First Principles Calculation of Topological Hall
Conductance in the Skyrmion lattice

Hsiao-Yi Chen
Center for Emergent Matter Science (CEMS), RIKENCollaborators: Ryotaro Arita (Riken & UTokyo), Takuya Nomoto (UTokyo)
E-mail: hsiao-yi.chen@riken.jp
RIKEN Symposium Second Workshop on Fundamentals in Density Functional Theory, Kobe

DFT2024

Hall effects
1. Ordinary Hall e ect: transverse Lorentz force v × B
2. Anomalous Hall e ect: Berry curvature+SOC

3. Topological Hall e ect: additional contribution attributed to a Berry phaseacquired by the electrons following adiabatically a smoothly varying non-coplanar
magnetization texturekzkykx

Exploration of Quantum Transport Phenomena via Engineering Emergent Magnetic Fields in Topological Magnets

Momentum space Berry curvature
PRL 126, 147203 (2021)

Real space Berry curvature

ff

ff

ff2

DFT2024

Topological spin texture: Skyrmion
Gd2PdSi3(T. Kurumaji, et.al. Science 365 6456 914)
Skyrmion phasevortex-like swirling spin

Skyermion

B. Gobel, Phys. Rep. 895 (2021) 1–28

On-site spin modulation
across several unit cell

GdRu2Si2(N. Khanh, et.al. Nat. Nanotech. 15 444 (2020))
Skyrmion phase-0.0

DFT2024

Topological spin texture: Skyrmion
Real space picture (λsk > > ls)
A. Neubauer et al., PRL 102, 186602 (2009)
R. Ritz et al., PRB 87, 134424 (2013)vortex-like swirling spin

Skyermion

B. Gobel, Phys. Rep. 895 (2021) 1–28

On-site spin modulation
across several unit cellTρ = P ⋅ RH ⋅ Bem
Polarization @Ef Normal Hallcoe cient

N. Verma, et. al. Sci. adv. 8 (2022)

Emergent B- eld

Momentum space picture (λsk ≳ ls)
K. Ohgushi, et al., PRL 62, R6065 (2000)
Z. Wang, et al., PRL 124, 207201 (2020)(Kubo formula)

How to solve it in DFT?
fiffi

Berry curvature of Bloch wave

First-principle method
Density functional theory

DFT2024

Square Skyrmion lattice(Kohn-Sham Equation)⃗ (x) ⋅ σ ⃗ ψ(x) = Eψ(x)−∇+V(x)
+B
KS
KS
[]
2

Independentfrom m(x)Can be solved in unit cell in FM

Self-consistent X

Skyrmion unit cell

Crystal unit cell

Spin texture have size larger than unit cell (10~102)

Any cheap way to obtain the BKS?

First-principle method
Density functional theory

DFT2024

Square Skyrmion lattice(Kohn-Sham Equation)⃗ (x) ⋅ σ ⃗ ψ(x) = Eψ(x)−∇+V(x)+BKSKS[]2⃗⃗
B KS(x) ≈ B KS(x, {Sr})Skyrmion unit cell

Double exchange model
Crystal unit cell

Local spin

flIn the double exchange model, the propagating
electron is in uenced by local spin in the form ofHeisenberg interaction

Spin texture have size larger than unit cell (10~102)

Any cheap way to obtain the BKS?

First-principle method
Density functional theory(Kohn-Sham Equation)

DFT2024

We can identify the KS-equation withdouble exchange model⃗ (x) ≈ J S ⃗ for x ∈ r
B KS,rH r⃗ (x) ⋅ σ ⃗ ψ(x) = Eψ(x)−∇+V(x)+B
KSKS[]2⃗⃗
B KS(x) ≈ B KS(x, {Sr})Double exchange model

Local spinfl

In the double exchange model, the propagating
electron is in uenced by local spin in the form ofHeisenberg interaction: Sr onsite spinxr

: Br(x) potential

First-principle methodDensity functional theory(Kohn-Sham Equation)⃗ (x) ⋅ σ ⃗ ψ(x) = Eψ(x)−∇+V(x)+BKSKS[]2⃗⃗
B KS(x) ≈ B KS(x, {Sr})Double exchange model

Local spin

flIn the double exchange model, the propagating
electron is in uenced by local spin in the form ofHeisenberg interaction

DFT2024⃗ (x)B KS,r≈ JH(x)Sr⃗

xr

⃗ (x)
B KS,rδ⃗≈
JH(x)Sr+δ
∑
δ

rr+δ

First-principle method

DFT2024⃗Sr+δ
⃗ (x) ≈B KS∑
δ⃗JHδ (x)Sr+δδJH(x)

rrr+δ

If we have the function JH in hand,
we can obtain the BKS when theon-site spin is given⃗ (x)B KS,rPatch over all unit cellin the supercell⃗ (x) ⋅ σ ⃗ ψ(x) = Eψ(x)−∇+V(x)+BKSKS[]
2KS-equation for the skyrmion latticeWork ow
scf-2

scf-n

1⃗
1⃗
S,B
( r+δ KS(x))

2⃗
2⃗
S,B
( r+δ KS(x))

⋮

scf-1

DFT2024

Properties/Bene tsn⃗n⃗S,B( r+δ KS(x))

1. JH is a function in unit cell,and respects the crystal
symmetryδ
2. JH decays very fast in terms⃗ (x) ≈B KS∑δ⃗JHδ (x)Sr+δ

fifl

1. Band structure
2. Wave function
3. Berry curvature
4. …..

δJH(x)+

Experimental
or modelof δ. Almost only the nearest
neighbor is required

3. No need to constructsupercell to cover the spintexture.

⃗ ⋅ σ ⃗ ψ(x) = Eψ(x)−∇+V(x)+BKSKS[]
2+(wannierization….)

4. Approach is general, it canbe extended to higher orderof local spin

Berry curvature from Skyrmion
Skyrmion latticeSr⃗

DFT2024δJH(x)⃗ ⋅ σ ⃗ ψ(x) = Eψ(x)−∇+V(x)+BKSKS[]
2ikx

ψnk(x) = e unk(x)Anm(k) = ⟨nk | ∂k | mk⟩

Current rst-principle way to compute berry connection and berry curvature (PRB 74 195118 (2006))

Maximally Localized Wannier Function
band

We can include the Skyrmion
potential into Wannier construction+⟨0n |∑r+⟨0n |

Berry connectionfiBerry curvature∑r⃗ (x) ⋅ σ ⃗ | Rm⟩
B KS,r⃗ (x) ⋅ σ ⃗ | Rm⟩
B KS,rParamagnetic

Paramagnetic+ tted B- eld

FM (DFT)

3.021a2.043⃗ (x) ≈B KS,r∑δ

Energy (eV)b↔
δ⃗
JH(x)Sr+δ
for x ∈ rBy(x)tted-predict

Bx(x)tted-predicttted-predict

Fitting using 128 scf datafi

DFTDFT

1.00.0

-1.0

Bz(x)
-2.0

Near Pd and Si

fifi

fi

fi

Gd2PdSi3

DFT2024

DFT-3.0GKM

GALH

The FM state can be rebuilt from paramagnetic orbital by modeled BAGd2PdSi3: Skyrmion lattice−7.15μB

DFT2024mz+7.15μB

super cell

Primitive cell
8.132Å

T. Kurumaji, et.al. Science 365 6456 914

σxy =

ρyx

| Qi=1,2,3 | = 1/32 + ρ2
ρxxyx
−1

≈ 400 Ω cm

Triple-q−1

3 times larger super cellmz = m⊥ < 0 ; η = 1

Gd2PdSi3: Skyrmion lattice−7.15μB

DFT2024mz+7.15μB

Energy (eV)super cell

Primitive cell
8.132Å

| Qi=1,2,3 | = 1/3
Triple-q

GKM

GALHA

3 times larger super cellmz = m⊥ < 0 ; η = 1

Gd2PdSi3: berry curvature

DFT20240.15kzEnergy (eV)

0.10

0.05

0.00

-0.05

-0.10

GKMG

Gd2PdSi3: Topological Hall conductance300

2001000P=0.10

Polarization−1Real space picture

400−1σxy(ω = 0)(cm Ω )Momentum space picture

DFT2024n↑ − n↓n↑ + n↓

P=0.022
0.05

0.00

-0.05

Nk = 144 × 144 × 216

0.000

0.005

Ef=14.2804
0.010

0.015

0.020

broadening δ (eV)T exp.
−1 −1Comparing σxy= 400 cm Ω

13.8

14.0

14.2

14.4

14.6

14.8

Energy (eV)Comparing Pexp. = 0.07

DFT2024

Conclusions• We have developed a rst-principles approach to compute the bandstructure and the topological Hall e ect of chiral-spin magnet.

• Without doing self-consistent calculation, we can accurately describe themagnetic potential of a large scale spin structure.

• For Gd2PdSi3, the computed dc-THC supports the momentum spacepicture over the real space picture, and the berry curvature contribution
are from generic points in BZ.

• Future work: Other materials, like GdRu2Si2, or 3D structure.

fffi17E-mail: hsiao-yi.chen@riken.jp