Electric-field Control ofTopological Spin Textures, Kitaev Spin Liquids, and Magnetic Orders

Masahiro Sato (Chiba Univ.)[1] K. Takasan and MS, PRB100, 060408(R) (2019).
[2] S. C. Furuya, K. Takasan and MS, PRR3, 033066 (2021).
[3] S. C. Furuya and MS, arXiv:2110.6503.

NQS2022＠YITP, Kyoto Univ. 11/25, 2022

[1] K. Takasan and MS, PRB100, 060408(R) (2019).
[2] S. C. Furuya, K. Takasan and MS, PRR3, 033066 (2021).
[3] S. C. Furuya and MS, arXiv:2110.6503.

Shunsuke C. Furuya (Univ. Tokyo)

Kazuaki Takasan (Univ. Tokyo)

Introduction tostrong DC electric fields in Mott insulators

Usual magnetic phase diagram
BParamagneticphase

Magneticallyordered phase

Strong DC electric fieldbeyond 1MV/cm can begenerated with current
experimental techniques.

kBTE(1) Field-effect transistor (FET)or Electric double layer transistor (EDLT)[Ueno et al., JPSJ (2014)](2) Scanning Tunneling Microscope (STM)

Our proposal
Magnetically Paramagneticphaseorderedphase(3) Interface between
different crystals

kBT
K. Takasan and MS, PRB100, 060408 (R) (2019).
S. C. Furuya and MS, PRR3, 033066 (2021).
S. C. Furuya and MS, arXiv:2110.6503.

[Hsu et al., Nat. Natnotech. (2016)]

(4) THz laser pulse may be also used as a quasi-static electric fieldHalf- or one-cycle
THz pulse~𝟏 − 𝟏𝟎 pico seconds (pico sec = 𝟏𝟎−𝟏𝟐 sec)
Typical time scale of electron dynamics of solids : ~𝟏 − 𝟏𝟎𝟎 femto seconds(femto sec = 𝟏𝟎−𝟏𝟓 sec)

Time evolution of electric field of THz pulse is much slower than electron motion.
Electric field of THz pulse ~ nearly static for electrons

In insulating systems, effects of DC electric fields have long studied,especially, in the fields of multiferroics and spintronics.
H. Katsura, N. Nagaosa, and Balatsky, Phys. Rev. Lett. 95, 057205 (2005),
M. Mostovoy, Phys. Rev. Lett. 96, 067601 (2006),
I.A. Sergienkoand E. Dagotto, Phys. Rev. Lett. 96, 067601 (2006),
F. Matsukura, Y. Tokura, and H. Ohno, Nature Nanotech. 10, 209 (2015),
H. Katsura, MS, T. Furuta and N. Nagaosa, Phys. Rev. Lett. 103, 177402 (2009),,,,,,, many papers

Many theoretical studies are based on
the symmetry arguments and the effective low-energy theories.

Theories from the microscopic viewpoint
have been not developed enough.
We have tried to develop the theory
which can predict E-field induced phenomena in a quantitative way.

Our target
Effects of DC electric fields in Mott insulators (spin systems)
Outline1, Electric field can change direct exchange interactions.
K. Takasan and MS, PRB100, 060408(R) (2019).

2, Electric field can change super exchange interactions.
S. C. Furuya, K. Takasan and MS, PRR3, 033066 (2021).

3, Electric field can change magnetic anisotropies.
S. C. Furuya and MS, arXiv:2110.6503.

Control of direct exchange interactions
with DC electric fields

K. Takasan and MS, PRB100, 060408(R) (2019).

Direct exchange interaction

Magnetic ion

Definition of “direct” exchange interaction in our study
Exchange interaction generated fromdirect hopping of electrons residing on neighboring magnetic ions

We theoretically treat the effect of DC electric field as the “voltage”.
Electron number operator−𝑒 ෍ 𝑉𝑗 𝑛𝑗Energy††𝑛𝑗 = 𝑐↑𝑗𝑐↑𝑗 + 𝑐↓𝑗𝑐↓𝑗𝑗𝑗=4
𝑗=3

𝑗=2
𝑗=1

Electric fieldSetup : Hubbard model + E-field
DC Electric fields = Spatial gradient of the on-site potential

U>E
Hamiltonianhopping

interaction

on-site potential

In the strong coupling regime ( t ≪ U ) and the half-filling case, without
Heisenberg ModelEffective spin model by large-U expansion
With on-site potential,(Remark)In 2-orbital case,
We obtain aspin-1 modelwith E-field dep.
Interaction.

→ Exchange coupling is enhanced !
For example, 1D case

Effective exchange coupling

At E~U, our treatment is not validWhat kinds of control is possible ?
Parallel to the E-field ： most enhanced
Perpendicular

not→ Control of spatial anisotropy of exchange interactions

Magnets where their spatial structure plays an important roleshould be nice for this scheme

1. Frustrated Magnets

2. Quasi-1D Magnets

Application 1: frustrated systems
Triangular lattice antiferromagnets
Spin Configuration (Classical G.S.)

E>0

E=0

1D AFM-likespiral120°structure

2D AFM-likespiralControl of spiral orders

Application 2: quasi-1D systems
Quasi-one-dimensional Magnets
E || 1D-chainE ⊥ 1D-chain
J : intra-chain interaction
Jy , Jz : inter-chain interaction
J >> Jy,zE || 1D-chain→ 1D nature is enhanced

E ⊥ 1D-chain→ 1D nature is suppressed (2D-3D nature is enhanced)“Dimensionality” can be controlled by changing the direction of the E-fieldPhase diagram of spatially anisotropic 2D spin-1 system(zero temperature, Jz=0)

𝐽𝑥

𝐽𝑦

Critical point between Haldane and Neel phases : Jy= 0.043Jx
Matsumoto et al., PRB 65, 014407 (2001)

E ||: 1D-chain
Edgespin-1/2

Haldane phase is favored (1D like)E ⊥1D-chain

Haldane phase is suppressed (2D like)→ Field-induced Haldane-gap phase (Control of topological order)

Control of super-exchange interactions
with DC electric fields

S. C. Furuya, K. Takasan and MS, PRR3, 033066 (2021).

Many magnetic materials have non-magnetic (ligand) atoms
between magnetic ions

(a)

(b)

Magnetic Non-magneticion
atom

Super-exchange interaction :exchange interaction between 2 magnetic ions
induced by hopping processes
with intermediate non-magnetic atomse.g) Alloul et al., RMP (2009)Set up:Single-band Hubbard-like models including non-magnetic atoms
Spin-1/2 (one-electron in each magnetic ion)

𝐻 = −∑ 𝑡𝑖𝑗𝑖,𝑗∑𝜎,𝜎 ′ =↑,↓

Electric potential (voltage)†(𝑐𝑖,𝜎𝑐𝑗,𝜎′ + H. c. ) + ∑𝑈𝑖 𝑛𝑖,↑ 𝑛𝑖,↓ + ∑∑𝑉𝑖,𝜎 𝑛𝑖,𝜎 + ⋯4th order Large-U expansion

𝑖𝑖 𝜎⃗

⃗

= 𝐽𝑆1 ⋅ 𝑆2 + ⋯M1,2: one electron in a d-orbit(transition metal such as Cu)
Δ2 − 𝑈𝑝
Δ1 − 𝑈𝑝

L : two electrons in a p-orbit(non-magnetic atom such as O)

Antiferromagnetic superexchange in the linear configuration⃗⃗𝐽A 𝑆1 ⋅ 𝑆2⃗

𝐸Δ2 − 𝑈𝑝

M1Δ1 − 𝑈𝑝

𝐽A= 2𝑡 4 (1+⃗Δ1 (𝐸 ) + 𝑈𝑑 − 𝑈𝑝+2𝑡 4 (1Δ2 (𝐸 ) + 𝑈𝑑 − 𝑈𝑝1⃗⃗⃗

1+(Δ1 (𝐸 )+𝑈𝑑 −𝑈𝑝 )2 Δ1 (𝐸 )−Δ2 (𝐸 )+𝑈𝑑1⃗⃗⃗)(Δ2 (𝐸 )+𝑈𝑑 −𝑈𝑝 )2 Δ2 (𝐸 )−Δ1 (𝐸 )+𝑈𝑑⃗

𝜟𝟏,𝟐 : potential energy at site 1 and 2⃗

Δ1 (𝐸 ) + Δ2 (𝐸 ) + 2𝑈𝑑 − 𝑈𝑝1⃗M21)2⃗L⃗⃗⃗
Δ2 (𝐸 ) − Δ1 (𝐸 ) = −𝑒𝐸 ⋅ 𝑟12

Ferromagnetic superexchange in the triangular configuration
Kanamori-Goodenough rule
𝑱𝑭 is generally ferromagnetic (𝑱𝑭 < 𝟎)

M1

⃗

𝐸L⃗

⃗

𝐽F 𝑆1 ⋅ 𝑆2M2⃗⃗⃗⃗
Δ2 (𝐸 ) − Δ1 (𝐸 ) = −𝑒𝐸 ⋅ 𝑟12

DC electric-field controls of AFM/FM superexchanges• Electric potential differences atthree sites.
• For 𝑈𝑑 = 5eV, 𝑈𝑝 = 1eV,

Δ𝑑𝑝 (0) = 2eV, 𝐽H = 1eV,
ℓ = 5Å, …
the O(1) MV/cm DC electric fieldis required to change thesuperexchange by O(1)%.
• The condition will be relaxed forthin films and organic compounds.

Application 1BaCdVO(PO4)2

Nath et al., PRB (2008)

Application 1⃗

𝐸BaCdVO(PO4)2

MU,1

𝐽1 < 0MD,1

Nath et al., PRB (2008)

MD,2

𝐽2 > 0MU,2

Application 1
Out-of-plane DC electric field changes the model

E field

Application 2⃗

𝐸• Spin-1/2 Heisenberg AFM chain.
• GS = critical (Tomonaga-Luttinger liquid)• DC electric-field drives dimerization of spins.
• Spin gap opens.
• Spin gap ∝ |𝐸⊥ |2Τ3

Control of magnetic anisotropies and spin textureswith DC electric fields

S. C. Furuya and MS, arXiv:2110.6503.

So far we considered exchange interaction modified by E-field.
isotropic in spin space

Electric field breaks inversion (parity) symmetry.
→ electron orbits are deformed (polarization appears).
→ Spin-orbit coupling is also changed/generated.
→ Magnetic anisotropy is generated by E field.
Case 1: Intra-atom SO coupling

Case 2: Inter-atom SO coupling

Kitaev magnets

E-field driven Rashba SO couplings

Large-U expansion leads to spin Hamiltonian
with E-field induced magnetic anisotropy

Rashba spin-dep. hopping, etc

Case 2 : Mott insulator with inter-atomic Rashba SOC
Building block (3 sites)⃗• Low-spin 𝑑 5 configuration.

𝑑1• One of the 𝑡2𝑔 orbitals, say, the 𝑑𝑥𝑦 orbital, carries spin-1/2.
⃗𝑑2
⃗

𝑧⃗

𝐸 = 𝐸 𝑒𝑧

𝑑𝑥𝑦Octahedron• Spin-dependent electron hoppings between magnetic ions andligand (λ ∝ 𝐸 electric field).
⃗⃗
⃗
⃗
†
†
𝜆 ∑ [𝑖𝑝𝑦,𝑠 (𝜎𝑠,𝑠′ × 𝑑1 ) 𝑧 𝑑1,𝑥𝑦,𝑠′ + 𝑖𝑝𝑥,𝜎 (𝜎𝑠,𝑠′ × 𝑑2 ) 𝑧 𝑑2,𝑥𝑦,𝑠′ + H. c. ]𝑠,𝑠 ′𝑒𝑔𝑡2𝑔• Three-site model w/ an intra-atomic SOC, the Rashba SOC.

• The out-of-plane electric field is assumed.

E-driven DM interaction⃗⃗𝑧⃗• The spin Hamiltonian w/ 𝐸 = 𝐸 𝑒𝑧 :𝑑1⃗⃗⃗⃗𝑧𝑧
𝐻spin = −|𝐽|𝑆1 ⋅ 𝑆2 − 𝐵 (𝑆1 + 𝑆2 ) +𝐷 ⋅ 𝑆1 × 𝑆2

⃗𝑑2

⃗

⃗

• 𝐸 -driven DM interaction.
⃗⃗𝑧
𝐸 = 𝐸 𝑒𝑧⃗⃗⃗

𝐷 = 𝐷 (𝑒𝑥 + 𝑒𝑦 )𝜆 ∝ 𝐸𝑧𝐷 = −4𝜆𝑡

3

𝑈𝑑 − 𝑈𝑝 + Δ𝑑𝑝4(𝑈𝑑 − 𝑈𝑝 + Δ𝑑𝑝 )2 − 𝐽H 2

Embedding to square lattice• Square lattice made of edge-sharing octahedra.
• Assume the Rashba SOC between M-L bonds.

• The resultant spin Hamiltonian is⃗⃗𝐻spin = −|𝐽|∑(𝑆 ⃗ ⋅ 𝑆 ⃗𝑟⃗𝑟⃗

⃗𝑟 +𝑒𝑥⃗+ 𝑆⃗ ⋅ 𝑆⃗𝑟⃗𝑟 +𝑒𝑦)

−𝐵∑𝑆 ⃗𝑧⃗𝑟⃗

SkX phase𝑧⃗

𝐸 = 𝐸 𝑒𝑧𝑟⃗⃗⃗

+𝐷∑(𝑒𝑦 ⋅ 𝑆 ⃗ × 𝑆 ⃗𝑟

⃗𝑟

B⃗

⃗

𝑟 +𝑒𝑥FM⃗⃗− 𝑒𝑥 ⋅ 𝑆 ⃗ × 𝑆 ⃗𝑟SkX⃗𝑟 +𝑒𝑦)

HLE

0E-driven Néel-type skyrmion lattice expected.

.

Other applications• Chiral soliton lattice.
• 1D version of the square-lattice model.

• Magnetic hedgehogs.
• A multiple-q state.
• Suppose double-q states.
• The electric field can drive a uniform DM
interaction that leads to another q vector.

Case 1 : Kitaev magnet with intra-atom SO coupling• Three-site models with two magnetic ionsand one ligand.
• Low-spin 𝑑 5 configuration.

• 𝐽eff = 1/2 doublet carries a (pseudo)spin1/2.

• Hubbard-like Hamiltonian with large onsite Coulomb repulsions.
• Our model is akin to the model used byJackeli and Khaliullin [PRL (2009)] torealize the Kitaev model on thehoneycomb lattice.

Large-U expansion : Kitaev-Heisenberg-Γ ′ -DM model• W/o the DC electric field:⃗⃗𝑧 𝑧𝐻spin = −|𝐽|𝑆1 ⋅ 𝑆2 + 𝐾𝑆1 𝑆2 − ℎ ⋅ ∑𝑆𝑗𝑗cf. G. Jackeli and G. Khaliullin, PRL (2009)⃗⃗𝑥⃗⃗𝑦⃗• W/ 𝐸 = 𝐸 𝑒𝑥 + 𝐸 𝑒𝑦 :⃗⃗𝑧 𝑧𝐻spin = −|𝐽|𝑆1 ⋅ 𝑆2 + 𝐾𝑆1 𝑆2 − ℎ ⋅ ∑𝑆𝑗𝑗⃗𝑧⃗
• W/ 𝐸 = 𝐸 𝑒𝑧 :
⃗⃗⃗⃗

⃗

⃗

⃗

⃗

+𝐷 ⋅ 𝑆1 × 𝑆2

⃗𝐻spin = −|𝐽|𝑆1 ⋅ 𝑆2 + 𝐾𝑆1𝑧 𝑆2𝑧 − ℎ ⋅ ∑𝑆𝑗𝑗𝑦𝑦+𝐷 ⋅ 𝑆1 × 𝑆2 + Γ ′ [𝑆1𝑧 (𝑆2𝑥 + 𝑆2 ) + (𝑆1𝑥 + 𝑆1 )𝑆2𝑧 ]⃗⃗⃗𝜞′ term and magnetic field cooperatively
opens Majorana gap and induces QPTs.
• Γ ′ interaction:

3𝑈𝑑 − 𝑈𝑝 + Δ𝑑𝑝32𝑡1′𝑧
2Γ =𝐼𝐸 ()
9𝑈𝑑 − 𝑈𝑝 + Δ𝑑𝑝 4(𝑈𝑑 − 𝑈𝑝 + Δ𝑑𝑝 )2 − 𝐽H 2

• Perturbative expansion:ℎ 𝑥 Γ′𝑥 𝑦 𝑧• Second-order term: −∑𝑆𝑖 𝑆𝑗 𝑆𝑘 .
𝐾
ℎ 𝑥 Γ′ℎ𝑥 𝐸 𝑧• The Majorana gap∝𝐾𝐾⃗• 𝐸 will change the Chern number of the chiralquantum spin liquid.

D. Takikawa and S. Fujimoto, PRB (2020)

DM interaction also leads to QPTs in Kitaev model

Kitaev interaction+ DM interaction+ Zeeman interaction

Ralko and Merino, PRL (2020)

SummarySummary 1[1] K. Takasan and MS, PRB100, 060408(R) (2019).
[2] S. C. Furuya, K. Takasan and MS, PRR3, 033066 (2021).

Setup : Mott insulator + DC Electric field(without current, No “Mott breakdown”)

Control direct- and super-exchange interactions with a DC Electric field(e.g.) Applications of Control of exchange interactions in many-spin systems

1. Frustrated Magnets

2. Quasi-1D Magnets

3. Frustrated ferromagnets

E-field-induced
magnetic order and QSL

E-field-induced
magnetic and top. order

E field changes models
E fieldSummary 2[3] S. C. Furuya and MS, arXiv:2110.6503.

Control magnetic anisotropies with a DC Electric field
Case 2: Inter-atom Rashba SO coupling• The DC electric field can yield DM interactions and drive quantum phase transitions fromcollinear phases to topological spin texture phases.
BFM0SkX

HLE

Case 1: Intra-atom SO coupling in Kitaev model• The DC electric field can yield DM interactions andoff-diagonal symmetric magnetic anisotropiesΓ ′ (𝑆 𝑎 𝑆 𝑏 + 𝑆 𝑏 𝑆 𝑎 ) to Kitaev materials such as 𝛼 -RuCl3.

⃗⃗𝑎 𝑎ℋ = ∑ ∑ (−𝐾𝑎 𝑆𝑖 𝑆𝑗 + 𝐽𝑆𝑖 ⋅ 𝑆𝑗 ) + Γ ′ ∑ ∑ ∑ (𝑆𝑖𝑎 𝑆𝑗𝑏 + 𝑆𝑖𝑏 𝑆𝑗𝑎 )𝑎 ⟨𝑖,𝑗⟩𝑎𝑎 ⟨𝑖,𝑗⟩𝑎 𝑏≠𝑎