物質中の周期外場駆動現象に対する理論研究の進展

佐藤正寛

茨城大理

S. Takayoshi, M. Sato, and T. Oka, PRB90, 214413 (2014).
M. Sato, S. Takayoshi, and T. Oka, PRL117, 147202 (2016).
M. Sato, Y. Sasaki and T. Oka, arXiv:1404.2010.
S. Higashikawa, H. Fujita, and M. Sato, arXiv:1810.01103.
H. Fujita and M. Sato, PRB95, 054421 (2017); PRB96, 060407(R) (2017).
H. Fujita and M. Sato, Sci. Rep. 8, 15738 (2018).
H. Fujita, Y. Tada, and M. Sato, arXiv:1811.10617.
H. Ishizuka and M. Sato, arXiv:1811.11515.
非平衡系の物理学-階層性と普遍性-@京大基研 27 Dec. 2018

自己紹介(最近の研究分野)
研究対象

物理現象

(量子)多体問題

非平衡物理学

緩和現象・輸送問題・平衡化
磁性体・誘電体・マルチフェロイクス
金属・半導体・超伝導体 ゆらぎ・周期外場系・定常状態
フロケ・エンジニアリング
トポロジカル量子相

スピントロ二クス

光物性科学

熱・温度勾配の効果

レーザー・光誘起相転移
分光学・共鳴現象
磁気光学効果・メタマテリアル

熱伝導・熱ホール効果
熱電効果・熱磁気効果(スピン)ゼーベック効果

外力その１外力その2光(電磁波)誘起現象

マルチフェロイクスのフロケ・エンジニアリング
MS, S.Takayoshi, T.Oka, PRL117 (2016).

熱輸送現象

スピノンのスピンゼーベック効果
D.Hirobe, MS, E.Saitoh, et al,Nat. Phys. 13 (2017).

スピン流

スピン流版太陽電池の理論
H.Ishizuka and MS, arXiv:1811.11515.

キタエフ磁性体候補a-RuCl3の熱伝導
D.Hirobe, MS, E.Saitoh, et al, PRB95 (R) (2017).

Outline
1, Introduction to laser science and Floquet engineering
2, Application of Floquet engineering to multiferroic magnets

3, Theory for Floquet engineering in (dissipative)classical (or ordered quantum) systems
4, Summary

Introduction tolaser science and Floquet engineering

Recent development of laser sciencelaser science and laser-driven systems
Laser science has largely developed in recent years.
We can now use THz lasers as an external field for condensed matter.

Electronics vs photo-science(spintronics)

1MHz

1GHz

electronics (spintronics)

1THz

1PHz

テラヘルツ

(meV)

(eV)

Frequency

Transistor(1947)

slide by K. Tanaka (Kyoto-U)

photo-science(quantum process)Energy scale of THz photons is comparable withe.g., magnetic excitations, phonons, superconducting gap, etc.

Structured (topological) light waves
Vortex beam (光渦): laser beam with an orbital angular momentum
重心運動(大域的)自転運動(局所的)

Spin (left or right handed)

Orbital angular momentum

Heating effect of vortex beam on chiral magnets

Skyrmionium

Skyrmion

Focused THz vortex beam

T. Arikawa, S. Morimoto, and K. Tanaka,Opt. Exp. 25, 13728 (2017)..

Application of a THz vortex beam to ferromagnets

m=2m=-2

Dt=1 : 0.1-1 pico second

H. Fujita and M. Sato, PRB95, 054421 (2017).

H. Fujita and M. Sato, PRB96, 040706(R) (2017).

Magnet meets light (opto-spintronics)
THz-laser driven magnetic resonancein an antiferromagnet HoFeO3

Laser induced (de)magnetization
in a ferrimagnetic alloy Gd22Fe74.6Co3.4
InverseFraday effect (?)
by a circularly
polarized laser(visible light)

Y. Mukai, H. Hirori, et al, New. J. Phys. 18, 013045 (2016).

C. D. Stanciu, Th. Rasing, et al., PRL99, 047601 (2007).

THz-wave transmission spectrum by magnetic
excitations in a 1D Ising-like magnet SrCo2V2O8

THz-wave driven electromagnon
resonance in a multiferroics Eu0.55Y0.45MnO3

A. Loidl, et al, Nature 554, 219 (2018).

Y. Takahashi, et al, Nature Phys. 8, 121 (2012).

What is Floquet thorem?
フロケの定理日本物理学会誌2017年11月号 p.783

Floquet theory and Floquet effective Hamiltonian
Floquet theorem = time version of Bloch theorem
Bloch theorem
Time-indep. Schrodinger equation for spatially periodic system
P.B.C.

SolutionwhereBloch function

Floquet theorem

Time-dep. Schrodinger equation for temporally periodic system

Floquet quasi energy

Solution

Time version of Bloch function
whereSchrodinger equation for temporally periodic system

Floquet mapping = discrete Fourier transform along time direction

Static eigenvalue problem

Floquet index

Floquet quasi energy

Time average of the original Hamiltonian H(t)Matrix form of the eigenvalue equation
Photon number -2
Photon number -1
Photon number 0
Photon number +1Photon number +2

Photon number -2
Photon number -1ℏW
Photon number 0
Photon number +1

Photon number +2

When W is larger enough and H+1,-1 are smaller enoughcompared with the energy scale of H0,
we can construct the perturbation theory in the sub-space

Unperturbed part
Perturbation

Effective Hamiltonian (1/W expansion)Commutator of H+1,-1Time averaged Hamiltonian

Effect of AC fields such as laser

Floquet-Magnus (1/w) expansion for the effective Hamiltonian 𝑯𝐞𝐟𝐟e.g., T. Mikami, N. Tsuji, T. Oka, et al, PRB(2016).

Floquet engineering
Applying a well-chosen AC field to a material,
we can control its physical properties and phases.

External AC field(Laser, etc.)
High frequency

Material A

Material A’

Temperature

Effective Hamiltonian for a periodically driven system (1/W expansion)

DisorderedphasesOrderedphasesLaser-driven
phasesLaser amplitude,
Laser frequency, etc.

Magnetic field,Voltages,Pressure, etc.

Predictions from Floquet theory
Floquet Topological insulator
Edge current

Laser-driven magnetization(Low-freq. Inverse Faraday effect)

Quantum spin systems
2D Dirac electronsT. Oka and H. Aoki, PRB(R)(2009).

𝑯−𝟏 , 𝑯𝟏
𝑯𝐞𝐟𝐟 ≃ 𝑯𝟎 ++ 𝑂 𝜔−2𝝎
Laser-driven spin chirality

Multiferroics

S. Takayoshi, H. Aoki, and T. Oka, PRB(2014). MS, S. Takayoshi, and T. Oka, PRL(2016).
S. Takayoshi, MS, and T. Oka, PRB(2014).

Remarks
Truncated effective Hamiltonian is valid only in a short time period.
T. Kuwahara, T. Mori, and K. Saito, Ann. Phys.(2016); PRL(2016).

In general, periodically-driven closed systems are eventually heated up.

Application ofFloquet engineering in multiferroic magnets
M. Sato, S. Takayoshi, and T. Oka, PRL117, 147202 (2016).

Shintaro Takayoshi(Max Planck Institute, Dresden)

Takashi Oka(Max Planck Institute, Dresden)

Magnetoelectric coupling
Dielectrics

Magnets+ + + + +Magneticmoment(electron spin)- - - - - Electric
polarization+ + + + +- - - - Response to magnetic field

Response to electric field

Multiferroics
Materials where electric polarization and magnetic
degree of freedom are strongly coupled with each other.
電気磁気効果Kimura, et al, Science (2003).

Magnetoelectric (ME) effects (cross correlation)Electric field

Magnetic order is controlled

Magnetic field

Polarization is controlledME couplings between electron spin and electromagnetic field
Anti-symmetric magneto-striction type
(Inverse Dzyalshinskii-Moriya (DM) effect)

Symmetric magneto-striction type(phonon-driven effect)

Katsura, Nagaosa, Balatsky (2005), Mostovoy (2006),
Tanabe, Moriya, Sugano (1965), etc.

Electric polarization

jVector spin chirality

kjk

Multiferroic materials are coupled to external electric fields
If we apply lasers to multiferroic magnets,
richer physics is expected compared to usual magnetic insulators.

TbMnO3 (S=2)

A multiferroic compoundwith anti-symmetric magneto
striction type couplinge.g., Yamasaki, et al, (2007)

RMnO3 (S=2)

multiferroic compoundswith dynamical symmetricmagneto-striction type couplingExperiments: Pimenov, et al, (2006).
Theory: Mochizuki, et al, (2010).
Miyahara, et al, (2008).
Cheong, et al, (2009).

Photo absorption spectra

Spiral order and polarization
appear simultaneously.

Electromagnons

We consider quantum multiferroic magnets with an inverse DM coupling

Anti-symmetric magneto-striction type(inverse DM effect)
Electricpolarization

Vector spin chirality

jk

Symmetric magneto-striction typejk

Typical interactions between two electron spins(1) Exchange interaction (inner product of 2 spins)

J>0 (antiferromagnetic)
J<0 (ferromagnetic)

Collinear spin structures favored
ジャロシンスキー・守谷(2) Dzyaloshinskii-Moriya (DM) interaction (outer product of 2 spins)
DM vector

Angle between 2 spins tends to be p/2
in the plane perpendicular to DM vector
Coplanar spiral spin structures favoredDz

Laser-induced DM interaction and spin current
Quantum multiferroics with anti-symmetric ME coupling

As a simple example, we consider a two-spin multiferroic systemz

Elliptically (circularly)
polarized laser

SzSy

circularlinear

Sx

Exchangecouplingxq
Electric polarizationyHamiltonian
Spin
Electric-field Magnetic-fieldsystem term
term
Coupling between polarization and electric field

Vector spin chirality
Dynamical Zeeman coupling

We assume that the ME coupling survives in finite-frequency regime.
Exchange interaction J = 0.1~10meV = 1~100T
Strong laser in THz regime b = 0.1~1T
Suitable multiferroic magnet a ~ ba/J and b/J canachieve the values of 0.1

Effective Hamiltonian for the case of circularly polarized laser

Right/left circularly polarization

Synthetic
Effective
DM interaction Zeeman term

Laser-induced terms
Dynamical system
Circularly
polarizedlaser

ME coupling

Static system
Floquet mapping
1/W expansion
Effective
magnetic
field

Synthetic
DM interaction

Synthetic DM interaction emerges due to circularly polarized laser.
Sign of DM vector can be changed by tuning helicity of laser.

Generalization to many-spin systems
example: 1D spin chain

Synthetic DM interaction

Antiferromagnetic chain
Spiral order is expected

Ferromagnetic chain
Magnetic field

Chiral-soliton-lattice order isexpected (in classical spin case)

Numerical analysis of Schrodinger equation
for multiferroic S=1/2 spin chains (finite size) under lasers

zSzSx
AFx

Note that the anisotropy or static magnetic field is necessary
in order to realize [H0,HB] = non-zeroNumerical results for finite-size spin chains with D=0.5 and H=0Time evolution ofvector spin chirality

Magnetic anisotropy

Cf: Effective Hamiltonian for large frequency W

Synthetic DM interaction

Numerical results for finite-size spin chains(1) Time dependence of <Vx>

D=0.5

d=0Clockwise

d=pAnticlockwise

d=p/2
Linearly polarized(2) Time average of <Vx>

The chirality <Vx> has a peakaround W ~ H when we apply
a static magnetic field H.
(Analog of magnetic resonance)

D=0.5 and H/J=0

D=1 and H/J=1.6

How to detect laser-driven DM interaction or vector chirality

Measurement of spin current
driven by spatially modulated laser

Pump-probe experimentSpace-time resolved observationof laser-driven magnetization(spin-wave) dynamics
T. Satoh(Kyushu Univ.)Laser-induced DM interaction and spin current in multiferroics
M. Sato, S. Takayoshi, and T. Oka, PRL117, 147202 (2016).

Dynamical system
MEcoupling

Static system
Effectivemagnetic field

Circularly
polarized laser

Synthetic DM term
FloquetMultiferroic magnet wheremappingzvector chirality V is coupled
to the electric field of laser

Multiferroic magnet
with synthetic DM term Vx

DM term Vx can be controlled by tuning ellipticity and helicity of laser

Theory for Floquet engineeringin (dissipative) classical (or ordered quantum) systems
S. Higashikawa, H. Fujita, and M. Sato, arXiv:1810.01103.

Sho Higashikawa(PhD student @ Dept. Phys. Univ. Tokyo)

Hiroyuki Fujita(PhD student @ ISSP, Univ. Tokyo)

Time evolution of Floquet engineering
Dynamics of a periodically-driven closed quantum system𝒊𝝏𝒕 𝝍 = 𝑯 𝒕 𝝍(𝑯 𝒕 + 𝑻 = 𝑯 𝒕 )is described by a static effective Hamiltonian

Frequency
𝝎 = 𝟐𝝅𝑻

Temperature𝒊𝝏𝒕 𝝍 = 𝑯𝐞𝐟𝐟 𝝍Laser,
Mechanicaloscillation,etc.

Material A𝑯(𝒕)New Material B

Disorderedphases
Orderedphases

Laser-driven
𝑯𝐞𝐟𝐟phases
Laser amplitude,Laser frequency, etc.

Magnetic field,Voltages,
Pressure, etc.

Floquet theorem
Time evolution of a periodically driven system 𝑖 𝜕𝑡 𝜓 = 𝐻 𝑡 𝜓= effective dynamics exp −𝑖𝐻eff 𝑡 + small oscillation 𝑒 𝑖𝐾 𝑡
𝑈 𝑡 ≔ 𝒯 exp

𝑡−𝑖 ‫׬‬0 𝐻 𝑡 ′ 𝑑𝑡 ′= 𝑒 𝑖𝐾 𝑡 𝐞𝐱𝐩 −𝒊𝑯𝐞𝐟𝐟 𝒕 𝑒 −𝑖𝐾 0≃ 𝐞𝐱𝐩 −𝒊𝑯𝐞𝐟𝐟 𝒕

𝑯𝐞𝐟𝐟 : Effective Hamiltonian Small, fast oscillation(time independent) M. Bukov, A. Polkovnikov, Adv. Phys. 64, (2015).
T. Mikami, N. Tsuji, T. Oka, et al, PRB(2016).
A. Eckardt, RMP (2017). etc.

Floquet-Magnus (1/w) expansion for 𝑯𝐞𝐟𝐟
For a harmonic drive: 𝐻 𝑡= 𝐻0 + 𝐻1 𝑒 −𝑖𝜔𝑡 + 𝐻−1 𝑒 𝑖𝜔𝑡

𝑯−𝟏 ,𝑯𝟏𝑯𝐞𝐟𝐟 ≃ 𝑯𝟎 ++ 𝑂 𝜔−2𝝎Time-averaged Hamiltonian

where 𝑯𝒏= 𝑻−𝟏

𝝎 = 𝟐𝝅𝑻𝑻‫𝒕𝝎𝒏𝒊𝒆)𝒕(𝑯 𝒕𝒅 𝟎׬‬Leading correction of O(1/w)

Equation of Motion for driven classical systems

Difficulty in classical systems
Equation of motion (EOM) is generally nonlinear and non-periodic
Floquet theorem is NOT applicable※ Floquet theorem is applicable to only linear differential equatione.g., Overdamped Langevin equation : 𝒙ሶ = −𝝁 𝜵𝑼 𝒙+𝝃Potential force:
Random Force:(generally)Nonlinear in 𝒙 Non-periodic in 𝒕
Even in quantum systems,
EOMs of ordered systems are often given by classical differential Eqs.
e.g., Gross-Pitaevskii Eq for BECs :𝝋ሶ = −𝒊𝜵𝟐 𝝋−+ (𝝁 + 𝒈|𝝋|𝟐 )𝝋𝟐𝒎

Extended Floquet technique to classical EOMs (if realized)
would be applicable to a broad class of physical, chemical, biological systems.

How to overcome the difficulty of classical EOMHow to overcome the difficulty : Set-up
Equation of motion for classical variables 𝝓 = 𝝓𝜶𝒅𝝓𝜶= 𝒇𝜶 𝝓, 𝒕𝒅𝒕+ σ𝜷 𝒈𝜶𝜷 𝝓, 𝒕 𝒉𝜷 𝒕Potential force, friction, etc.: Random force: Diffusion
𝒉𝜶 𝒕 𝒉𝜷 𝒕′ = 𝟐𝑫𝜹𝜶𝜷 𝜹 𝒕 − 𝒕′deterministic motion
Periodicity: 𝒇 𝝓, 𝒕 = 𝒇 𝝓, 𝒕 + 𝑻 , 𝒈 𝝓, 𝒕 = 𝒈 𝝓, 𝒕 + 𝑻e.g., Underdamped Langevin equation 𝝓 = 𝒗, 𝒙𝒅𝒙
= 𝒗
𝒅𝒕
𝒅𝒗
𝒎 = −𝜸𝒗 − 𝛁𝑼 𝒙
𝒅𝒕+𝝃Deterministic motion Random motion
Damping, potential force Diffusion

How to overcome the difficulty : Use of master equation!
Time-periodic equation of motion (EOM)𝜙ሶ = 𝑓 𝜙, 𝑡 + 𝑔 𝜙, 𝑡 ℎ 𝑡→ non-linear & non-periodic

Time-periodic master equation
𝜕𝑡 𝑃 = 𝜕𝜙 ℱ 𝑡 𝑃 + 𝜕𝜙 𝜕𝜙 𝒟 𝑡 𝑃→ linear & periodic(neglecting higher-order terms)

Floquet theorem
FM expansion

Effective EOM𝝓ሶ = 𝒇𝐞𝐟𝐟 𝝓 + 𝒈𝐞𝐟𝐟 𝝓 𝒉 𝒕

Effective master equation
𝝏𝒕 𝑷 = 𝝏𝝓 𝓕𝐞𝐟𝐟 𝑷 + 𝝏𝝓 𝝏𝝓 𝓓𝐞𝐟𝐟 𝑷(neglecting higher-order terms)

Master equation(non-Hermitian Schrodinger equation)“Hamiltonian”

Fourier transformationFloquet–Magnus expansion
for the time-evolution operator(1/w)0(1/w)1(1/w)2

Application of Floquet method
for classical systems

Example 1 : First-order FM expansion
General form of EOM for classical variables 𝝓 = 𝝓𝜶𝒅𝝓𝜶= 𝒇𝜶 𝝓, 𝒕 + σ𝜷 𝒈𝜶𝜷 𝝓, 𝒕 𝒉𝜷 𝒕𝒅𝒕
Potential force, friction, etc.
Random force (from environment)

EOM of classical particles (or field) under a harmonic drive w/o diffusion𝝓ሶ = 𝒇 𝝓, 𝒕 = 𝒇𝟎 𝝓 + 𝒇𝟏 𝝓 𝒆−𝒊𝝎𝒕 + 𝒇−𝟏 𝝓 𝒆𝒊𝝎𝒕𝒊→ Effective equation of motion : 𝝓ሶ = 𝒇𝐞𝐟𝐟 𝝓 = 𝒇𝟎 𝝓 + 𝒇𝟏 𝝓 , 𝒇−𝟏 𝝓𝝎𝑨, 𝑩 𝐜𝐥 ≔𝝏𝑩⋅𝝏𝝓𝑨−𝐜𝐥𝝏𝑨⋅𝝏𝝓Analogous to a quantum system
𝑯 𝒕 = 𝑯𝟎 + 𝑯 𝟏 𝒆−𝒊𝝎𝒕+ 𝑯−𝟏 𝒆𝒊𝝎𝒕

𝑯−𝟏 ,𝑯𝟏→ Effective Hamiltonian : 𝑯𝐞𝐟𝐟 ≃ 𝑯𝟎 +𝝎𝑩Example 1 : First-order FM expansion
Kapitza Pendulum (逆立ち振り子)

From YouTube

FM (1/w) expansion
for the master eq.

Effective EOM

EOM for Kapitza Pendulum :Friction

Fast oscillation𝜃ሶ = 𝑣𝑎𝜔𝑣ሶ = −𝛾𝑣 − 𝜔02 sin𝜃 − 2𝑙𝜔
02sin 2𝜃New potential (time-indep)

Example 1 : First-order FM expansion
Kapitza Pendulum (逆立ち振り子)
Nonequilibrium steady state (𝒕 → ∞)Short-time dynamics
𝜃=𝜋

𝜃=0
𝜃=𝜋

𝜃=0

FM expansion of open classical systems converges even in a long time,
if systems have sufficiently strong friction or thermal fluctuation.
(in contrast to FM expansion of closed quantum systems)

Example 2 : Classical many-spin systems
Landau-Lifshitz-Gilbert(LLG) equation
Effective field Random field Gilbert damping(Torque)(Temperature)

tJ/ℏ=1 : 0.1-1 pico second

Our set-up: laser-induced magnetization
Circularlypolarizedlaser
2D ferromagnetic
Heisenberg model

Staticmagnetic field

𝑩𝒅 𝑡 = 𝑩1 𝑒 −𝑖𝜔𝑡 + 𝑩−1 𝑒 𝑖𝜔𝑡

Quantum version
S. Takayoshi, H. Aoki, and T. Oka, PRB(2014).
S. Takayoshi, MS, and T. Oka, PRB(2014).

Example 2 : Classical many-spin systems
FM expansion up to O(1/w2)

∆𝑆 𝑧
∆𝑆 𝑧

Short-time dynamics

Nonequilibrium steady state (𝒕 → ∞)

マルチフェロイクス(強誘電磁性体)

Example 3 : Driven multiferroics described by LLG eq.
FM expansion up to O(1/w)
H t = − ෍ 𝐽 𝒎𝒓 ∙ 𝒎𝒓′ + 𝑔𝑚𝑒 𝑬 𝑡 ∙ 𝒆𝒓,𝒓′ × 𝒎𝒓 × 𝒎𝒓′ − 𝑔𝜇𝐵 𝑩(𝑡) ∙ ෍ 𝒎𝒓𝒓,𝒓′𝒓

Magnetoelectric (ME) coupling(inverse DM effect)

Zeeman coupling

Circularlypolarizedlaser

Quantum version
MS, S. Takayoshi, and T. Oka, PRL(2016).

(𝒎𝒓 × 𝒎𝒓′ )𝒙

Nonequilibrium steady state (𝒕 → ∞)SummaryLaser-induced DM interaction and spin current in multiferroics
M. Sato, S. Takayoshi, and T. Oka, PRL117, 147202 (2016).

Dynamical system
MEcoupling

Static system
Effective
magnetic field

Circularlypolarized laser

Multiferroic magnet wherevector chirality Vz is coupledto the electric field of laser

Synthetic DM term
Floquetmapping

Multiferroic magnet
with synthetic DM term Vx

DM term Vx can be controlled by tuning ellipticity and helicity of laserWe develop FM expansion for periodically-driven classical systems
S. Higashikawa, H. Fujita, and M. Sato, arXiv:1810.01103.

・FM expansion of classical EOM→ obtained from FM expansion of master equations𝝓ሶ = 𝒇 𝝓, 𝒕 = 𝒇𝟎 𝝓 + 𝒇𝟏 𝝓 𝒆−𝒊𝝎𝒕 + 𝒇−𝟏 𝝓 𝒆𝒊𝝎𝒕𝝓ሶ = 𝒇𝐞𝐟𝐟 𝝓 = 𝒇𝟎 𝝓 +𝒊𝒇𝟏 𝝓 , 𝒇−𝟏 𝝓𝝎(1) Wide application from classical to quantum
e.g. Langevin, Gross-Pitaevskii (GP), time-dep Ginzburg-Landau (GL),Landau-Lifshitz-Gilbert (LLG) eqs.

(2) Convergent FM expansion for driven dissipative case→ accessible to non-equilibrium steady state𝐜𝐥