Interna'onal*workshop*on*"Axion*physics*and*dark*ma8er*cosmology”***20=21*December*2017,*Osaka*University,*JAPAN1 M N CAN-‐‑‒I

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“Axion” in material science
Axion is a common research themein the particle physics, cosmology, and material science

Par'cle*Physics

Cosmology
Axion

Material*science**

“Axion” in material science
IM MC
KMC
MI K I O S MKIHA -‐‑‒

KI-‐‑‒ H CU

M

IK . K 2 MM K

Spontaneous**
symmetry*brakingAxion*mode*

Par'cle*Physics

Cosmology
Dark*Energy*******(68%)

Axion

Ordinary*ma8er*(5%)

Material*science**
Bi1-xSbx

S, CIHU
C I M

CIH C
KMC
CH MI I IAC

Dark*Ma8er*(27%)

Planck*2013*results

:
M KC

Topological*materials*=*materials*categorized*by*topology*
It’s*very*hot*theme*for*material*science
Theory*(*~*2000*)*j@discovery*of*theory*about*“topology”*in*the*condensed*ma8er************************************************************2007*:*Science*138(2007)*766*[*2D*topological*insulator]***********************************************************2008*:*Nature*452(2008)*970**[*3D*topological*insulator]

2016*Noble*Prize*in*Physics*
The*Nobel*Prize*in*Physics*2016*was*awarded*with*one*half*to*David*J.*Thouless,*
and*the*other*half*to*F.*Duncan*M.*Haldane*
and*J.*Michael*Kosterlitz*"for%theore)cal%
discoveries%of%topological%phase%transi)ons%
and%topological%phases%of%ma5er".
Mahmoud*David*J.*Thouless*

Mahmoud*
F.*Duncan*M.Haldane*

J.*Michael*
Kosterlitz

Topological insulator!@Ph`_MoTbfh`Xgce
!@Whe\XY[M@U[gX``^Y

V[X`@fdXY[Uca[bgha@fdXY[ 
Phys
   S[`^YX`@[Z][@fgXg[Wd^bAZ[d[bZ[bg@geXbfdceg
EQ@Q^eXY@Ycb[Superlattice of topological insulator (TI) and normal insulator (NI)SuperlaEce*of*GeTe*(NI*)/*SbTe*(TI)

Ref: J. Tominaga, et al., Adv. Mater. Interfaces 1, 1300027(2014)

Sb
Ge
Te

This material has been already applied in interfacial phase change memory

. To detect it, we focus on characteristicc responses due to the DM-axion fields.
tic responses are basically caused by the in
weenEffective
axionelectromagnetic
fields as
Lagrangian andin a topologicalmaterials is described byγ,

Ref:*F.*Wilczek,*PRL*58,*1799*(1987).

(Topological*insulator,*Weyl*semimetal,*Dirac*semimetal,*etc…)

X.[L.*Qi,*T.*L.*Hughes,*and*S.[C.*Zhang,*PRB*78,*195424*(2008).

A.*M.*Essin,*J.*E.*Moore,*and*D.*Vanderbilt,*PRL.*103,*259902*(2009).

La = gaγγ αθ(x, t)E · B, “axion-like”
particle (or field)

Coupling constantFine structure= 1/137

Electric field
and magnetic field

α, θ, E, and B denote the couplingThis Lagrangian is similar to the Lagrangian in Field theory (QCD-axion)
fine structure,axion field, an electric field,soc field,“θ”-termrespectively.
is called as “axion” field Unfortunately, theant gaγγ is tiny, so that its detection is diﬃ
es a challenging problem. So far, several t
periments, which are axion-halloscope, ax

Anomalous Hall effect term(Hall effect w/o B)
Chiral magnetic
effect termkθ and ∂tθ couple with electromagnetic fields
Trigger characteristic transport:
Ref:*F.*Wilczek,*PRL*58,*1799*(1987).
X.[L.*Qi,*T.*L.*Hughes,*and*S.[C.*Zhang,*PRB*78,*195424*(2008).
A.*M.*Essin,*J.*E.*Moore,*and*D.*Vanderbilt,*PRL.*103,*259902*(2009).

Study*of*“Axion”*in*material*scienceWhy*?**
The*reasons*are*…
1. Interac'on*between*“Axion”*and*electromagnehc*ﬁelds*in*materials**science*could*be*large*(compared*with*that*in*QCD[axion)*
2. Controllable*“Axion”*by*using*knowledge*of*material*science*
3. There*are*some*analogy*between*parhcle*physics*(cosmology)*and*material*science*******(e.g.,*majorana*fermion,*axion*..)*

par'cle*physics**&*cosmology

Material*science

High*energy*physics

energy*scale

Low*energy*physics*
0.1*meV*1*eV*Strong*CP*
Dark*maker

context

Unconven'onal*
electromagne'c*eﬀects*

Pure*research*

Pure*or*appliedPure**&*Device*applica'on**
via*the*electromagne'c*eﬀects*

ic responses due to the DM-axion fields.
stic responses are basically caused by the iween axion and electromagnetic fields as
Property*of*“Axion”**in*the*parhcle*physics*vs*that*in*the*material*science*

La = gaγγ αθ(x, t)E · B,
Partciel*physics*+*cosmology*

Material*science*

Type*of*axion

QCD*axion,*ALPs*

Topological*axion

Interac'on*strength**

Very*small*(~10=19)

Large*(~1)

Character*

Dynamical*

Detec'on*

Halo*scope*(cavity)**+*Primakoﬀ*eﬀect*γ , α, θ, E, and B denote the couplingfine structure, axion field, an Not*dynamical
electric fieldc field, respectively. Unfortunately, theDynamical*Axion*signals*in*the*both*ﬁeld*have*not*seen*yet!!**tant gaγγ
is
tiny,
so
that
its
detectionisdiﬃ
First,*let’s*try*to*consider*about*the*axion*response*in*material*science.*
mes a challenging problem. So far, several
xperiments, which are axion-halloscope, aIn*this*topic

DE

Topological*materials*are*…
good*experimental*planorms*for*teshng*axion*physics,**
because*of
Controllable*parameters*such*as*coupling*constant*and*“Axion”*is*controllable*in*lab…*
axion*mass***
We*can*challenge*a*low[energy*physics
so
axion*is*not*“dark”*in*material*science!*DM-axion is expected to influence the form
se. To detect it, we focus on characteristi
etic responses
due
to
the
DM-axionfields
Naively*queshon:*ristic responses
are basically caused by theIs*there*true*of*the*eﬀechve*Lagrangian*even*in*material*science*?*etween axionand electromagnetic fields asγγ ,La = gaγγ αθ(x, t)E · B, “axion-like”
particle (or field)

Coupling constantFine structure= 1/137

Electric field
and magnetic field

α, θ, E, and B denote the coupline fine structure, axion field, an electric fiel
tic field, respectively. Unfortunately, thstant gaγγ is tiny, so that its detection is dElectrical*evidence:*Quantum*Hall*eﬀect

M.*Mogi,*et*al.,*Nat*Mater*advance*on,*(2017).

TI*with*mag.**
Topological**insulator*(TI)*

An'=parallel**magnehc*conﬁgurahon*
There*is*cancelahon*(=1/2*[*1/2)*of**
each*half[quantum*Hall*eﬀect**
on*top*and*bokom*surface*of*TIs

Parallel**magnehc*conﬁgurahon*
There*is*no[cancelahon
half[quantum*Hall*eﬀect**(=1/2*+*1/2)

Op'cal*evidence*in*TIs*:*Giant*(half=quan'zed)*Kerr*eﬀect“Quanhzed*Faraday*and*Kerr*rotahon*and*axion*electrodynamics*of*a*3D*topological*insulator”**Liang*Wu*et*al.,*

Maxwell*equahon*from*Euler[Lagrange*Equahon** However…
In*usual*TI,*as*θ*is*stahc,*it*does*not*have*hme[dependence.*(and*θ*=*π)*

Chiral magnetic
effectYou*can*not*see*the*electromagne'c*eﬀect*following*Maxwell*equa'on!!
We*dare*to*think*the*dynamical*θ*from*the*analogy*of*par'cle*physics**

However,*There*is*the*source*term**generated*by*the*hme[dependent*axion*ﬁeld*
n@How*can*we*make*such*dynamical*axion*ﬁeld*?*
DJ

Our*idea**“Generahng*the*hme[dependent*θ*using*the*spin[ﬂip*“

DK

	
  
Conventional works consider static “axion”-response
Our work shows that …
[1] how to drive dynamical “axion”-response[2] the electromagnetic effect via dynamical “axion”-responseθ term is mathematically described when there is gap

Gapless (massless)

X.-L. Qi, T. L. Hughes, and S.-C. Zhang, PRB 78, 195424 (2008).[See Eq.
P. Hosur, S. Ryu, and A. Vishwanath, PRB 81, 45120 (2010).
R. Li, J. Wang, X.-L. Qi, and S.-C. Zhang, Nat Phys 6, 284 (2010).

Gap

θ term is obtained from Hamiltonian with PT-symmetry

φ4 corresponding the band gap in φ5=0

Gap

φ5 corresponding the breaking P and T simultaneously

Hamiltonian with PT-symmetry

Theta termθ = Arctan(φ5 /φ4)

[Review] P. Hosur, S. Ryu, and A. Vishwanath, PRB 81, 45120 (2010).
K.*Fujikawa*and*H.*Suzuki,*“Path*Integrals*and*Quantum*Anomalies”*Clarendon*Press,*Oxford,*2004

Model*EETo control θ = Arctan(φ5 /φ4)
we consider the superlattice of topological insulators (TIs)
and magnetic insulators (for P and T symmetry breaking )EFaking symmetry influences θ. Therefore,
oes not aﬀect θ to the lowest order. Using

(b)!

ΔM"

ΔM"

Weyl!
semimetal!

Δ

ΔT"

T!
!
2
2
mF − mAF

Axion!
insulator!

Axion!
insulator!

QAH!
!
0! m2 − m2
F

AF

ΔM!

online) (a) Schematic illustration of superd MIs, where ∆T and ∆M are tunneling beottom of the same (b) The phase diagram ofof TIs and MIs for several tunneling ∆T and− m2
AF indicates the critical points from thephase into the Axion insulator phase and thelous Hall phase (QAH) into Weyl semimetal

QAH
(gap)

Weyl semimetal(gappless)Axion insulator(gap)

We consider axion insulator

ΔT(inversely proportional to
thickness of TI )

How to derive ∂tθ ≠ 0
Anti-magnetic configuration is important for manipulation of θ
The magnetic order can be manipulated by an applied magnetic field Bex“Hard” M-H loop

M-H loop in multilayer

Q.*L.*He*et.*al.,*Nat*Mater*16,*94*(2017)

Spin*control*of*super[laEce*device
Assume:**M1*sow* M1*m@M2
M2*hardθ = Arctan(φ5 /φ4)b1< Bex< b2

Simple form θ=0jstarhng*poin¤

EI

EJ

How to derive ∂tθ ≠ 0
Anti-magnetic configuration is important for manipulation of θ
The magnetic order can be manipulated by an applied magnetic field Bex“Hard” M-H loop

M-H loop in multilayer

Q.*L.*He*et.*al.,*Nat*Mater*16,*94*(2017)

2

µ

2π!

Next, in ord
Lagrangian
haveconsider thwhere θ ≡ θ0 + δθ(x, t) can be decomposed into Therefore,
the static we weMI1/TI/MI2,
axion field θ0 and the dynamical axion field δθ(x, t).
#
L = LMaxwell + LAxion + Le
(1)
2
the$direc
Here, je is the charge current, which isdetailed given∂where
2
2
2
−
c
∇
Bo
MI2
(M
)
1in insulators
1e2
2 are inasje = ϵ∂t P and je = σE, respectively,22∂t= ϵ0 E −B +θE · B + je · A(2)absence of appl
andσdenoteelectric
permittivity
and conduc2where ϵ 2µ2π!
0antiparalleleac
tivity, respectively.
µ0 cα It isµ0c
Fujikwa’s
method, we obtain the eﬀective Lagrangianasfound that
fo
Here,
is
estimatedbyMaxwellWe
equationππassume the following Lagrangianθ has a finiteva
is inducedby theAxion field isdefinedby prhand,
in theL = LMaxwell + L∂Belectric
polarizati
Axion + Le
!ELECTROMAGNETIC"
along isthe
layer
×E =−
(3)
III. ∇EFFECTIN AN
izationcaused1
1
α∂t2m
INSULATOR
rationofM1 an=
ϵE 2 − AXION
B
+
θE
·
B
+
j
·
A
(10)
e
θ
=
arctan
field
in
the
axion
∂D
2 ×H = µm
the parallelma∇+ j 2π!
(4)
Next,
in order∂tcharge current
by H
ap
A.
Magnetic field
due to the time-dependent
axion weflipped
permittivity
considerthewhere θ ≡ θ0 + Electricδθ(x,t) can be decomposed into the static
as theschematicall
We assume MI1/TI/MI2,
thattime-deri
and magnetic permeability fieldere
asaxionAssume
field θ0 and the dynamical axion field δθ(x,
t).
plied
magnetic
estimated bywhere the directi
Here, Bje=isµ the
chargecurrent,
which is detailed
given
(H
+
M
)
→
µ
H
,
(5)
θ →
the
m
0
0
We consider
an axion in- MI2(M02via) areopp
in insulators
as jethe= electromagneticϵ∂t P and je =eﬀect
σE, inrespectively,
gered ofwithinthD = ϵ̃ϵ→superlattice
ϵ0 E
(6)
θ(t
sulatorof0 E
theusing Lagrangian.
From absence
applied
where
ϵ and
σ
denoteelectric bypermittivity
and conducθ̇ =theLagrangian,Maxwellequations
are
given
by
j
≡
j
+
j
(7)
antiparalleleachτoeafltivity, respectively.
Maxwell equations
θ has a finite valu
re, je and ja are givenby∇ × E = −∂t B(11) hand, in the
1. presIn th∇ × H = ∂t D + je + jaEFFECT IN AN (12) along the layered
III.
ELECTROMAGNETIC
je = σE(8)Bration
tof
<M
τflip
) ≈(
ind (x,AXIONcϵINSULATOR
α
Using
Eq.
!
"
1 and
0
cϵ0 α ja = −[(∇θ) × E + (∂t θ)B](13) theduced
parallelmagn
by thetja = −(∇θ) × Eπ+ θ̇B
(9)
π
B
(t
=
τ
)
≈
indan byflip . elflippedHofapplied
ap A
A.
Magnetic field due
the c:
time-dependentaxion
α: fineto
structure,
velocity of light
where ja is charge
current
due to an applied electric field. as equation
schematically
s
becom
field
s the current density[A/m2 ].
≈
From forthese
we obtain
Wave equation
theMaxwell
induced equations,magnetic field
Bind the wave equa- plied magnetic fition asθ → 0 via the mag# consider$ electromagnetic eﬀect in an axion inWe
the
2
! 2
"
∂ of the1 bygered withinthe2
2superlattice
2.
In
8
2 tht
sulatorusing
Lagrangian.
From
∂
1
−
c
∇
B
=
∇
×
j
(10)
If
we
take
|j|
≈
10
A/m
an
ind
2
2
2
−
c
∇
B
=
∇
×
j,
(14)
∂t
ϵ
ind
0
the Lagrangian, Maxwell
equations are given by2

Derivation

∂t

!ϵ0

"

1 µ0 cα∇ × E = −∂t1
B(11) α1. willIn thediscuss
absence ofthe
an applied
electricfi
1 (8)2
WeB
propertyje = σE
2
(x,
t
<
τ
)
≈
−
ind
flip
ϵE
B
+
θE · B + je · A
(10)
∇×H
==
∂t D + "jefield.
+ ja −
(12)
!
2 spa
π
where Bindcϵis0 the
inducedmagnetic
Here,
therightthepresenceofthestatic
and
2
µ
2π!
α
cϵ0 α 1
Using Eq. (***), we consider the magnet
ja =− of the (∇θ)
1 µwe
jE− θ̇B [(∇θ)
+ (∂
θ)B]
(13)
hand
side
above×equation,∇ ××Ej,denotes
the
a =+
t(9)
fieldbyHthe
ẑ. From Eq.
(16),
ap ||
duced
time-dependent
axion
field
in0 cα
th
π ϵ0
π
B
(t
=
τ
)
≈
−
ind
flip
where
θ0source+ δθ(x,t)iscanbe decomposed
intothestaticsource term for nonzero
Bindθ. ≡The
term
repreofboundary
an applied electricfield in B
theindsystem.
Thencondition
(t ≤
2 0)
π=
where jaxion
current due
toanappliedelectric
field.
a is charge field
equation
becomes
θ0 and the dynamical
2 Maxwell equations,senteddensityby
0 as axion field δθ(x,1t).
µ0 cαFrom these
we obtain the wave equaj is the current
[A/m
].
Here,
j
is
the
chargecurrent,which
isdetailed
given
≈
−
e
tion as for B
obtain the forwavetheequation
WaveWeequationmagnetic" field
2 π
1
1 induced
1cαµin insulators
as B
jeind= ϵ∂t P and je2. =In theσE,respectively,!cα20of an external
electric
Bind (t ≤ τf )presence=−∂t θ(je − ∂ [(∇e+
2 ·2E)∇θ −1 (∇θ) · ∇E]#ϵ ∇2× j = ϵ ∇$× jwhereϵandσdenoteelectric
permittivity
and
conduc−
c
∇
B
=
∇
×
j,
(14)
ind
π
2 π
∂t2
ϵ0
∂
2 2cαµ tivity,1respectively.
2 of Bind of E
will |j|discuss
the8property−c ∇=t D +∇j×
j ja ) ,(10) (15)If weWe
take
≈
10
A/m
and τflip ≈−whereBB∂ind
θ
(∂
+
t
e
2
magnetic field. Here, the rightthe presence of the static and spatially uniform∂t0
π ind is theϵinduced
hand side of the above equation, ϵ10 ∇ × j, denotes thefield Hap ||ẑ. From Eq. (16), we obtain Bind
The first and second0)term
denot
sourceterm forfield
nonzero Bind
. Thetermsource term is repreSourceboundary condition Bind (t ≤−2= 0 and ∂t Bin
Inducedmagnetic
where
the
first
and
second,
third
term
of
right-handside
ELECTROMAGNETIC
EFFECT
IN
AN
axion
field
an
Assume ∇ × je = 0 andsented
Ė =by0. III.
We have
Bind0 the(t
τflip ) ≈ 4 × 10
T≫geom
as =time-dependent

denote the source term due to the conventionalchargeAXION
INSULATOR
field, respectively.
It is found th
&
1
1% 0
cα
1
cαµ
0
0
2
current (je ), theα spatial-dependent
field
(∇θ),
and Baxion-induced∇ × j = ∇ × je − 2 axion[(∇ · E)∇θ− (∇θ)· ∇E]∂t θ(je + ∂field
ind (t ≤ τf ) = −t D)t +
magnetic
isµH
al
ϵ
ϵ
π
2
π
∇
×
j
=
−
θ̇j
+
o
∇
θ̇,
θ̇
(11)
e
the time-dependent
axion Magnetic
field
field dueSince
to theized
time-dependentaxion
cαµ (∂t θ), respectively.
current jp = ∂t D andthe macπ A.
−∂t θ (∂t D + je + ja ) ,(15)
field field is determined by j and ∂there is no charge current πin the insulator we consider
The first and second term denotepthe magnet
but there is finite
electric
polarization,
we
take
j
=
0
e sidewhere the first and second, third term of right-hand
Notethat m5 axion
depends
on thetheappliedapp
thetime-dependent
field and
Assumedenote theWesource
term
due
to
the
conventionalchargeand ∂t D ̸= 0 generally.
When
E
is
spatially
uniforminfield,
It is
foundthat thedirectconsider the electromagnetic
eﬀect
in
an
axion
inbutrespectively.
m
4 is basically independent
current(je ),
thespatial-dependent
axion
field
(∇θ),
and
j
=
0
and
E
is
spatialuniformeaxion-induced
magnetic field is along
the dielec
the superlattice,thesulator
wave
equationfor theleading orderof theby using
Lagrangian.
From
determinedfromthetime-depen
the time-dependent
axion
field (∂superlatticet θ), respectively. Since
ized current jp = ∂t D and the magnitude of thof ∂t θ becomesthere isthe
Lagrangian,Maxwellequations
are givenno charge
current in the insulator
we consider
netization
flip bydueto. Phenofield
is determined
jpby
andH
∂tap
θ =
∂t tan−1
Result:butthatofmtheon the applied flip
magnetic! 2" there is finite electric polarization, we take je = 0 Note
5 depends
scale
magnetization
is
and
∂
D
=
̸
0
generally.
When
E
is
spatially
uniform
in
t
∂
cαµ
but
m
is
basically
independent
of
H
.
∇
×
E
=
−∂
B
(11)
4
ap He
0
t
2 2
cal
magnetization
flip
time
τ
. 4Th
the superlattice,thewave
equationfortheleading
order
−
c
∇
B
=
−
∂
θ
(∂
D
+
j
+
j
)
.
ind
t
t
e
a
determined from the time-dependent of fm
o
∂t2
π
∇
×
H
=
∂
D
+
j
+
j
(12)
of ∂t θ becomesteaproximately
as ∂t θ(0 ≤netization
flip due estimatedto Hap . Phenomenologicall
! 2
"
(16)
cϵ
α
scale
of thetime-dependent
magnetization flip is determinedb0
so×theaxion-ind∂cαµj
0a = −
2 2
[(∇θ)
E
+
(∂
θ)B]
(13)
t
cal
magnetization
flip
time
τ
.
Therefore,here− c ∇ Bind = −∂t θ (∂t D + jeπ+ ja ) .
f∂t2
π
proximately estimated as ∂ θ(0 ≤ t ≤ τ ) ≈ −θ
(16)

t

f

so the time-dependent axion-induced magnet

where ja is charge current due to an applied electric field.
From these Maxwell equations, we obtain the wave equation as!
"where the first and second, third term of right-hand side
denote the source term due to the conventional charge
current (je ), the spatial-dependent axion field (∇θ), andthe time-dependent axion field (∂t θ), respectively. Since
there is no charge current in the insulator we consider
but there is finite electric polarization, we take je = 0
Main message:and ∂t D
̸= 0 generally. When E is spatially uniform in
dynamical
θ is driven by applied magnetic field (Bex)the superlattice, the wave equation for the leading orderof ∂t θ becomes
Result:
! 2
"
∂
cαµ0
2 2
−
c
∇
B
=
−
∂t θ (∂t D + je + ja ) .
ind
∂t2
π
(16)

the time-dependent axion field andfield, respectively. It is found tha
axion-induced magnetic field is alon
ized current jp = ∂t D and the mag
field is determined by jp and ∂t θ
Note that m5 depends on the appli
but m4 is basically independent odetermined from the time-depende
netization flip due to Hap . Phenom
scale of the magnetization flip is decal magnetization flip time τf . The
proximately estimated as ∂t θ(0 ≤ t
so the time-dependent axion-induc

When we apply the external magnetic field,
the dynamcial θ is induced
andthe dynamical θ drives the electromagnetic effects(Case 1) in the presence of only the external magnetic field(Case 2) in the presence of the external magnetic field and charge current (layered direction)(Case 3) in the presence of the external magnetic field and charge current (inplane)

(d)

B / Bex(c)
4

j =107A/m2

Bex

V

j

V

FI1
TI
FI2
NI

Bex j < 0

Bex j < 0j

Bex j > 0

2

B

0

−2
−4
0.0

2.0
Time (ns)

FI1
TI
FI2
NI

4.0
Bex j > 0B

flipped by Hap . As a result, m5 and θ are going to zero,
where the first and second, third term of right-hand side
as schematicallyshown
in term
Fig. due
3 (d)-(e).
Thus, the apdenote thesourceto the conventionalchargeplied magnetic
plays
a role to change
̸= 0(∇θ),
into andcurrent field(je ), thespatial-dependent
axionθ fieldθ → 0 viathe
thetime-dependent
magnetizationaxion
flip, and∂t θ is trig-Since
field nonzero(∂t θ), respectively.
gered within
time-scaleof the inmagnetization
τf .
theretheis nocharge current
the insulator flip
we consider

the time-dependent axion field and the applied magnetic
field, respectively. It is found that the direction of theaxion-induced magnetic field is along the dielectric polarized current jp = ∂t D and the magnitude of the magnetic
field is determined by jp and ∂t θ = ∂t tan−1 (m5 /m4 ).
but there is finite electric polarization, we take je = 0
Note that m5 depends on the applied magnetic field Hap
and ∂t D ̸= 0 generally. When E is spatially uniform in
but m4 is basically independent of Hap . Hence, ∂t θ issuperlattice,wave
equationforof
theleadingorder
1. theInthe absenceofthe
anappliedelectric
field
(Case
1) inthepresence
only
theexternal
magnetic
Bex
determinedfrom thefieldtime-dependentof m4 or the magof ∂t θ becomes
netization flip due to Hap . Phenomenologically the timescale of the magnetization flip is determined by the typi2
Using Eq.! ∂(***),
we"consider the
cαµ0magnetic field in2 2
cal magnetization flip time τf . Therefore, here ∂t θ is ap− c ∇ Bind axion= − field∂t θin(∂ttheD +absence
je + ja ) .
duced by the ∂ttime-dependent2πproximately estimated as ∂t θ(0 ≤ t ≤ τf ) ≈ −θ(t = 0)/τf ,of an applied electric field in the system. Then, the wave (16)so the time-dependent
axion-induced magnetic field beθ

equation becomes
! 2
"
∂
α2
2 2
−
c
∇
B
=
(∂t θ)2 Bex .
ind
2
2
∂t
π

(17)

where we have used ϵ0 µ0 = 1/c2 . Thus, the magnetic
field is additionally induced by the applied magnetic field
Bex . The direction of the magnetic field is parallel to
Bex direction and its magnitude is proportional to (∂t θ)2and Bex . When (∂t θ)2 cannot be ignored, (∂t θ)2 also
contributes to the wave equation. If the applied magnetic
field is spatial uniform, the time-revolution of the induced
magnetic field is described by
1 # cαµ0 $2 θ02 2
Bind (t < τf ) ≈t Bex ,
(18)
2
π
τf2
1 # cαµ0 $2 2
Bind (t ≥ τf ) ≈θ0 Bex(19)2
π2.

In the presence of an external electric field

We will discuss the property of Bind of Eq. (16) inthe presence of the static and spatially uniform magnetic
field Hap ||ẑ. From Eq. (16), we obtain Bind within theboundary condition B (t ≤ 0) = 0 and ∂ B (t = 0) =ORU@j@RU
@@@aHmC@@@j@aHNC

0

t0

t0+

t

Additionally induced
magnetic field δB
Besides, charge current
via chiral magnetic effect(Case 3) in the presence of the external magnetic field and charge current (inplane)

B ,x (mT)

B|| (mT)bycal magnetization flip time τf . Therefore, here ∂t θ
proximately estimated as ∂t θ(0 ≤ t ≤ τf ) ≈ −θ(t =
! 2
"
2
∂
cαµ
α
0
so the time-dependent axion-induced magnetic fi− c2 ∇2 Bind = −∂t θ (∂t D + je ) + 2 (∂t θ)2 Bex .
2
comes
∂t
π
π
(20)
1 µ0 cα θ0Contribution
Contribution
Bind (t < τf ) ≈(je + ∂t D)t2 + µHap ,2 π τf
fromBind within the boundaryfrom Bex
From Eq. (16), we obtain
condition Bind (t ≤ 0) = 0chargeand ∂t B
current
ind (t = 0) = 0 as
where θ0 ≡ θ(t = 0) denotes the axion field in the#$state. In the case of t ≥ τf , there is no magnet
1 cαµ0
150
2
2
(b)
B
(t
≤
τ
)
=
−
∂
θ(j
+
∂
D)t
+
µH
+
o
(∂
θ)
.
ind
f
t
e
t
ap
t
dynamicsof the magnetic insulators, so ∂t θ becom(a)
Bex = 0.12T π
Then, unlike Bind (t < τf ), the induced magnet(21)z
Bind (t ≥ τf ) is represented by|| FI1
The first and second term denote the magneticx field ofTI
1 µ0 cαthe time-dependent axion field and the applied magnetic
FI2
BNI
θ0 (je + ∂t D)τf + µHap .
ind (t ≥ τf ) ≈j  of the
2 π100 field, respectively. It is found that the direction200 axion-induced magnetic field is along the dielectric polarThese above Eqs. (22)-(23) are main results in thisized current jp = ∂t D and the magnitude of theVmagnetic
Finally, we discuss how to experimentally det−1field is determinedj =10
by7A/mjp2, andj ||∂+x(m5 /m4 ).
t θ = ∂t tan
0 Note that m depends
time-dependent axion field-induced magnetic fieldj || −x
j =107A/m
on2,the appliedmagnetic field Hap
5axion materials we consider has been realized in th
but m4 is basically independent of Hap . Hence, ∂t θ istilayer of the TIs film and Cr-doped TIs film4 ,−200 determined from the time-dependent of m4 or the magTherefore, the obtained magnetic field should bnetization flip due
the time0
2to Hap . Phenomenologically4
exsured by using Gauss maters. The obtained mscale of the Timemagnetization(ns) flip is determined by the typifield can be also electrically detected via the planeﬀect, because the Hall voltage of the planer Hal
depends on the induced magnetic field. Because(c)!
M
induced magnetic field is parallel to the applied(a)!
current je + ∂t D, the Hall voltage due to the co
M1
M2
tion of the induced magnetic field is distinguished

B

B

B

How*can*this*eﬀect*be*detected*?
[1].*Planar*hall*eﬀect*
Planar*hall*voltage*lM2*sinθ
j@provably*µV*order*and*it’s*possible*
to*detect*the*nsec*response*
[2].*Chiral*magnehc*eﬀect**

Chiral*magne'c*eﬀect*

Probably,*device*resistance*should*be*~MΩ,*it’s*possible*to*detect*<*nA*current.**
FH

Summary[1] We study the multilayer of topological insulators (TI) and magnetic insulator (MI)[2] For “axion”-response in materials,
we consider axion insulator and how to drive ∂tθ≠0

joUse magnetization flip by Bex[3] Main result: We find that ∂tθ and an applied electric field trigger magnetic field

(b)



(a)
FI1MI1

TI

ΔT

FI2 
Normal insulator (spacer)

FI1 
TI

M1

FI2 

M2

ΔM

z

B|| FI1

x
j

TI
FI2
NI

V

B

Normal insulator (spacer)θ = Arctan(φ5 /φ4)Bex

[Electromagnetic effect]

Summary***
" Axion*is*also*important*in*the*material*science,*specially*in*
topological*insulator** θ = Arctan(φ5 /φ4)
" *By*this*collabora'on,*we*phenomenologically*found*new*method*and*response*for*dynamical*axion*response**" *Next*plan*is*to*ﬁrst*detect*this*response*and*connect*this*collabora'on*to*par'cle*physics*and*cosmology*

Please*join*us*if*you*are*interested*in*such*achvihes*!!

FL