2020/9/24

物質科学と高エネルギー物理が拓く非平衡物理のフロンティア

岡 隆史
東大 物性研究所 (2020/1/1より)マックスプランク複雑物理研究所（10月末まで）
マックスプランク固体物理化学研究所（10月末まで）

物質科学と高エネルギー物理が拓く非平衡物理のフロンティア
非平衡物理が大切な理由： 状態の劇的な変化と制御

平衡：半導体

非平衡：非線形デバイス
や LED

応用：コンピュータ etc

物質科学と高エネルギー物理が拓く非平衡物理のフロンティア
非平衡物理が大切な理由： 状態の劇的な変化と制御

グラフェン

平衡：Dirac電子

応用：
光コンピュータ
非平衡：光誘起ホール効果（フロッケ・トポロジカル絶縁体） 超高速エレクトロニクス？

物質科学と高エネルギー物理が拓く非平衡物理のフロンティア

今日の話では次の３分野の連携を例に見ていく

物性実験

物性理論

高エネルギー物理

また、連携には• 面白いサイエンス• 出会いの場（人と人のつながり）
• 辞書
が必要

物性物理の３つの柱

物質合成

革新的実験技術

量子物質（ディラック・ワイル半金属等）

超高速分光、時間分解ARPES
アイデア

光誘起相転移 (那須先生)
分割・普遍化 （かみくだく）
熱化、フロッケ・エンジニアリング、相制御、非エルミート制御

連携
•
•
•

冷却原子系
高エネルギー物理
数理物理・分類理論→ 学術変革時間分解角度分解光電子分光（ARPES)

2D Dirac電子@ トポロジカル絶縁体（Bi2Se3）表面
電子状態の観測と格子振動
Shen group @ Stanford

V.

CORRELATED ELECTRONS DRIVEN BY ELECTRIC FIELDS

時系列に分解“distance” from equilibrium

Ultrafast phenomena by short pulses
photo-doping
2

Floquet prethermalization
heating

4 oscillation

prethermalizationg.s.

thermalization
intra&inter-band relaxation
doublon decay

熱化

laser

state

open system
Relaxation (electron + phonon)

8

During laser pulse
el-ph scattering
Floquet states
5 coherent phonon pumping
multi-photon absorption
and tunneling (Schwinger mechanism) 1 lattice structural dynamics
fs = 10-15 s

レーザー照射中
フロッケ状態トンネル励起THz pulse and non-linear transport (Ω<<ΔMott)

tance” from equilibrium

3

g.s.

ps = 10-12 stime (log)Franck-Condon

Off state

On state

開放系としての振る舞い
非エルミート性On state
6 Mott RRAM

electric fieldstate

Excitation heating
impurity state ionization

Off state
Carrier multiplication heating
数字については下の論文を参照
7 electron avalanche (cascade)structuraldestortion
Oka,
Kitamura, Annu. Rev. Condens. Matter Phys.
10, 387(2019)
Relaxation

フロッケ状態（周期駆動系）
YoutubeKapitza’s pendlum

•
•
•

状態の制御
量子、古典
線形、非線形物性理論（場の理論） (a)高エネルギー物理(b)（超弦理論・重力）

time

time

物性実験

固体電子系
Wang, Gedik et al. Science ’13
Mciver, et al. Nat. Phys. ‘20

q

q

E-field

冷却原子系
Jotzu, Esslinger et al. Nature ’14

Oka-Aoki
2009
Figure 1. Schematic illustration of the Schwinger mechanism. (a) In static electric fi

the quark-antiquark pair is created from the vacuum and are pulled away from each o
2D Dirac+ 円偏光by the field. (b) In rotating electric fields, the created quark and antiquark rotate aro
each other as they are separated.
(a)

(b)
E

E

quark

meson
polarization
gluon

gluon

Figure 2. Schematic illustration of the solutions. (a) Minkowski embedding (insula

３D Dirac+are円偏光
ホログラフィック・フロッケ状態
Quarksconfined and only meson
excitation exists in the fermion sector. Because orotating electric field, vacuum polarization is induced as a coherent excitation of mes
Ebihara, Fukushima, Oka,
2016
Hashimoto, Kinoshita, Murata, Oka, 2017
which rotates with the same frequency as the field. This rotation induces an oscilla
Bucciantini, Roy, Kitamura,Oka, 2016 Kinoshita,
Murata,2018
current. (b) Blackhole embedding(metal):GaplessOka,
excitationof quarks and anti-qu
Takayoshi, Wu, Oka, 2020
forms a plasma and the system is fully deconfined.

物性理論（場の理論） (a)高エネルギー物理(b)（超弦理論・重力）

time

time

物性実験

固体電子系
Wang, Gedik et al. Science ’13
McIver, et al. Nat. Phys. ‘20

q

q

E-field

冷却原子系
Jotzu, Esslinger et al. Nature ’14

Oka-Aoki
2009
Figure 1. Schematic illustration of the Schwinger mechanism. (a) In static electric fi

the quark-antiquark pair is created from the vacuum and are pulled away from each o
2D Dirac+ 円偏光by the field. (b) In rotating electric fields, the created quark and antiquark rotate aro
each other as they are separated.
(a)

(b)
E

E

quark

meson
polarization
gluon

gluon

Figure 2. Schematic illustration of the solutions. (a) Minkowski embedding (insula

３D Dirac+are円偏光
ホログラフィック・フロッケ状態
Quarksconfined and only meson
excitation exists in the fermion sector. Because orotating electric field, vacuum polarization is induced as a coherent excitation of mes
Ebihara, Fukushima, Oka,
2016
Hashimoto, Kinoshita, Murata, Oka, 2017
which rotates with the same frequency as the field. This rotation induces an oscilla
Bucciantini, Roy, Kitamura,Oka, 2016 Kinoshita,
Murata,2018
current. (b) Blackhole embedding(metal):GaplessOka,
excitationof quarks and anti-qu
Takayoshi, Wu, Oka, 2020
forms a plasma and the system is fully deconfined.

フロッケ理論 (ブロッホの定理)
時間周期系

離散フーリエ変換

フロッケハミルトニアンの固有値問題
e: 擬エネルギー~ m 光子吸収由来

フロッケスペクトル: Dirac電子 + 円偏光
coupling to AC field

時間依存シュレディンガー方程式

truncated at m=0,+1, -1 for display

フロッケスペクトル: Dirac電子 + 円偏光

TO, Aoki 20090-photon absorbed state0-photon absorbed statekx

フロッケスペクトル: Dirac電子 + 円偏光

TO, Aoki 2009

1-photon absorbed state0-photon absorbed state

1-photon absorbed state0-photon absorbed statekx

フロッケスペクトル: Dirac電子 + 円偏光

TO, Aoki 2009

1-photon absorbed state0-photon absorbed state-1-photon absorbed state

1-photon absorbed state0-photon absorbed state-1-photon absorbed statekx

フロッケスペクトル: Dirac電子 + 円偏光

TO, Aoki 2009

1-photon absorbed state0-photon absorbed state-1-photon absorbed state

1-photon absorbed state0-photon absorbed state-1-photon absorbed stateフロッケスペクトル: Dirac電子 + 円偏光

TO, Aoki 2009

1-photon absorbed state0-photon absorbed state-1-photon absorbed statenear Dirac point

2k

1-photon absorbed state0-photon absorbed state

Dirac gap

-1-photon absorbed stateフロッケ・トポロジカル絶縁体
Dirac電子 + 円偏光 ＝ 量子ホール状態（Chern絶縁体）

円偏光照射によってトポロジーを制御→ フロッケ・エンジニアリング

Oka, Aoki ’09,
Lindner, Rafael, Galitski ’11,
Kitagawa, TO, Fu, Brataas, Demler ’11

超高速エレクトロニクスへの応用
McIver, et al. Nat. Phys. ‘20

円偏光レーザー

グラフェン

ホール電流
ホール伝導率物性理論（場の理論） (a)高エネルギー物理(b)（超弦理論・重力）

time

time

物性実験

固体電子系
Wang, Gedik et al. Science ’13
McIver, et al. Nat. Phys. ‘20

q

q

E-field

冷却原子系
Jotzu, Esslinger et al. Nature ’14

Oka-Aoki
2009
Figure 1. Schematic illustration of the Schwinger mechanism. (a) In static electric fi

the quark-antiquark pair is created from the vacuum and are pulled away from each o
2D Dirac+ 円偏光by the field. (b) In rotating electric fields, the created quark and antiquark rotate aro
each other as they are separated.
(a)

(b)
E

E

quark

meson
polarization
gluon

gluon

Figure 2. Schematic illustration of the solutions. (a) Minkowski embedding (insula

３D Dirac+are円偏光
ホログラフィック・フロッケ状態
Quarksconfined and only meson
excitation exists in the fermion sector. Because orotating electric field, vacuum polarization is induced as a coherent excitation of mes
Ebihara, Fukushima, Oka,
2016
Hashimoto, Kinoshita, Murata, Oka, 2017
which rotates with the same frequency as the field. This rotation induces an oscilla
Bucciantini, Roy, Kitamura,Oka, 2016 Kinoshita,
Murata,2018
current. (b) Blackhole embedding(metal):GaplessOka,
excitationof quarks and anti-qu
Takayoshi, Wu, Oka, 2020
forms a plasma and the system is fully deconfined.

場の理論

重力

ホログラフィ

••••熱化
カノニカル分布の普遍性
情報の喪失（励起の詳細を忘れる）
輻射と緩和

•
•
•
•

ブラックホールの形成
No-hair theorem
情報の喪失
ブラックホールの消失R[dρ2 + ρ2 dΩ23 +222
ρ + w1 + w2
場の理論
重力
where R is the AdS radius. Hereafter, we will take a unit of R = 1 to s
expressions.
+

曲がった５次元時空の非線形電磁気学
Dynamics of the D7-brane is describedby the Dirac-Born-Infeld (3D Dirac粒子＋ボソン

quark(electron)
meson(exciton)q

S = −T7!

"d σ −det[hab + 2πα! Fab ] ,8anti-quark
4(hole)Pair annihilation of Weyl points with opposite chiralities is one way to dehab: AdS metric

Another mechanism is interaction: It was proposed that forward scattering canq [48].
spectrumgluon(Coulomb force)

運動方程式（非線形マクスウェル方程式）

7

D3/D7系 Karch-Katz ‘02c.f. reviews
Erdmenger et al. “Mesons in gauge/gravity duals”
夏梅誠「超弦理論の応用」

地平線の形成と有効ホーキング温度
非線形マクスウェル方程式

線形化
光の向き（光円錐）
電場により変化cf) BH形成 (Hawking-Page transition)

inside the BH

arrows = -z direction light coneboundary

AdS direction
Hashimoto TO JHEP ‘13

物性理論（場の理論） (a)高エネルギー物理(b)（超弦理論・重力）

time

time

物性実験

固体電子系
Wang, Gedik et al. Science ’13
McIver, et al. Nat. Phys. ‘20

q

q

E-field

冷却原子系
Jotzu, Esslinger et al. Nature ’14

Oka-Aoki
2009
Figure 1. Schematic illustration of the Schwinger mechanism. (a) In static electric fi

the quark-antiquark pair is created from the vacuum and are pulled away from each o
2D Dirac+ 円偏光by the field. (b) In rotating electric fields, the created quark and antiquark rotate aro
each other as they are separated.
(a)

(b)
E

E

quark

meson
polarization
gluon

gluon

Figure 2. Schematic illustration of the solutions. (a) Minkowski embedding (insula

３D Dirac+are円偏光
ホログラフィック・フロッケ状態
Quarksconfined and only meson
excitation exists in the fermion sector. Because orotating electric field, vacuum polarization is induced as a coherent excitation of mes
Ebihara, Fukushima, Oka,
2016
Hashimoto, Kinoshita, Murata, Oka, 2017
which rotates with the same frequency as the field. This rotation induces an oscilla
Bucciantini, Roy, Kitamura,Oka, 2016 Kinoshita,
Murata,2018
current. (b) Blackhole embedding(metal):GaplessOka,
excitationof quarks and anti-qu
Takayoshi, Wu, Oka, 2020
forms a plasma and the system is fully deconfined.

Ω

i=1,2,3

フロッケ・ワイル半金属
H="

i=1,2,3→ k3 σ3 + (k+ )γ 0 γ i (p̂i − Ai )A(t) = A(cos Ωt, sin Ωt)
Bucciantini, Roy, Kitamura, Oka, ‘172n∓1 +

H(t)
= −J
2n±1
−

σ + (k− )σH(t) = −J

+
−
H
=
k
σ
+
k
σ
+
k
σ
Weyl
3
3
+
−
e
フロッケ
擬エネルギー!"e−iAij (t) c†i cj + h.c.

!ij"

!"

e−iθ(t)A cos Ωt c†i+1 ci + h.c.

ipy = 0

jy

pz

#

En

ディラック点がワイル点に分裂し、他のワイル点と対消滅

#

H=

"

i=1,2,3

ホログラフィック・フロッケ・ワイル半金属
σi (pˆi − Ai )
重力

場の理論
E
J"
→ J00( i )JH=γ γΩ (p̂i − Ai )3

i=1,2,3

A(t) = A(cos Ωt, sin Ωt)→ k3 σ3 + (k+ )2n∓1 σ + + (k− )2n±1 σ −

•
•
•

#
•
!"
円偏光レーザーによる周期駆動−iA (t) †H(t) = −Jeci cj + h.c.
ホール応答
!ij"
HWeyl = k3 σ3 + k+ σ + + k− σ −ギャップの制御（絶縁体・金属転移） •#•!"−iθ(t)A cos Ωt †ij

H(t) = −Jeci+1 ci + h.c.

回転電場による周期駆動（回転座標系で静止）
ホール応答
D7ブレーンの埋め込みの変化
Minkowski embeddingi

1.5

bedding

Critical em

py = 01Black hole embedding
Effective
horizon

0.5

0

jy

0

0.5

1

1.5

2

2.5

3

Hashimoto,Kinoshita,
Murata,2017
Figure 6. Typical profilesof the D7-branein the (ρ, w)-plane
for Oka,
E = 1 and
Ω = 1. F
Kinoshita,
Murata,Oka,
2018 horizon.
black hole embeddings,
they are shown
only outside
the effectiveEn

1. The brane reaches ρ = 0, the axis of S 3 wrapped by the D7-brane, and th

ホール応答Hashimoto-Kinoshita-Murata-TO JHEP‘17

E,W
j=sxyffparallel current
Hall current

parallel current

Hall current

σ ∝ TH(a)非平衡相図
E-field(b)

time

Figure 1. Schematic illustration of the Schwinger mechanism. (a) In static electric fields,
the quark-antiquark pair is created from the vacuum and are pulled away from each otherby the field. (b) In rotating electric fields, the created quark and antiquark rotate aroundeach other as they are separated.

DC-Schwinger limit 0.6
c.f. S. Nakamuraq

0.5

Conductive phase

time

quantum tunneling

(a)

q

(b)

0.4
E-field(plasma)
E

E

0.3

quark

meson

Schematic illustration of the Schwinger mechanism. (a) In static electric fields,
antiquark pair is created from thevacuum and are pulledpolarization
away from each other(hadron)d. (b) In rotating electric fields, the created quark and antiquark rotate around
gluon
gluon
as they are separated.

0.2

0.1 Insulator phase

Figure 2. Schematic illustration of the solutions. (a) Minkowski embedding (insulator):(b)
Quarks are confined and only meson excitation exists in the fermion sector. Because of therotating electric field, vacuum polarization is induced as a coherent excitation of mesons,Ewhich rotates with the same frequency as the field. This rotation induces an oscillating
current. (b) Blackhole embedding (metal): Gapless excitation of quarks and anti-quarks
meson resonanceforms a plasma and the system is fully deconfined.

(a)

0.0
0
E

2

4

6

8

10

(exciton energy)

meson

quark

q̄ pairs canbe created
without any phase”
energeticに、
cost and,thus, the system
is を
alwaysFigure 6.
”conductive
phase”
を”metallic”Insulator
phase””insulatpolarization
in a metallic (conductive) phase [9] (Fig. 2 (b)), and2 an anomalous quantum Hall
phase”に Nonequilibriumphase diagram in the (Ω/m, E/m )-space. The phase bounstate, i.e., a state where a current can be induced that flows perpendicular to thegluon of critical embedding solutions. The region above t
gluonline) is givenary (solid
by aelectricfamilyappliedfield, was realized since the rotating electricfield breaks
timereversal
S Kinoshita,
K Murata,
T Oka
JHEP ‘18

物性実験

物性理論
（場の理論）高エネルギー物理（超弦理論・重力）固体電子系
Wang, Gedik et al. Science ’13
McIver, et al. Nat. Phys. ‘20
冷却原子系
Jotzu, Esslinger et al. Nature ’14

Oka-Aoki 2009

2D Dirac + 円偏光

円偏光中のSchwinger limit?
0.6

Conductive phase

0.5
0.4
0.3
0.2
0.1 Insulator phase0.00

非断熱遷移における幾何学効果
c.f. Fukushima, Shimazaki 2020

３D Dirac + 円偏光Figure 6.

2

4

6

8

10

ホログラフィック・フロッケ状態”conductive phase”を”metallic phase”に、”Insulator phase”を”inphase”に Nonequilibrium phase diagram in the (Ω/m, E/m2 )-space. The phaseary (solid line) is given by a family of critical embedding solutions. The region absolid line is in the metallic phase and that below is the insulator phase. The dashe
represents E/m2 = π2 (Ω/m)−2 .

Ebihara, Fukushima, Oka, 2016
Hashimoto, Kinoshita, Murata, Oka, 2017
Bucciantini, Roy, Kitamura, Oka, 2016 Kinoshita, Murata, Oka, 2018
Takayoshi, Wu, Oka, 2020

the axis ρ = 0, and for the black hole embedding b0 and w0 at the effective hFigure 2. Schematic illustration of
the statesin the3. (a) Insulator
quark
mass
mqphaseand diagram
quark Fig.
condensate#Om $ in the boundary theory
phase: Only mesons and gluons can be excited since the quark-antiquarks are gapped.
√"#1/2
Vacuum polarization is induced as a collective excitation of mesons and follows
λ the externalNc λ=E(t) , JSQCD (t) = 5/2 3 J (t)rotating electric field. This induces a non-dissipativeErotatingi.e., polarization
SQCD (t)current,
22π
2 πcurrent, in the xy-plane with no Joule heating. (b) Conductive phase: Gapless excitation√"#of quarks and antiquarks is possible and a current is induced byλthe external field. The Nc λmq =mcondensate:, #Om $ An
= − 3/2 3 c .
current is dissipative with finite Joule heating. (c) Vector
meson Floquet
2π 2
2 π
insulator solution that has a coherently rotating current even when the external electric• フロッケ状態、非エルミート状態のトポロジーの分類理論
field is absent.

まとめ

物質科学と高エネルギー物理が拓く非平衡物理のフロンティア

Note that we use the complex notation for the electric field and cu
imaginaryis parts
ofbut
E and
J correspondIn the case of QCDs, vacuum polarization
nothinga coherent
excitation toof x- and y-components of
(c)
(a)
(b)
and current,such
as E a=rotating
Ex + iEJ = Jx + iJy .
y andfield,
vector mesons !ψ̄γi ψ". In the insulator
phase, byapplying
electric# will follow it as P
# ∝ E.
# The insulator phase of D3/D7 is a perfect no rotating electric field
the polarization P
EE• 非平衡系特有の秩序状態
2.2is zero.
Co-rotating
and reduction
insulator and the DC-conductivity
However, in framea time dependent
field, a to an 1-dimensional
can exist
given by condensate)、フロッケ時間結晶
the time derivative of the polarization
1
0.3 polarization current
振動秩序(Floquet

(a)

(b)

0.4
0.2

(c)

Jx + iJy = !ψ̄(γx + iγyP)ψ" = |J|eiΩt−iδ .
J

0.1

(d)

quark
We focus on therotating
external electric meson
field that is described by
5
meson
J

(1.2)

P

E(t) = EeiΩtJ ,Kinoshita,
Murata,0.5
gluon
gluon field, i.e.,
This is perpendicular 0.2
to the electricδ 0= π/2.
Here,Oka,δ is 2018
introduced
to

gluon

generically parameterize the phase delay of the current relative to the electric field.
where
E is aillustration
complexof constantargument
represents
the initial
Figure2. Schematicthe states
inwhosethe phase
diagram Fig.
3. (a) InsulatorIntheinsulator
phase,sincethecurrent
isalwaysperpendiculartotheelectric
field,
00phase: electric
Only mesons
andat
can Thebe excitedsincefield
the quark-antiquarks
are gapped. with afield
t = 0.
electric
rotates anti-clockwise-5gluons
Joule heating is absent and the system is dissipation-less. A time dependent
0 oscillatis induced
as a collective
excitation
ofboundary
mesons and follows
the external
0
5
10
0
0.1
0.2
0.3
0 Vacuum
0.2 polarization
0.4
0.6• condensate
高エネルギー物理への波及？
(x, y)-plane.
Eq. (2.7),
thesystems
condition
for the gauingsuch as Eq. (1.2) theis genericallyfound Fromin periodicallydrivenrotating electric field. This induces a non-dissipative rotating current, i.e., polarizationand is called the Floquet condensate[38].
A prominent
examplestudiedtheoreti-phase: Gapless excitation
boundary= Joule∞becomescurrent,AdS
in thexy-planewithρ noheating.
(b) Conductive
cally in condensed matter physics
the antiquarks
superconducting
order
drivenby the external field. Theof quarksis andis possible
andina periodicallycurrent is induced
lattice models [39–42]. Thus,
in isthisarticle, with
we calltime
periodiccondensate
current
dissipative
finitetheJouleheating.
(c) Vector
meson iEAna|
= FloqueteiΩt condensate:.
ρ=∞
回転電場によるホログラフィック・フロッケ状態
expressed as Eq. (1.2) the electric
inducedcondensateof vector
mesons.
insulatorfield
solutionthat Floquet
has a coherently
rotating
current
even whenΩ the external electric
is absent.
However, this state in the field
insulator
phase is nothing special nor surprising; It is aWe
definebytwo
real variablesρ) example
and χ(t, ρ) as
state where the order parameter is induced
an external
field. カーブラックホールとの共通点or違い？
A b(t,
similarlasercase of magnetic
QCDs, vacuum
polarization
is nothing but
would be a paramagnet in In
an theoscillating
field where
the magnetization
willa coherent excitation ofi[Ωt+χ(t,ρ)]
vector mesons !ψ̄γi ψ". In the insulator phase,a rotating
electric
重イオン衝突時のa(t,byρ)applying= b(t, ρ)e, field,
simply follow the field. 生成される粒子分布###
電場は回転しているthepolarization
P willcriticalfollow value,
it as Pan∝ insulator-to-conductor
E. The insulator phase of D3/D7 is a perfect
When the field strength
exceeds
a threshold
はカイラル対称性を破る
回転座標系（内部空間）
insulator
andtakestheDC-conductivity
is zero.
However,
in aandtimethedependent
a respec
where
b and
χ In
represent
the amplitude
phase field,for ξa,transition(i.e.,
dielectric breakdown)place.
the conductor
phase,
asPillus3D masslessDiracfermion:(a)-(c)The
production
probability
chirality=+pξ (q̃) for
Takayoshi,
Wu,
Oka,
2020
polarization current can exist given by the time derivative of the polarization

have beatter [14,electrons
ects suchhand, theprocesses
s such as
unterdiahe tunnelthe Diracg electric
the field

nonadiabatic
geometric
in a tw
FIG. 1.
Twistedeffects
LZ model:modelwith Landau-Zenera parameterq definedby
LZ tunnelingwith curvatureintaneous energy of the Hamilton
1(0.1, 1, 1) and schematicz
x picture
Ĥ(q)
=
mσ̂
+
vqσ̂
+
κ
Tunneling probability P (F )2 fo
sweep q = −Ft obtained
nu
2
(m +
vF/4)
k
j )=
(F
.(0,
1,
0),
where Pσ̂are(j(m,=expv,x, κy,! )⇡z)=is(0.1,
the
Pauli
m
|vF |
dashed,blue slope.
dotted andand v theenergyWeredusesolt
Landau-Zener formula

1. If we regard the coefficientsofm2
P (F1 ) = exp⇡.
2 2x(q) = (vq, 2 κ! v q , m)|vFas| a traje
sweep q = −F t in Eq. (1) is gdimensional (3D) space, it defines a!
curvature around the⌦ gap minimum
(
P=(F
= exp
space [Fig. 1(a)]. Theof κ!−=π0⇠E)caseLandau-Zener (LZ) Hamiltonian [7,y, nonadiabatic
nt Hamiltonian
13].

namics have besed matter [14,n that electrons
otic effects suchother hand, theabatic processes
xamples such as
and counterdiaider the tunneled by the Diracrotating electric
tory of the fieldd linearly polarts are induced,

FIG. 1.
Twistedmodel
LZ –model:(a)TwistedLandau-Zener
We demonst
TwistedLandau-Zenermodel
nonadiabatic
geometric
effects
in a two level
LZ tunnelingwithcurvaturein Hamparwithtaneousa parameterq defined
by Hamiltonian
energyof the(0.1, 1, 1) and schematic1 picture fo
z
x
2 2 y
Ĥ(q)
=
mσ̂
+
vqσ̂
+
κ
v
σ̂ , E! ) q forTunneling probability P2 (Fsweep q = −F t obtained numer
j
where
σ̂
(j =v,
x,κy, z)is thePaulimatrix,miscurvatureare (m,
)
=
(0.1,
1,
0),
(0,
1,
1)
!
and v the energy slope. We use the unit h̄dashed, blue dotted and red solid li
1. If we regard the coefficients of the Pauli
Tunneling formula1(vq, 2 κ! v 2q 2 , m) as a trajectory2x(q) =in th+ k vF/4)
dimensional (3D) space, (mit defines
a curve and κ
P (F ) = exp⇡.
curvaturearoundthegapminimuminthepa|vF |(1) is give
sweep q = −F t in Eq.
space [Fig. 1(a)]. The case of κ! = 0 correspondLandau-Zener (LZ) Hamiltonian[7,11]. effect
The i!
Nonadiabatic
geometric
m2is plotted(mneous energy of this Hamiltonianin F+
P (F ) =exp⇡exp. −orderBerry1990 (firstP(F)=
π in k) pa
The tunneling probability P (F|vF) for| a linear
Takayoshi, Wu, Oka 2020

(with general form)