(a)(b)(c)

Twisted Schwinger effect
5
0.4
and
geometric effects in quantum tunneling
0.2
0.3

Takashi Oka
0
ISSP U-Tokyo, Max Planck institute (PKS, CPfS)
0.2

0.1

0

0.5

0
Shintaro Takayoshi(PKS→ Konan Univ.)
Jianda Wu (TD Lee ins. @ Shanghai Jiao -5
Tong Univ.)

0
0.1

0.2

0.3

0

0.2

0.4

0.6

0

1

5

10

0

laser

massless Dirac fermion: (a)-(c) The production probability

Take home message(non)-linear optics

0.3

1. Energy-momentum conservation(b)(c)
2. Optical selection rule

(a)

1

(d)

5

0.4
0.2
0.2

0.1

0

0.5

e.g. Q. Ma et al. Nat. Phys. ‘17

0

0
0

0.1

0.2

0.3

0

-5
0.2

0.4

0.6

0

0
H0 + 10E(t)X

5

non-perturbative optics

H(t)1. Energy-momentum conservation
2. Nonadiabatic Geometric effectlaser⌅k (t) = h 0 ( (t))|

Takayoshi, Wu, Oka 2020p(t)iBerry' 19901

Pk ei˜kĤ( )n =0

d h n |i@ | n i

Geometric effects in near “adiabatic” time evolutionĤ( ){"n ( ), | n ( )i}R
i 0t dt"n ( (t)) i n| (t)i ' | n ( (t))ieeSolution of the td-Schrodinger equation at slow parameter change
Z t{"n ( ), | n ( )i}d h n |i@ | n i single tunneling approximation
n =
Rt
p
p
0
i 0 dt"Zn ( t(t)) i˜n| (t)i ' 1 P | n ( (t))ie
e + O( P )d h n |i@ | n i
n =
Rt
0
n
zone
1
' | n ( (t))ie i 0 dt"n ( (t)) eiBrillioin
˜n =electronn +(ii)correction
intraband effect
Example:CB
Insulator in E-fieldRtidt"n ( (t)) i˜nĤ(k +0A(t))

Rt

hole

| (t)i ' | n ( (t))ie i 0 dt"n ( (t)) ei np
| n ( (t))iee Z +t O( P )tunneling˜ = (i) dp
h n |i@ |(iii)n i +... Rtunneling(creation and annihilation)
A=-Et
t
VB
i
dt"n ( (t)) i˜n
Ĥ( )
0'
0
|
(t)i
1
P
|
(
(t))iee +On˜ =+ correctionnn{"n ( ), | n ( )i}

-//d

k

//d

LZ interference

F

E

˜n =

n + correction

n

n

n

0

R
Geometric effects in near “adiabatic”
evolution
R i t dt"n ( time(t))
i| (t)i(t))iei dt"0 ( (t)) eie n| (t)i''| | nn (( (t))ie
t
0

n

n

single tunneling approximationpp' 1| (t)i| (t)i '1

p| n ( (t))iei

P | n ( (t))ie
2

˜n =

Rt
0

i

dt"
R n ( (t)) i˜n

e
t
dt"
(
n
0

n + correctionp+ O( p)(t)) i˜ne1p+ O( P )Berry 1984
correction
1. (nonadiabatic)˜Geometricphase
n = n+
Aharonov-Anandan 1988Z t
˜=d h n |i@ | n i + . . .

Z

0

t

Applications: polarization, quantum Hall effect, TI,
˜ = geometricd h neffecti + .…. .
@|i@ |in n
HHG,
X(E)
=
ReL
0
@E
2. Nonadiabatic Geometric effect Berry 1990

(E)/Ld =E(2⇡)d 1 h✓Ev◆(d 1)/2 X
1

1⇡nE /ES
Takayoshi,
Wu,eOka 2020(d+1)/2n=1nalso Kitamura Nagaosa Morimoto 2019

Proc. Roy. Soc. London 430, 405 (1990)

ics have bematter [14,at electrons
effects suchr hand, theic processes
ples such ascounterdiathe tunnely the Diracting electricof the field

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

1. If we regard the coefficientsof thm2
P (F1 ) = exp⇡.
2 2x(q) = (vq,
κ
v
q
,
m)
as
|vF | a trajecto!
2
sweep q = −F t in Eq. (1) is giv
dimensional (3D) space, it defines a cu!
curvature around the⌦ gap minimum
i
(m
P=(F
= exp
space [Fig. 1(a)]. Theof κ!−=π0 co⇠E)caseLandau-Zener (LZ) Hamiltonian [7, 11

mely, nonadiabaticndent Hamiltonianes [13].

dynamics have bedensed matter [14,own that electrons
exotic effects suchhe other hand, theadiabatic processes
w examples such as
19] and counterdiaonsider the tunnelribed by the Diraca rotating electric
jectory of the field
and linearly polarffects are induced,

geometric effects dictate the physics.
FIG. 1.
Twistedmodel
LZ –model:(a) ScTwistedLandau-Zener
We demonstrate
TwistedLandau-Zenermodel
nonadiabatic
geometric
effects
in a two level
LZ tunnelingwithcurvaturein Hamilt
param
withtaneousa parameterq defined
by Hamiltonian Eqenergyof the(0.1, 1, 1) and schematic1 picture for q
z
x
2 2 y
Ĥ(q)
=
mσ̂
+
vqσ̂
+
κ
v
σ̂ , Eq.
! ) q forTunneling probability P2 (Fsweep q = −F t obtained numeric
j
where
σ̂
(j =v,
x,κy, z)is thePaulimatrix,misthecurvatureare (m,
)
=
(0.1,
1,
0),
(0,
1,
1),
!
and v the energy slope. We use the unit h̄ =dashed, blue dotted and red solid line
1. If we regard the coefficients of the Pauli ma
Tunneling formula1(vq, 2 κ! v 2q 2 , m) as a trajectory2x(q) =
in the t+ k vF/4)
dimensional (3D) space, (mit defines
a curve and κ! i
P (F ) = exp⇡.
curvature
the gap
param|vF |(1)inisthesweeparoundq = −Ft inminimum
Eq.
given
space [Fig. 1(a)]. The case of κ! = 0 corresponds t
Landau-Zener (LZ) Hamiltonian[7,11]. effect
The inst!
Nonadiabatic
geometric2+κmneous energy of this Hamiltonian is plotted(min Fig.
P (F ) =exp⇡exp. −orderBerry1990 (first
in k)
P
(F
)
=
π
The tunneling probability P (F|vF) for| a linear param
Takayoshi, Wu, Oka 2020

(with general form)|

Derivation by unitary transformation

Twisted LZ
LZ

Gap minimum“Untwist”+ higher order
LZ model with an effective gap

Twisted Landau-Zener model
P (F ) = exp

(m + k vF/4)2
⇡
.
|vF |

Effective gap depends
on2the speed F


m
⇡
.
|vF |

P (F ) = exp

LZ

Rectification:=

TLZ (m=0)

⌦
⇠E

for

TLZ (m≠0)

Perfect tunneling:Rii 0⌧ E X̂(s)ds
L(E) = lim
lnh0|T̂ e at
|0i
t!1 tVCounter diabaticity:Z

dk ˜ (k)ReL(E) = For
E large d ,2⇡
BZ (2⇡)ImL(E) =

E

Z

dk

1

Takayoshi, Wu, Oka 2020

ln[1

p(k)]

Schwinger effect
In strong E-field

vacuum

Dirac 1928
Sauter 1931
Heisenberg-Euler 1936
Schwinger 1951

1. pair creation
2. induced polarization+ -1. pair creation+ + 2 . polarization++ -Electric field E-

E(E)/Ld =(2⇡)d 1 h✓Ev◆(d 1)/2 X
1Schwinger effectn=1

1. pair creation

+ +
k

-

k

1

n

⇡nES /Ee(d+1)/2

m2 c3
ES == 1.3 ⇥ 1018 V/me~Creation probability (k=0)✓◆
ES
p = exp⇡
E

Schwinger limit
E >
2m
m2 c3
ES =
= 1.3 ⇥ 1018 V/me~◆
J n = @k E n✓ES
p = exp⇡
E

k ! k EA(t)> 2m
E=@t A @V
J n = @k E nn⌦k>! kA(t)Schwinger’s work
Heisenberg-Euler Effective Lagrangian

Schwinger 1951=
generating function of nonlinear responses
Vacuum decay rate (sum over tunneling channels)
pair creation+
Vacuum Polarization
X(E) =

@
ReL
@E

+ -

-

|vF |⌅(t) ⇠ eitV L(E)
Loschmidt echo and the Heisenberg-Euler
Effective Lagrangian

2
m
R
Rt
P (F )i =t exp⇡.
H((s))dsi
E0 (A(s))ds⌅(t) = h 0 ( (t))|T̂ e 0| |vF
0 ( | (0))ie 0⌅(t) ⇠ eitV L(E)For dc E-field

L(E) =

many-body ground state

R⌧
⌦
i
i
+ A(t))
limĤ(k=lnh0|T̂ e 0 E X̂(s)ds |0i⇠E
t!1 tVR
R
R
ii
H( (s))ds i ⌧ E X̂(s)ds
i : position
operatort) = h 0 ( (t))|T̂ e| 0 ( 0(0))ieĤ( )E (A(s))ds
Oka, Aoki 2005
E) = lim
lnh0|T̂ e
|0i
Z
R
i
dk
˜E(k)
t!1 tVpi
X̂(s)ds
L(E) =
lim
lnh0|
T̂
e
|0i
ReL(E) = E
t
0

t
0

0

⌧
0

i˜k
⌅k (t) = t!1
h 0 ( tV(t))| (t)i '
1
P
e
d
k
(2⇡) 2⇡{"n ( ), | BZ
n ( )i}Ĥ(k +polarization
A(t)) = nonadiabatic
Inducedgeometric phaseZ

1 ˜ (k)Z ImL(E) = JZEZ= Jdk Zdk
dk
Jdln[1 p(k)]˜ (k)@ReL(E)
=
E
dk
˜
(k)
=
d
h
|i@
|
(2⇡)
4⇡
n
ni =
ReL(E) = nE
d
BZ
X(E)
ReL
(2⇡) BZ2⇡ (2⇡) 2⇡@E
ReL(E) = E Ĥ( )
d 2⇡
(2⇡)
BZeffect Z= Landau-Zener
Schwinger
tunneling

L(E) =

t

5

⇠=+1

BZ 0

⇠= 1

d

Z
R tk vF/4)2
(m
+
itV
L(E)
dk
1
i 01
dt"n ( ✓(t)) ◆
1
dk
⌅(t)
⇠
e
P
(F
)
=
exp
⇡
.i(dn 1)/2 X
|
(t)i
'
|
(
(t))ieeEE
1n
ImL(E)
=
E
ln[1
p(k)]
= (2⇡)
E(E)/L
ln[1
P
(k)]
|vF
|
d 4⇡d ={"n ( ),ImL(E)| n ( )i}e ⇡nES /Ed
BZd1(d+1)/2
4⇡ h vn
BZ (2⇡)(2⇡)
n=1

Z
Consistent with
Schwinger
1951, agrees
with the Berry phase theory

dk
1
R
R
R m
Z
p
p
of
polarization
Resta,
King-Smith
Vanderbilt
at
E→
0.
i
H(
(s))dsi
E (A(s))ds
i
dt"
(
(t))
i˜
P
(F
)
=
exp
⇡
.
⌅(t)
⇠
e
E =⌅(t)ln[1p(k)]= h '0 ( 1(t))|p|T̂ en ( (t))ie itV L(E)||vF0 (| m(0))ie| (t)i+ O( p)ce
d h d |i@ | it
itV 0L(E)tnnn⌅(t) ⇠ e

t
0

2

n

E =

2 3

n

t
0

0

= 1.3 ⇥ 1018 V/m

Schwinger effect by circularly polarized field“Twisted Schwinger effect”

Takayoshi, Wu, Oka 2020

+ rotating E-field(circularly polarized field)

+ -

+ + -

E = E0 (cos Wt, sin Wt, 0)

(8)

tion
(k)
see
atio
folthe

(9)

as

(10)

fermions.
We which
introduce
as gauge potenproductionin the weak
field regime,
leads theto anfield
asymptotic
tial A = A(− sin(Ωt), cos(Ωt)) [electric field E =form
FIG.in4.rotating
3D massless
Dirac fermion: (a)Application
to Diracsystems
E-field
E(cos(Ωt),
sin(Ωt))
(E
=
AΩ
>
0)],
and
the
effective#
strength parameters (a) Ã = veE/Ω2 = 0.00!
|Ω|
Hamiltonian
foreE
the fermions with chirality ξ = ± is given
Γξ $
|Ω|m
(optically allowed).
axis andlaserthe(11)probability for particles with ch2
as2D gapped
(4π)
v
1.
Diracvsystem
at which perfect tunneling occursis(Eq.
(8)). T
independ
for fixedchiralityk̃yy = 0. (d)x forbiddenz The total production
On the otherhand,
opticallyinThegrapheneĤ =v[ξ(kthe
+
eA
)σ̂
+
(k
+
eA
)σ̂
]
+
mσ̂
,
(3)
x
x
y
FIG. 4.ylines
3D masslessDiracfermion:(a)-(c)product
represent
their
asymptotic
power
law b
ξΩ < 0 shows the same behavior strength
as the parameters
low frequency(a) Ã = veE/Ω2 = 0.001, When(b) Ã = the
1/8,
regime, i.e.,
the eSchwinger
continues
tocharge,
increasev forasaxis and theprobability
with chirality
ξ = −imb
is r
TMD,
where
(> 0) is limit
the elementaryis particles
the Fermi
at which perfect tunneling occurs (Eq. (8)). The lower panels|Ω| increase.
grow(velocity, and m (> 0) is themassparameter.
This
model ratetially
Circularly
polarized
laser
with
the
3D
wavenumber
q
=
for
fixed
k̃
=
0.
(d)
The
total
production
andchiralcuy(a)(b) effect in 3D(c) massless
TwistedSchwinger
Dirac
(d)
1
has implication to valleytronics
in 2D
such
as
The
prod
0.3
µ
lines represent
theirmaterials
asymptotic
power
law γ̂
behavior
Eqs.
(14)
A(−
sin(Ωt),cos(Ωt),
0).
(µ
=
0,!x
5
!
"
fermions–
Next,
weproceedtoananalysisof
3D
mass0.4
monolayer transition metal dichalcogenide (TMD) and0 I Fig. 2(b).
0.2jless Diracgraphene
fermions[36,subject where
to rotating
electric
degammafields
matricesγ̂ 0 =, γ̂wavenu=
x=1 37], x=-1
laser-induced
valleypolarizationthe0 kz ), A =
0.2
0.5
0
with
the 3D wavenumber
q = I (k,scribed by
theHamiltonian
0.1
2. 3D gapless Dirac system A(− sin(Ωt),
cos(Ωt),0). !Hamiltonianγ̂ µ "(µ = 0,!x,isy, dividedz) are
thein
x,
y,
z).
This
"
'
j
0
0 j
0
I
0
σ̂
0
0
j
designated
by
ξ
=
±,
each
of
which
Ĥ3D = vγ̂ γ̂ (q-5gammaγ̂ 0 (12)=, γ̂ =(j =j + eAmatrices
j)
j
I
0
−σ̂
0
0
5
10
0
0.1
0.2
0.3
0
0.2
0.4
0.6
the
Weyl
Hamiltonian Eq. (3) with thj=x,y,zx, y, z). This Hamiltonian is divided into two sectors
m =assumethat thez . ±,Below,
designated
by vk
ξ =
each ofwe
which
corresponds
tois Hamiltonianat the Diracandtheexploit
the s
the WeylEq. point(3) with
identificationm = vkBelow, we assume
that theFermi
energyz . rewritingtry
the model
withvariablesis at theDiracpoint Thenand exploit
thesetscalingsymmeq̃ =vq/|Ω|.
we canthe frequen
try rewriting the model with variables2 t̃ = |Ω|t andlaser
and Ã = veA/|Ω| = veE/Ω is the uniqueq̃ = vq/|Ω|. Then we can set the frequency |Ω| to unity
that=characterizes
the field
strengthand Ã =eter
veA/|Ω|
veE/Ω2 is the uniquescaling
param-

x=1

and theparticles
witheffective
chirality ξ = − is rE(cos(Ωt),axissin(Ωt))(Eprobability= AΩ >for0)],
and theat whichtunnelingoccurs (Eq.
lower panels
Hamiltonian
for theperfectfermions
with chiralityξ =(8)).
± isThegiven
for 2D
fixedgappedk̃y = 0. Dirac(d) Thesystem
total production rate and chiral caslines represent their asymptotic power law behavior Eqs.
(14)
is indein graĤ = v[ξ(kx + eAx )σ̂ x + (ky + eAy )σ̂ y ] + mσ̂ z ,(3)
When
Circularly
polarized
laser
with
wavenumberkz ), ATMD=
where e (>0) isthethe 3Delementarycharge, qv is=the (k,
Fermi
µ
A(−
sin(Ωt),cos(Ωt),0).
γ̂
0,!modelx, y, z) arethetiallyvelocity, and m (> 0) is the mass parameter.
This
!
"(µ =
"
j
0
I
0
σ̂
0
j
has implication
valleytronics
gammato matricesγ̂ =in 2D materials, γ̂ = such jas
(jThe
=
I
0
−σ̂
0
monolayer transition metal dichalcogenide (TMD) and
Fig. 2“Nonperturbative
Valley-selective
circulardichroism”
x, y,37],z). where
This laser-induced
Hamiltonian valley
is divided
into two sectors
graphene [36,polarization
the w
designated by ξ = ±, each of which corresponds tothe Weyl Hamiltonian Eq. (3) with the identificationm = vkz . Below, we assume that the Fermi energy
is at the Dirac point and exploit the scaling symmetry rewriting the model with variables t̃ = |Ω|t andx=-1q̃ = vq/|Ω|. Then we can set the frequency |Ω| to unityand Ã = veA/|Ω| = veE/Ω2 is the unique scaling parameter that characterizes the field strength. In Figs. 4(a)4(c), we plot the production probability Pξ (q̃) for ξ = +and Ω > 0 evaluated by setting m = vkz in Eq. (4). It
Takayoshi,
Oka 2020
is rotationally symmetric aroundthe k̃Wu,
The proz axis.
duction probability shows peaks around the wavenumber

E(cos(Ωt), sin(Ωt)) (E = AΩ > 0)], and the effective
Hamiltonian for the fermions with chirality ξ = ± is given
2D gapped Dirac systemasĤ = v[ξ(kx + eAx )σ̂ x + (ky + eAy )σ̂ y ] + mσ̂ z ,(3)

where e (> 0) is the elementary charge, v is the Fermi
velocity, and m (> 0) is the mass parameter. This model
has implication to valleytronics in 2D materials such asmonolayer transition metal dichalcogenide (TMD) andgraphene [36, 37], where laser-induced valley polarization

x=1

x=-1

: Optically allowed: Optically forbidden
Note: Selection rule is replaced
by nonadiabatic geometric effects

is indein graWhen
TMD
tiallyThe
Fig. 2the w! ) deviatesve
gap mfromthatthe Γξ ∝ E 3/2 behavio
eff closes at one of the valleysveE'Ω) peaks> 0, out
andandperfect tunneling2DPξgapped(k)2 = 1Diractakeshavior at weak fields. A
system
2 + m2 .
where we defined M = v (|k| − eE/(|Ω|))heFig.for2(b), only the pos= −4m/(κ! v)
PT wavenumbers
In Fig. 2(b), we plot the production probability
d by an arrow inseveral Ω. We see that there is a strong chirality
de- weak field regi
in
the
!
eE
1pendence, and the sign of ξΩ determines whether exci= as ± tationsm(ξΩ/4− m).allowed” (ξΩ >(8)form(F )|k|decreasesare“optically0)or“optically|Ω| v
e of m = 0, theforbidden” (ξΩ < 0). This difference originates from the!
decreasing func|Ω|Total
production
rategeometric
amplitude
factor.
In
thissense,
theopticalsene the total fermion-antifermion production
|Ω|m
ξ $
"
lection rule [28, 29] in |Ω|perturbative
optics is replacedΓby2(4π) vnit
of time We
and volume as Γξ ≡dkPξ (k)ac fermions–3the nonadiabatic geometric(2π) effects when nonperturbative
lds in 2D Diracstrong field
excitations
are considered.
The ratio of theitagainstfrequency
in
Fig.
3(b).
We
see
On the other hand,
as gauge potenproduction rates between the two chiralities
rectificationtric
field E = effect where the imbalance ratioξΩ < 0 shows the sam$%
P+large
(k)
πξΩm
and
the effective exponentially forγ increases
Ω/E
folγ
=
=
exp
(5) i.e., the Schwi
x=1
x=-1
regime,ity ξ = ± is given
P− (k)
veE(5). In the low frequency region, it takes the|Ω| increase.
is independent of the wavenumber. In the gapless case, as
Twisted Schwingerywithchiraldichroismin graphene,measurable
γ is unity and
there
is no
valley dependence.
] + mσ̂ z ,(3)# When the$ gap parameterfermions– Next, we p
m is finite, as in monolayer%
S,ξ
TMD, imbalance E
becomesfinite and the ratio exponene, v is the eE
Fermi eEless Dirac fermions su
Γ
$
exp
−
π
.
(9)
2
tiallygrows
ordecayswithincreasing|Ω|/E.
eter.ξ This
model
(2π)
v
E
scribedmaterials such as
The production probabilities have peaks as shownin by the Hamilt
nide (TMD)
andFig. 2(b).
understandthe peakefine
the Schwinger
limitWeofcanfieldstrength
as structure fromalley polarization
the wavenumber dependent effective mass parameter inĤ3D = v
2

2

x=1

satisfies
ξΩ at
> 0,
perfect tunneling Pξ (k) = 1 takes
havior atplace
theandwavenumbers
Fiplace at the wavenumbers
Fig. 2(b),in
2D gapped
Diracsystem!
eE1
in|k| = 1 ! ±m(ξΩ/4 − m).
(8)the wefo
eEv|k| = Dirac±in|Ω|m(ξΩ/4
− m).
(8)
form
3D gapped
dc-E
field
|Ω| v
We define the total fermion-antifermion production|Ω| "We ratedefinethetotalfermion-antifermion
production
per unit of time and volume as |Ω|Γξ ≡" (2π)3 dkPξ (k)Γξ $ (4rate per unit of time and volume as Γξ ≡ (2π)3 dkPξ (k)and plot it against frequency in Fig. 3(b). We seeOnand plotitagainstfrequency
in
Fig.
3(b).
We
see
On theξΩoclearly the rectification effect where the imbalance ratio
clearlyΓtherectification effect where the imbalanceratioξΩ < 0res
largeΩ/E
fol+ /Γ− = γ increases exponentially for1951
Γ+ /Γ− = γ increases exponentially for largeSchwingerΩ/E foli.elowing Eq. (5). In the low frequency region, it takes regime,the|Ω
lowing Eq. (5). In the low frequency region, it takes the|Ω| increa
form [35]form [35] 2D gapped, rotating E
Twisted
#
fer
$
%
#eE
fermions–
x=-1
eE
E
$
% S,ξ
eE
eE
E
les
Γ
$
exp
−
π
.
(9)
S,ξ
ξ
2
less Dirac
Γξ $
. E
(9)
(2π) exp v − π
2
sc
(2π)
v
E
scribed byHere
we define
the Schwinger
field strengthas
Here wedefinethe Schwinger
limit oflimit
fieldofstrengthas
Schwinger
limit
2
2
E
≡
(m
)
/(ve)
=
(m
−
ξΩ/4)
/(ve),
2
2
S,ξ
eff,ξ
ES,ξ ≡ (m(10)
eff,ξ ) /(ve) = (m − ξΩ/4) /(ve),Vanish at W=4m (for x=1)
What does this mean?

(10)and converge to a Γ+ ∝ E 1/2 behavior. In Fig.
summarize the tunneling behaviors into a (E,
2D gapped Diracdiagram,systemwhich we explain below.
Low frequency (|Ω| < 4m): The Schwinger lES,ξ [Eq. (10)] characterizes
3 the crossover from”phasediagram”
field exponentially suppressed regime to the Γξ
behavior at strong field. Increasing Ω from
Eq. (4) defined bywhere meff,ξ is the effective massSchwinger
at the limit
gap E
minimizS,ξ for the optically allowed
meffing= Mwavenumber− ξΩm/(4M ). Eq. (7). Equation
(6)
(ξΩ >
decreases
and becomes zero at |Ω| = 4(9)0) isan extension
perfect tunneling starts to happen. In contrastof Schwinger’s
production
rate evaluatedoriginally for
The peaks are dictatedby the wavenumber
minimizoptically forbidden chirality (ξΩ < 0), ES,ξ m
ing the effective massand their
properties
qualitatively
a DCelectric
field tothe case ofcallyrotating
electric
increaseagainst field.
|Ω|. This suppression of tchange depending onwhetherthefrequency
Ω
is
below
For Ω = 0, Eq. (9) coincides with
2D versionofis thetheconsequenceof counterdiabaticity
in theor above 4m. For Ω < 4m, the distributions have a peaktunneling.
Schwinger’s result [1, 13] with theLZQED
Schwinger limit
atHigh frequency≥ 4m): For the optically
ES = m2e c3 /(h̄e) obtained by replacingm → me(|Ω|c2 andchirality ξΩ > 0, the effective gap closes at wave
eE/|Ω|,(7)v |k|→=h̄c.
Eq. (8) and the Schwinger limit vanishes due t3(b) hand,showselectrictunneling.
field dependenceof rate shows a crosso
x=1 meff is nonzero.
where
On the otherfor theΩ ≥ 4m,x=-1 Figure
The production
the effective gap mtheat one ofrate
the valleys
that opticallythe Γξ ∝allowed
E 3/2 behavior
at strong field to a Γξ ∝production
with thechirality
eff closes
satisfies ξΩ > 0, andperfect
tunnelingPξ (k) frequencies.
= 1 takes
haviorweak fields.
As allshown in the right mos(ξΩ> 0)
for several
For atstrong
fields,&place at the wavenumbers
Fig. 2(b), only the positive branch of Eq. (8) coeEeE
curves!converge to the dashed line
Γ
in
the
weak
field
regime,which
leads to an as2ξ →large E(2π)
v deeE1|k| = scribed±m(ξΩ/4 − m).
(8)form|Ω| v by the asymptotic form of Eq. (9) independent#! behavof
Ω.
For
weak
fields,weobservetwodifferent|Ω|eE small E
exponential We define the total fermion-antifermion productionΓ$|Ω|m(optically allowedξ" (Ω < 4m) curves
2 v below the
|Ω|
iors.
The
low
frequency
drop
(4π)
v
W>4m
suppression
rate per unit of time and volume as Γξ ≡ (2π)3 dkPξ (k)lineinfollowing(9)seedue to the exponential supand plot it againstdashedfrequency
Fig. 3(b).Eq.We
On the other hand, the optically forbiddenpressionoftunnelingatweakfields.
curvesclearly the rectification effect where the imbalance ratioξΩ <In0 contrast,shows the same
behavior as the low f
Γ+ /Γ− = γ increases
for largeΩ/E
fol-turnregime,
for exponentiallyhigh frequency(Ω ≥4m)
abovei.e.,thethedashedlinelimit continues to in
Schwinger
lowing Eq. (5). In the low frequency region, it takes1/2
theFIG. 3. 2D gapped Dirac fermion: (a) The total pair
production rate per unit of time and volume. We fix ma/v =0.5 and eEa2 /v = 1. (b) The electric field dependence of thetotal production rate Pξtot . (c) (E, Ω)-phase diagram of thetwisted Schwinger effect.

fermions– Next, we proceed to an analysis ofstrength
3D massparameters (a) A = veE/Ω = 0.00
less Dirac fermions subject to rotating electricfieldsaxis
anddethe probability for particles with ch
3D
systemwithgaplessthe 3DatDiracwavenumber= (k, occurs
kz ), A(Eq.
= (8)).pro
which
perfectqtunnelingT
scribed by the HamiltonianµA(− sin(Ωt), cos(Ωt),0).k̃(µ (d)= The
0,!x, y,z) aretheandfor fixed
total
productionr!y γ̂= "0.
"'jz0 σ̂ power lawJb+
asymptotic
Ĥ3D = vγ̂ 0 γ̂ j (q(12)0 I theirgammamatrices=, γ̂ j =(j =j + eAj ) linesγ̂ 0represent
j

I 0
−σ̂ 0
of
x, y, z). This Hamiltonian is divided into two sectors
indesignated by ξ = ±, each of which corresponds toÃ == Two 3D Weyl fermions
with
x=1,
-1
=
two
2D
Dirac
(m
↔
k
)
z
with theEq. 3D
wavenumberq = (kthe Weyl Hamiltonian(3) with
the identificationm = vkwe assumethat theFermiA(− sin(Ωt),
cos(Ωt),0). !
γ̂ µ "energy(µ = 0,!xΓ
Circularly polarizedlaser
z . Below,
is at the Dirac point and exploit the scaling
0 Isymme0
matricesγ̂ = t̃ = |Ω|t, andγ̂ j =
try rewriting gamma
the model
with variables
I 0
−
q̃
=
vq/|Ω|.
Then
we
can
set
the
frequency
|Ω|
to
unity
“Circular photo-Galvanic effect”
J
x,
y,
z).
This
Hamiltonianisdividedin2and Ã = veA/|Ω| = veE/Ω is the unique scaling paramdesignatedby ξstrength.
= ±, Ineachwhicheter that characterizes
the field
Figs. of
4(a)Th4(c), we plot theprobability PξEq.
(q̃) forξ =with+theproduction
Weyl Hamiltonian
(3)
the
exc
and Ω > 0 evaluatedby setting
m = vkEq. (4).that
It thez in
m = vk.
Below,
weassumez
Diris rotationally symmetric around the k̃z axis. The prois at the Diracpoint
and exploit theands
orderperturbation
duction probability shows2nd
peaks
aroundthe wavenumberTh
rewritingthevariables
satisfying the try
perfect
tunneling
condition
Eq. with(8)
theory Chanetmodelal. PRB‘17 plottedin= vq/|Ω|.
Then
can Eq.
set (8)
thedefrequenas black solid q̃curves.
For eachcrosswesection,las
2‘17a radius
Q.
Ma
al.(=
Nat.
Phys.
fines a circle centered
k̃|,
k̃zet
)=
Ã,
1/8)
with
and Ã at= (|veA/|Ω|
veE/Ω
is the uniqueW
1/8. A crossover
eteroccursthat atcharacterizes
the
field
strength.
cur
TaAs
4(c), we Ãplot
the production probability
P
(13)
bro
c = 1/8,Weand Ω > 0 evaluated by setting m = vkzwhere for Ã < Ãc the circle is incomplete and approaches
whisrotationallysymmetricaroundthek̃za semicircle in the small Ã limit, while it is complete forexc
j=x,y,zfermions– Next, we proceed0.4
to an analysis of 3D mass- 0.4
Ãrotating≥ Ãc electric. Thefields
widthof the distribution aro
less
Dirac
fermions
subject
to
de2
0.2
3D
gapless
systemwiththe 3D Diracwavenumber= (k,scalingkz ), A =pro
Gaussianwithaµ qwidthas Ã
scribed by the Hamiltonian is aA(− sin(Ωt), cos(Ωt), 0). ! γ̂ "(µ = 0,!x, y, z) are
the
and
"
' 0.2
0
0.2
that
the distribution
for j the ξ0 =σ̂j − chirali
z
0 j
0
I
J
0
+
Ĥ3D = vγ̂ γ̂ (q(12)
gammamatricesγ̂ =, γ̂ =(j =j + eA
j)
1
0.1
j
I
0
−σ̂
0
of
j=x,y,z a reflection of ξ = + around the k̃z = 0 pla

0
0

0.1

0.2

(b)

in
Ã =

We define the total production rate and#along the z axis as Γ =dq̃Pξ Γ0.61#
− (c) dq̃P (q̃), respectively, w(d)J

lifetime of the pairs [35]. We define the

0.4

0.2

0
0

x, y, z). This Hamiltonian is divided into two sectors
designated by ξ = ±, each0of which corresponds to0= Two 3D Weyl
fermions
with x=1, -1 = two 2D with
Diracthe
(m ↔
k4z)
0
the Weyl Hamiltonian Eq. (3)identification|Ω|
3D-5
m = vkz . Below, we assume
Fermi
4 v 3energy
ξ0 that the
(2π)
0
0.1
0.2
0.3
0.2
0.4
0 scaling symme5
0.3
0 is at 0.2
thepoint and0.6
exploit the
4 Dirac 0.4
2eτ |Ω|
k̃z with variables t̃ = |Ω|t and!
try 4rewriting
the
model
ξ
(2π)
v 2 cone2
Weyl2q̃ = vq/|Ω|.
Then(x=1)
we
can
set
the
frequency |Ω| to unityk̃ +k̃z
1
and Ã = veA/|Ω| = veE/Ω2 is the unique scaling param5eter that characterizes the field strength. In Figs. 4(a)4(c), we plot the production probability Pξ (q̃) for ξ = +and Ω > 0 evaluated by setting m = vkz in Eq. (4). It
is rotationally symmetric around the k̃z axis. The pro0.5 the wavenumber
0duction probability shows peaks aroundsatisfying the perfect tunneling condition Eq. (8) plottedas black solid curves. For each cross section, Eq. (8) delaserfines a circle centered at (|k̃|, k̃z ) = (Ã, 1/8) with a radius
1/8. A crossover occurs at

-5

0.2

0.4

0.6

0

5

0
Ã10
c = 1/8,(13)kz→ where-kz forfortheÃWeylwith isx=-1< Ãccone
the circle
incomplete and approaches
a semicircle in the small Ã limit, while it is complete for

Th
exc
Dir
and
Th
in
las
W
cur
bro
We
wh
exc

0.3

is a Gaussiana widthscalingm = vkz . Below,
we assume withthat theFermi energy
is at the Dirac
point
and
exploit the scaling
symmethat
thedistribution
for
the
ξ=−
3D gaplessDiracsystemtry rewriting the model with variables t̃ = |Ω|t anda reflection of ξ = + around the k̃z =q̃ = vq/|Ω|. Then we can set the frequency |Ω| to unity
definetotal scaling
productionra2
scaling parameterand Ã = veA/|Ω| We= veE/Ω
is thethe uniqueparam#
4|Ω|
3D
eter that characterizes
the z
field
strength.
Figs. 4(a)along theaxis
as Γξ In =(2π)4 v 34(c), we plot the 2eτ
productionPξ (q̃) for ξ = +|Ω|4 # probabilityk̃z− (2π)4bydq̃ !m2= vkP(q̃),
2 setting
Eq. respectiv(4). Itz ξinv(b)and Ω > 0 evaluated
(c)
2
= 1/8
+k̃z
(d)
1
is rotationally symmetric around k̃thek̃z axis.
The pro5
lifetime
thearoundpairs the[35].
We definduction probability
showsofpeaks
wavenumber
0.4
satisfying the perfect tunneling condition Eq. (8) plottedas black solid curves. For each cross section, Eq. (8) de0.2
0.5
0fines a circle centered at (|k̃|, k̃z ) = (Ã, 1/8) with a radius
1/8. A crossover occurs at

(a)

0.2
0.1

0

0
0

laser

0.1

0.2

0.3

0

-5
0.2

0.4

0.6

0

Ãc = 1/8,
5

10

0

(13)

where for Ã < Ãc the circle is incomplete and approaches
a semicircle in the small Ã limit, while it is complete forÃ ≥ Ãc . The width of the distribution around the peakis a Gaussian with a width scaling as Ã1/2 . We notethat the distribution for the ξ = − chirality particles isa reflection of ξ = + around the k̃z = 0 plane.
We define the total production rate and the currents|Ω|4 #
along the z axis as Γ3D=
dq̃Pξ (q̃) and Jξz =ξ(2π)4 v 3
2eτ |Ω|4 #− (2π)
dq̃ ! k̃2z Pξ (q̃), respectively, where τ is the
4 v2k̃ +k̃z2

lifetime of the pairs [35]. We define the total (chiral)m = vkz . Below, we assume that the Fermi energy
(d)
1
0.3
is at theDirac point and exploit the scaling symme5 3D gapless Dirac system
0.4
try rewriting the model with variables t̃ = |Ω|t and
0.2q̃ = vq/|Ω|. Then we can set the frequency |Ω| to unity
0.2
0.5
0
scaling parameter
0.1
and Ã = veA/|Ω| = veE/Ω2 is the unique scaling parameter that characterizes the field strength. In Figs. 4(a)0
1
0
-5
4(c),
we0 plot the production probability Pξ (q̃) for ξ = +
5
0
5
10
0
0.1
0.2
0.3
0
0.2
0.4
0.6
and Ω > 0 evaluated by setting m = vkz in Eq. (4). It
is rotationally symmetric around the k̃z axis. The production probability shows peaks around the wavenumber
0.5
0satisfying the perfect tunneling condition Eq. (8) plottedas black solid curves. For each cross section, Eq. (8) deaser
fines a circle centered at (|k̃|, k̃z ) = (Ã, 1/8) with a radius-5
1/8. A crossover occurs at
0
0
5
10
0.4
0.6
distribution1Ãc = ξ1/8,
sless Dirac fermion:shifted(a)-(c)inThe= + plotted for several(13)field
kz production probability Pξ (q̃) for chirality

(a)

(b)

(c)

(c)

(d)

ers (a) Ã = veE/Ω2 = 0.001, (b) Ã = 1/8, and (c) Ã = 10. They are rotationally symmetric around the k̃zwhere for Ã < Ã the circle is incomplete and approaches
ability for particles withchirality
currentJ5z ξ = − is reflected as k̃z → −k̃z . cThe solid black curve denotes wavenumber!
asemicircle
in
the
small
Ã
limit,
whileitiscomplete
unneling occurs (Eq. (8)). The lower panels show the fermion-antifermion pairs on the Dirac cone E = ± for|q̃|Ã are≥ Ãplotted
width
thereddistribution
the dashedpeakc . Theas
J5 = J⇠=+1J⇠=
1 current(d) The total production
rate andchiralblue ofandsolid curvesaround
while
the
a Gaussian with a width scaling as Ã1/2 . We note
eir asymptotic power law behavior Eqs. (14) and is(15).

that the distribution for the ξ = − chirality particles isa reflection of ξ = + around the k̃z = 0 plane.
2
3Dproduction
3D
3D and3D the 3D
3D(m+Ak vF/4)
We definerates
the astotal
ratecurrents
wavenumber
q
=
(k,
k
),
=
production
Γ
=
Γ
+Γ
(Γ
=
Γ
−Γ
z
tot
+
−
+
− ),
5
#
4
P (F
)
=
exp
⇡
.
|Ω|
3D
z
z
z andz J z=alongz axisas Γξcurrents= (2π)as(q̃)|vF the|4 v 3 J dq̃PΩt), 0). ! γ̂ µ "(µ = 0,!x, y, z) areandthethetotal(chiral)ξ 5 =tot =ξJ+ + J− (J"z 2eτ |Ω|z4 #z
0 I
0 σ̂ jJ−
Duethesymmetryξ → −ξ,kzτ →! k̃to0j+ − J−2 ). dq̃Pξ (q̃),respectively,
where
is −k
thez ,4γ̂ =, γ̂ =
(j
=
(2π)
v
2
j
3D k̃ +k̃z2
3D
z
I
0
−σ̂
0
of
P
(q̃),
Γ
=
J
=
0
holds.
We
plot
Γ
and
J
z

ξ
tot
5
5
ξ
The production probabilityPξ (q̃)for chirality ξ = + plotted for severa
m2
lifetime of the pairs [35]. We define the total (chiral)

i

Rt

i n
| (t)i ' | n (Summary(t))ie 0 dt"n ( (t))
eTakayoshi, Wu, Oka 2020

1. Slow is not adiabatic!

| (t)i 'p

P (F ) = exp

0.3



(a)

P | n ( (t))ie

1

i

Rt

dt"n ( (t)) i˜n
0

e

p

1

+ O( P )(m + k vF/4)2= large⇡. +Adiabaticity
˜=
correction
n
n
|vF |
≠ slow parameter change F

P (F ) = exp
0.2



(b)

(c)

2

m
0.4
⇡
.
|vF
|
Z

1

(d)

5

0
2. Twisted
Schwinger0.2 effect = Nonperturbative
optical effect
0.1

˜=

t

d h n |i@ | n i + . . .

⌦0=“Valley-selective“Circular
0 photo0⇠E
circular dichroism”
Galvanic
0
0.1
0.2
0.3
0
0.2 effect”
0.4
0.6
L(E) =

0.5

Rii 0⌧ E X̂(s)dslim
lnh0|T̂ e|0it!1 tV

-5

0

5

10

0

Guiding principle

1. Energy-momentum conservation
2. Nonadiabatic Geometric effect
Berry 1990

laser

ReL(E) =EZ

dk ˜ (k)d 2⇡
BZ (2⇡)

Z fermion: (a)-(c) The production probability Pξ (q̃) for chirality ξ = + plot
FIG. 4. 3D massless Dirac