆φ 2π• Global constraint
Pmcb ∆φb,cb = −b · 2π n mod 2π• Applications
Double Weyl points in TaAs
Quasi-symmetry-protected quadruple
Weyl points
Crossings with Chern number ν = 5
Topological nodal planes
Chiral nodal lines■ 21

Summary - Thank you!
• Local constraintn
mod n
ν = ∆φ 2π• Global constraintPm
cb ∆φb,cb = −b · 2π n mod 2π• Applications
Double Weyl points in TaAs
Quasi-symmetry-protected quadruple
Weyl points
Crossings with Chern number ν = 5
Topological nodal planes
Chiral nodal lines■ 21

Topology in a Weyl semimetal• Chern number ν : Obstruction to a smooth gauge for eigenstates.
• Wilson loop : Its eigenvalues are the Wannier charge centers.

Band structure of MnSi
Topological nodal planes in ferromagnetic MnSi (mSG 19.27 P 2′1 21 2′1 )• Several twofold (WP) and fourfold (FP) Weyl points• Band degeneracy protected at the Fermi surface: Topologicalprotectorate (TP)M.A. Wilde, M. Dodenhöft, A. Niedermayr, A. Bauer, M.M.H., K. Alpin, A.P. Schnyder, C. Pfleiderer (2021). Nature, 594(7863), 374-379.

Symmetry eigenvalues in MnSi
Does the picture of the eigenvalue winding argument hold?
a2a3Energy

a1k y = -πky = 0ky = π

k y = -πky = 0ky = πk y = -πky = 0ky = π

Y1

ΓY

R1UR

S1XSb1b2

˜yC 2 eigenvalue

1+iE-E F (eV)0.5
+1-10-i- 0.5-1k y = -πky = 0ky = π

k y = -πky = 0ky = πY1

Γ

Y

R1U

R

M.A. Wilde, M. Dodenhöft, A. Niedermayr, A. Bauer, M.M.H., K. Alpin, A.P. Schnyder, C. Pfleiderer (2021). Nature, 594(7863), 374-379.

Topological nodal planes in MnSi
Ferromagnetic MnSi (mSG 19.27 P 2′1 21 2′1 )Fermi surface sheets

de Haas-van Alphen spectroscopy
provides evidence for two gaplessnodal planes

M.A. Wilde, M. Dodenhöft, A. Niedermayr, A. Bauer, M.M.H., K. Alpin, A.P. Schnyder, C. Pfleiderer (2021). Nature, 594(7863), 374-379.

Surface states from DFT for MnSi
WP (generic model)
π a1

FP (generic model)
b1DFT at EF
c1kz (1/c)

π/2DOS0(a.u.)-π/2

1.00.8-πkz (1/c)

π a2b2c2

0.6

π/2

0.4

00.2

-π/20-π-π-π/20

π/2kx (1/a)π -π-π/20π/2kx (1/a)π -π-π/20π/2

πkx (1/a)• Generic tight-binding model: Characteristic surface states• ferromagnetic MnSi: Several overlapping bands, many Weyl points⇒ Need for simpler systems to study topological nodal planes.
M.A. Wilde, M. Dodenhöft, A. Niedermayr, A. Bauer, M.M.H., K. Alpin, A.P. Schnyder, C. Pfleiderer (2021). Nature, 594(7863), 374-379.

De Haas-van Alphen data
• Oscillations measured using magnetic torque• Several orbits passing the nodal plane• Focus on the single frequency M1 .

M.A. Wilde, M. Dodenhöft, A. Niedermayr, A. Bauer, M.M.H., K. Alpin, A.P. Schnyder, C. Pfleiderer (2021). Nature, 594(7863), 374-379.

Angle-resolved spectrum

Band structure for rotated fields
baR
R1Ukz
YX1
Y1ΓS

B1 B2

kyB3

X
R2

E-E F (eV)

0.4 c1

0.4 e1

d1

0.

0.
-0.2

-0.4
0.4 c2E-E F (eV)

f1

0.2

0.2

-0.2

-0.4
0.4 e2

d2

f2

0.2

0.2
0.

0.

-0.2

-0.2

-0.4
0.4 c3E-E F (eV)

kx

-0.4
0.4 e3

d3

f3

0.2

0.2
0.

0.

-0.2

-0.2
-0.4

-0.4
Y1

ΓY X1

ΓX

R1

U

R R2SRBerry curvature of the genericmodelb10.40.2

Berry curvature

Berry curvature

a10.10.0-0.1

0.20.0-0.2

8π
109π
10π

11 π
1012 π108π10kx

9π10π

11 π
1012 π10kx

ΩxΩyΩzLog |Ω|-0.5-1.0-1.5-2.0-2.5-3.0