Automatic generation and topological classification of low-energy
Hamiltonians at multi-fold degeneracies
Kirill Alpin

Max Planck Institute for Solid State Research, Stuttgart, Germany
Yukawa Institute for Theoretical Physics, Kyoto - November 2022

Topological band crossingsfrom [Setyawan et al. 2010]

Band crossings of CoSi [Huber et al. 2022]

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Topological band crossings
Simplest is a Weyl point (WP):• Two band crossings at single k-point• Sources and sinks of Berry curvature Ωn (band n)• Ωn = ∇ × An with Berry connection An = i hn|∇|niR

1• Topological invariant/charge νn = 2π
Ωn · dS ∈ Z overSsphere S around Weyl point 6= 0 =⇒ Weyl point stableagainst perturbations• Can only be removed by annihilation with WP of opposite
chargePhysical implications: Anomalous Hall effect, Fermi arc-states, nonlinear optic responses, quantum oscillation
Other kinds of topological crossings are possible:• Multi-fold crossings (crossings of >2 bands)• Higher dimensional crossings (crossings on lines or planes inthe BZ)
Today: Multi-fold crossings. Goal: Classify (compute νn ) all multi-fold crossings in all space groups
Space groups: collection of symmetries of lattices (example: rotations Cn )2

Irreducible representations
Multi-fold crossings at high-symmetry points possible (where gk = k). Example: R
Need build local Hamiltonian to compute Ω and ν
Low-energy Hamiltonian must be symmetric in the little group G∀g ∈ G : H(gk) = D(g)H(k)D(g −1 )(1) from [Setyawan et al. 2010]

Representation D(g) simple example of C2 with two orbitals (no lattice)

Exchange of orbitals: σx in orbital position space
Project rotated orbital onto original basis: exp(−iφJ) rotation with angular momentum operator
So: D(g) = σx ⊗ exp(−iπJ)3Irreducible representations
Multi-fold crossings at high-symmetry points possible (where gk = k). Example: R
Need build local Hamiltonian to compute Ω and ν
Low-energy Hamiltonian must be symmetric in the little group G∀g ∈ G : H(gk) = D(g)H(k)D(g −1 )(1) from [Setyawan et al. 2010]

Simultaneous block-diagonalization of all D(g) in little group G:Representations D(g) can be decomposed into irreducible representations (irreps) D(g) = D1 (g) ⊕ D2 (g)...
dg is the dimension of an irrep Dn (g)mass terms are not allowed by Dn (g)=⇒ H is dg degenerate
Look up irreps in all space groups (BILBAO [Aroyo et al. 2006]), search for all dg > 2 (multi-fold degeneracy)
General low-energy Hamiltonian at high-symmetry points (expansion in k) n, m ≤ dg :Hnm (k) =X
hp1

1
αh,1 Hhp
kp1 +1 nm

X2αh,2 Hhp
kp1 kp2 + O(k 3 )1 p2 nm(2)

hp1 p23

The algorithm: generation of symmetric Hamiltonians
We only look at linear k terms H 1
Input: Space group, Irrep
Output: All possible terms of a general low-energy Hamiltonian1• Start by generating some number (30) random hermitian matrices H̃hpnm(0 ≤ h < 30)• Symmetrize with [Gresch et al. 2018]
1
Hhvij=1 X −1
1
gvp D(g)in H̃hpnm
D(g −1 )mj
|G|

(3)

g∈G

• Find all linearly independent terms via Gram-Schmidt orthogonalization
Discard all other terms• Problem: H 1 is ugly. Random numerical values.
Solution: New terms from superpositions of old terms. Matrix of all nonzero entries in Hh1 per h Mhq withq = (v , i, j). Reduce q entries such that M is invertible and quadratic. Build up new H terms
Hh10 pnm =X

1
Mh−10 h Hhpnm(4)h(redefinition αh =Ph0αh0 Mh−10h )4The algorithm: generation of symmetric Hamiltonians
We only look at linear k terms H 1
Input: Space group, Irrep
Output: All possible terms of a general low-energy Hamiltonian1• Start by generating some number (30) random hermitian matrices H̃hpnm(0 ≤ h < 30)• Symmetrize with [Gresch et al. 2018]
1Hhvij=1 X −1
1
gvp D(g)in H̃hpnm
D(g −1 )mj|G|(3)g∈G• Find all linearly independent terms via Gram-Schmidt orthogonalization
Discard all other terms
ExampleH = α0 (0.41kx σx + 0.2ky σz )+0.41=⇒ M =−0.2
α1 (−0.2kx σx + 0.41ky σz )M −1 M =0.20.41=⇒

1001=⇒

H =α0 kx σx +α1 ky σz

4Input to the algorithm
Example: Irrep Γ̄6 Γ̄7 with dimension 4 of space group 198
Input (from BILBAO [Aroyo et al. 2006]):

{2001 |1/2, 0, 1/2}
Γ̄6 Γ̄7 i5π/12Γ̄6 Γ̄7ee i5π/12

00−i
 0
 000
i0000
i000
0−i

{3+111 |0, 0, 0}e −iπ/120
e i11π/1200 e −i5π/120 e −i5π/12

{2010 |0, 1/2, 1/2}
01
00−10000001

 Time-reversal T 0100

e iπ/12e −i11π/1200
−10000001−10000−1005

Output of the algorithm
Example: Irrep Γ̄6 Γ̄7 with dimension 4 of space group 198

√
H =α0 2kx σx τz + ky − 3σx τ0 − σy τ0 +hkz σx τx +αn are parameters√3σy τxi+h

α1 −2kx σy τz + ky −σx τ0 +σ and τ are Pauli matriceskz√h3σx τx − σy τxi√

3σy τ0 ++2α2 kx σz τx + ky σ0 τy + kz σz τzi6

Topological classification
What now? Find an analytical mapping from parameters α to topological invariant ν : α0 , ..., αN → ν =⇒topological phase diagramProblems:• Classification must be done manually :(1• Can not analytically integrate ν = 2πRSΩ(k, H) · dS in general

Strategy:• Need to only find points in parameter
space with H gapless away form k = 0analytically. Only places wheretopological phase transition canhappen.
• Color spaces in between with computed
Chern number at one single pointαnαm7

Topological classification
What now? Find an analytical mapping from parameters α to topological invariant ν : α0 , ..., αN → ν =⇒topological phase diagram
Problems:• Classification must be done manually :(1• Can not analytically integrate ν = 2πRSΩ(k, H) · dS in general

Strategy:• Need to only find points in parameter
space with H gapless away form k = 0analytically. Only places wheretopological phase transition canhappen.
• Color spaces in between with computed
Chern number at one single pointαnαm7

Topological classification
What now? Find an analytical mapping from parameters α to topological invariant ν : α0 , ..., αN → ν =⇒topological phase diagramProblems:• Classification must be done manually :(1• Can not analytically integrate ν = 2πRSΩ(k, H) · dS in general

Strategy:• Need to only find points in parameter
space with H gapless away form k = 0analytically. Only places where
topological phase transition canhappen.
• Color spaces in between with computed
Chern number at one single point

αnαm7

Topological classification of all multi-fold crossings
3-fold crossings: points are all equivalent in all space groups.
Hamiltonian consistent with [Bradlyn et al. 2016]. ν = ±22

12.0
1.5
1.0

10.5

00.00.51.0
1.511012.008

Topological classification of all multi-fold crossings
3-fold crossings: points are all equivalent in all space groups.
Hamiltonian consistent with [Bradlyn et al. 2016]. ν = ±22

12.0
1.5
1.0

10.5

0

0.0
0.5
1.0
1.5

1

100

12.0

https://www.safetyemporium.com/01614

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Topological classification of all multi-fold crossings
3-fold crossings: points are all equivalent in all space groups.
Hamiltonian consistent with [Bradlyn et al. 2016]. ν = ±22

12.0
1.5
1.0

10.5

0

0.0
0.5
1.0
1.5

110012.0

https://www.safetyemporium.com/01614

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Topological classification of all multi-fold crossings
6-fold crossings: all crossings are equivalent. Hamiltonian consistent with [Bradlyn et al. 2016]. ν12 = ±49

Topological classification of all multi-fold crossings
4-fold crossings: many different models.
Without SOC:Space group
1992
96
198
212/213212/213

Irrep
R1 R1
A1 A2
A1 A2
R1 R3 /R2 R2
R1 R2R3

ν1-

ν2-ν12±2±2±2±2±2±2Model
977556Transformation
kx → ky ,ky → −kx ,α2 = 0,α3 = 0
ky → −ky ,α2 = 0,α3 = 0-10

Topological classification of all multi-fold crossings
4-fold crossings: many different models.
With SOC:Space group
18
19
19
19
90
92/94/96
92/96
92/96
94
195/196/197/198/199
195
197
198
199
207/208/209/210/211/212/213/214
207/208
211
212
213
214

Irrep
S̄5 S̄5 /R̄5 R̄5S̄5 S̄5T̄5 T̄5Ū5 Ū5Ā6 Ā7 /M̄6 M̄7M̄6 M̄7R̄5 R̄5Ā7 Ā7Ā6 Ā7Γ̄6 Γ̄7R̄6 R̄7H̄6 H̄7M̄5 M̄5H̄6 H̄7Γ̄8

ν1±3±3±3±3±3ν2±1,∓5±1,∓5±1,∓5±1,∓5±1,∓5ν12±2±2±2±2±2±2±2±4±2±4,∓2±4,∓2±4,∓2±2±4,∓2±4,∓2

Model
3333
7
738
7
1
1
13
12Transformation
kx → ky , ky → kz , kz → kx
ky → kz , kz → ky
kx → ky , ky → kz , kz → kx
kx ↔ kyU = σz τx-R̄8H̄8M̄6 M̄7M̄6 M̄7H̄8±3±3±3±1,∓5±1,∓5±1,∓5±4,∓2±4,∓2±2±2±4,∓222442ky → −ky
U = σ0 τx ,γ → −γ

10Finding topological phase transitions

1.0

1.00.50.5

0.00.0E21012E• Characteristic polynomial of H isχ(E ) = E 4 + aE 2 + det(H) =⇒spectrum is particle-hole symmetric

EExample: Irrep Γ̄6 Γ̄7 with dimension 4 ofspace group 198
Main points of derivation:

0.5
1.0

0.5

0.0k0.5

1.0

1.00.51.0

0.50.0k0.5

1.0

1.01.00.50.0k0.51.0• =⇒ can become gapless when
E2 = E3 = −E2 = 0 or E1 = E2
• Happens when det(H) = E1 E2 (−E1 )(−E2 ) = E12 E22 = 0or Q = a2 − 4det(H) = 0• χ(E ) is rotationally invariant in phase space around the α2 axis=⇒ topological phase diagram is rotationally invariant• det(H) ≥ 0 =⇒ Don’t find det(H) = 0. Easier, find minimum with ∇α det(H) = 0• Same can be shown for Q• Use scale-invariance of H in α and k (H is linear)11

Topological phase diagram of Γ̄6 Γ̄7Example: Irrep Γ̄6 Γ̄7 with dimension 4 of space group 198. ν2 ∈ {−5, −1, 1, 5}

22=12=152
20+ 1

1-1
2=0

52=5-50212

Topological phase diagram of Γ̄6 Γ̄7Example: Irrep Γ̄6 Γ̄7 with dimension 4 of space group 198. ν2 ∈ {−5, −1, 1, 5}
Weyl point production via phase transitions

νn = {-3,5,-5,3}
ν1/2 = {-1,1}

νn = {-3,-1,1,3}
ν3/4 = {-1,1}

ν2/3 = {1,-1}

ν2/3 = {-1,1}

νn = {3,1,-1,-3}

Creation of total of 26 WPs12ν = 5 phase material searchMaterial search was performed to find ν = 5 phase near Efermi
0.75
0.50
0.25
E Ef [eV]

0.00
0.25
0.50
0.75
1.00
1.25
1.50

BaAsPt in SG 198

X|M{3, -5, 5, -3}{2, -2}{1, -1}{-1, 1}{-2, 2}{-3, -1, 1, 3}{-4, 0, 4}{-15,15}

RM{21,-21}{-9,9}{-21,21}{-5,5}

XR{3,-3}{-3,3}{9,-9}
S.DOS energy

Full topological classification of BaAsPt with Γ̄6 Γ̄7 in ν = 5 phase13ν = 5 phase material searchMaterial search was performed to find ν = 5 phase near Efermi

MBaAsPt in SG 198

R1-1

M

2-2-4

Surface DOS with Fermi arcs and projected charges

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Conclusion• Automatic generation of low-energy Hamiltonian at multi-fold crossings• Analytical tools for topological phase diagrams• Applied to all space groups• Found unusually high Chern number of ν = 5• Material BaAsPt

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References iHuber et al. (2022). “Network of topological nodal planes, multifold degeneracies, and Weyl points in CoSi”.
In: Physical Review Letters 129.2, 026401.
Setyawan and Curtarolo (2010). “High-throughput electronic band structure calculations: Challenges and tools”.
In: Computational materials science 49.2, 299–312.
Aroyo et al. (2006). “Bilbao Crystallographic Server. II. Representations of crystallographic point groups andspace groups”. In: Acta Crystallographica Section A: Foundations of Crystallography 62.2, 115–128.
Gresch et al. (2018). “Automated construction of symmetrized Wannier-like tight-binding models from ab initiocalculations”. In: Physical Review Materials 2.10, 103805.
Bradlyn et al. (2016). “Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventionalcrystals”. In: Science 353.6299.

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