熱場の量子論とその応用
8/22/2016 RIKEN

トポロジカル物質における電磁結合
Electromagnetic coupling in topological matters

Kentaro Nomura (IMR, Tohoku)

collaborators
Yasufumi Araki, Koji Kobayashi, Daichi Kurebayashi,
Ryota Nakai, Yuya Ominato, Akihiko Sekine,
Shinsei Ryu, Akira Furusaki, Naoto Nagaosa

熱場の量子論とその応用
8/22/2016 RIKEN

トポロジカル物質における電磁結合
Electromagnetic coupling in topological matters

Kentaro Nomura (IMR, Tohoku)
outline• Introduction to topological insulators• Electromagnetic and thermal responses• Charge-spin coupling in Weyl semimetals

What is a topological insulator?
E

Normalinsulator

_

+

EF

Topologicalinsulator

_

+

n= 0

n= 1

Quantum Hall insulator
K. von Klitzing 1980

B

2DEG

*
𝑒
&' = 𝑁𝜎"# = 𝑅"#ℎ𝑁 ∈ ℤ = { 0 0, −1,0,1, 2,0 0 0}

n= 1

Z2 Topological insulators
T-symmetry broken

Z = {..,-1, 0, 1, 2, …}
Quantum Hall effectT-symmetry preserved

Z2 = { 0, 1 }even or odd

Quantum spin Hall effect(Z2 topological insulator)

Basic idea
Quantum spin Hall effect (QSHE)TheoryKane-Mele (2005)
Bernevig-Zhang (2006)
Bernevig-Hughes-Zhang (2006)
Experiment
Molemkamp goup (2007) HgTe QWEkSpin-Orbit Coupling (SOC)
Moving electrons feel
an effective magnetic field

SOC in an atom+

S_

Beff = −H so = −g µ B S ⋅ Beff

v
E

−e!
=
p×E⋅S
2 2
2m c
= λ r × p⋅S

Hso = λ L ⋅ S

v ×E
v2
1− 2
c

Spin-Orbit Coupling (SOC)
Moving electrons feel
an effective magnetic field

SOC in an atom+S_

H so = −g µ B S ⋅ Beffv
E

Analogy with a magnetic field(p − eA)2H=
2m

Beff = −

v ×E
v2
1− 2
c

−e!
=
p×E⋅S
2 2
2m c
= λ r × p⋅S−e
H B−orbital = p ⋅ Am−e=r × p⋅B
2m

1
A = B×r
2

2dTI in HgTe/CdTe quantum welltsp=0
s-bandEn (k )En (k )kk

p-band

4.2. 2 次元トポロジカル絶縁体

E

(a)

(b)

H so = λ L ⋅ S

tss
tsp

s

+i
-1

p

63

λ 2 2 2J −L −S
+12

=

-i

-tpp

s

px+ipy()

2dTI in HgTe/CdTe quantum well
Weak SOC

Strong SOC
tsp=0

tsp=0

En (k )

s-bandkp-band

4.2. 2 次元トポロジカル絶縁体

E

63

Bernevig,
Hughes, Zhang (2006)
(b)

(a)

tss
tsp

s

+i
-1

p

+1
-i

-tpp

s

px+ipy2dTI in HgTe/CdTe quantum well
Weak SOC

Strong SOC +sp hybridization

tsp=0

tsp=0

s-bandp-band

4.2. 2 次元トポロジカル絶縁体

E

63

Bernevig,
Hughes, Zhang (2006)
(b)

(a)

tss
tsp

s

+i
-1

p

+1
-i

-tpp

s

px+ipy2dTI in HgTe/CdTe quantum well
Normal insulator

Topological insulator
tsp=0

s-band第3章

量子 Hall 絶縁体

らないが、これは図 3.5(b) にあるエッジモードに他ならない。一方、

p-bandな m > 0 の領域と真空の間ではエッジモードは現れない。この意味

xy = 0 の状態は真空とトポロジカルにつながっている。

(a)with boundary

(b)
2

m<0

1

x

E(kx) 0
-1

y

-2

0

kx

tsp=0

2dTI in HgTe/CdTe quantum well
Normal insulator

Topological insulator
tsp=0

s-band第3章

量子 Hall 絶縁体

らないが、これは図 3.5(b) にあるエッジモードに他ならない。一方、

p-bandな m > 0 の領域と真空の間ではエッジモードは現れない。この意味

xy = 0 の状態は真空とトポロジカルにつながっている。

(a)with boundary

(b)
2

m<0

1

x

E(kx) 0
-1

y

-2

0

kx

tsp=0

2dTI in HgTe/CdTe quantum well

2 次元トポロジカル絶縁体

63(b)Normalinsulator

E

ロジカル絶縁体

63

(b)

+i
-1
s-band+1+Topologicalinsulator
_

-i

s+i
-1

+1

_

px+ipyp-band

+

-i

グの様子。
s 軌道および p 軌
px+ip(a)yweak SOC

x + ipy 状態の波動関数の異方性。

(a) s 軌道および p 軌の波動関数の異方性。

, s, σz ⟩⟨R, s, σz |σz |eµ ,pσpzσ, zσ,z |σz ⟩⟨R, pσz , σz |⟨R,zσz ⟩⟨R, pσz , σz ||R + eµ , s, σ#z ⟩⟨R, pσz , σz |pσz , σz ⟩⟨R, s, σz |σz|R + eµ , pσ , σz ⟩⟨R, s, σz |#strong SOC

2dTI in HgTe/CdTe quantum well

2 次元トポロジカル絶縁体

63(b)Normalinsulator

E

ロジカル絶縁体

63

(b)

+i
-1
s-band+1Topologicalinsulator
_

+

-i

s+i
-1

+1

_

px+ipyp-band

+

-i

グの様子。
s 軌道および p 軌
px+ip(a)yweak SOC

x + ipy 状態の波動関数の異方性。

strong SOC(a) s 軌道および p 軌の波動関数の異方性。

, s, σz ⟩⟨R, s, σz |σz |eµ ,pσpzσ, zσ,z |σz ⟩⟨R, pσz , σz |⟨R,zσz ⟩⟨R, pσz , σz ||R + eµ , s, σ#z ⟩⟨R, pσz , σz |pσz , σz ⟩⟨R, s, σz |σz|R + eµ , pσ , σz ⟩⟨R, s, σz |#Normalinsulator

Topologicalinsulator

2dTI in HgTe/CdTe quantum wellTime-reversal symmetry

Ekh = E-ki
2-fold degeneracy at k=0 is protected by symmetry

Normal0Topological

k

0

k

0

From 2d to 3d
Moore-Balents, Roy, Fu-Kane-Mele, …

E
ky
kx

Topological insulatorEk2d
HgTe QW

3d
Bi-Sb, Bi2Se3, Bi2Te3, …

Impurity effects
Hsurface = vF(sypx - sxpy ) + V0(r) + s .V(r)non-magnetic impurities

magnetic impurities

E

E

3D
Topological insulator

3D
Topological insulator

Impurity effects
Hsurface = vF(sypx - sxpy ) + V0(r) + s .V(r)non-magnetic impurities

PRL
99,99,
146806(2007)
PRL
146806(2007)

KN,impurities
Koshino, Ryu, PRL (2007)
magnetic

Topologically protected from
Anderson localization
week
endingweek
endingPHYSICAL REVIEW LETTERS
PHYSICAL REVIEW LETTERS

E

E

5 OCTOBER
2007
5 OCTOBER
2007

EnergyEnergyEnergyEnergy(a)massless
Dirac
model
(b)
random
SOSO
model
(a)
massless
Dirac
model
Topological
Conventional(b)randommodel
expectedsince
when
g isglarge,
thetheZ2 Z
topologicaleffectin in
expectedsince
when
is large,
effect2 topological
0.2 0.2
0.3 0.3
thetheNL!Mis small.
NL!Mis small.
WeWeobservethatthat
thethesingleparameter
scaling
holds
observe
singleparameter
scaling
holds
0.1 0.1
reasonably
wellin bothmodels
as as
shown
in in
Fig.Fig.
1 [24].
reasonably
wellin bothmodelsshown
1 [24].
In In
a sharpcontrast
to tothetheconventional
case,thethebeta
a sharpcontrast
conventional
case,
beta
0 0
0.25 0.25
function
of
the
Dirac
model
monotonicallyincreaseswithfunction of the Dirac model monotonically increases witha 2D
masslessDiracdecreasing
g wellbelow
g! "" 1:4:
a 2D
masslessDiracdecreasing
g wellbelow
g! 1:4:-0.1 -0.1
fermioncannotbelocalizedbyarandomscalarpotential
fermion cannot be localized by a random scalar potential
0.2 0.2[25].
numerical
betabetafunctionof ofthetheDiracmodel
is is 3D
[25].
numerical
function
Dirac
model
-0.2 -0.2
3DTheThe
wellwellfittedby bythetheone-loop
betabetafunctionof thesymplectic
fittedone-loopfunctionof thesymplectic
Topologicalinsulator
Topologicalinsulator
class
even
in thestrongly
disorderedregime
g&
class
even
in thestrongly
disorderedregimeg 1.
& 1.
-0.3 -0.3
TheTheabsenceof oflocalization
in inthetheDiracmodel
cancanabsencelocalization
Dirac
model
intuitively
be beunderstoodby byexaminingthethespectralflow
intuitively
understoodexamining
spectralflow
inducedby bytwistingboundary
conditions.
LetLetus usconinduced
twistingboundary
conditions.
con- FIG. 2 (color
Z2spectra
even
Z2online).
odd
evolutionof ofenergyas as
a aFIG. 2 (color
online).TheThe
evolutionenergyspectraImpurity effects
Hsurface = vF(sypx - sxpy ) + V0(r) + s .V(r)non-magnetic impurities

magnetic impurities

E

E

3D
Topological insulator

3D
Topological insulator

Ideal uniform caseHsurface = vF(sypx - sxpy ) + V0(r) + s .V(r)E
non-magnetic
impuritiesmmbz (k) = E
3222 k +m

d 2k
1
=
b
(k)
=
∫
z
2 3D
2πinsulator
2e / Topologicalh

σ xymagnetic impurities

E

3D
Topological insulator

Quantum Anomalous Hall EffectQuantum Anomalous Hall Effect
TheoryExperiment (201３)

KN, Nagaosa (2011)

Low T

sxy

EF

sxx
0
EF

Quantum Anomalous Hall Effect
TheoryExperiment

KN, Nagaosa (2011)

PHYSICAL REVIEW LETTERS

106, 166802 (2011)
1.5

L=4 hvF /m
L=6 hvF /m
L=9 hvF /m

ME

1

j=c

XM E

σxx
0.5

0
-0.5

Checkelsky, Yoshimi, Tsukazaki, et al. (2014)

0

σxy

0.5

3 (color online). Scaling flow of !xx and !xy as increasstem size, indicating ð!xx ; !xy Þ ! ð0; !1=2Þ, in units of

week
22 AP

shown in Fig. 1(a) and the inset in Fig. 3. Sin
moments Eqs. (3) are protected by the bulk gap (remains intact) and insensitive to disorder on theEqs. (4) can survive even though all surface st
localized. With the above results, the existence
surface quantum Hall states generalizes the axion
dynamics effects to disordered systems. Precisely,
ergy region for the topological ME effect, whoriginally limited in the surface gap, is enlarged
bulk gap by the localization effect of surface stat
will greatly facilitate the experimental realization.
Even though the 2D surface Dirac Hamiltonia
such bulk properties by the construction, nonva
half-quantized Hall conductivity is represented asanomaly. A similar situation can be seen in the cochiral anomaly. For the one-dimensional edge ch
the quantum Hall system, the chiral anomaly repres

熱場の量子論とその応用
8/22/2016 RIKEN

トポロジカル物質における電磁結合
Electromagnetic coupling in topological matters

Kentaro Nomura (IMR, Tohoku)
outline• Introduction to topological insulators• Electromagnetic and thermal responses• Charge-spin coupling in Weyl semimetals

Response to ElectroMagnetic fields
in normal insulators

M = cm B

P = ce E(magnetization)(Electric polarization)
E

B

M

P

Response to ElectroMagnetic fields
in topological insulators(magnetic moment)

M = am E(electric field)(electric polarization)

P = ae B(magnetic field)
B

E

M

P

Response to ElectroMagnetic fields
B = (4p/c) j
3D TI + magnetic impurities

E

j

M =

e2
2 hc

E

Qi, Hughes, Zhang ’08
Essin, Moore, Vanderbilt ’09

On the surfaces perpendicular to the magnetic field, theLandau level structure is formed. When Fermi energy resides in the mass gap (jEj < jmj), electric charge density

Response to ElectroMagnetic fields
H E

ρ = +(e2/2h)B

(a)

+
EF

- - 𝜎"# > 0

ρ = - (e 2/2h)B
y

(b)

+ +
+ + +

z

σxy

Surface QH statesρ = +(e 2/2h)B

𝑗@ = 𝜎"#𝜖 @BC𝐹BC

H

E

-

B E

-

-

+ +

𝜎+"#+<+0 +σxyx𝜌 = 𝜎"# 𝐵>
𝒋 = 𝜎"#𝑬×𝒛;

Response to ElectroMagnetic fields* 𝜃𝑒
𝐸HI = − J 𝑑 L 𝑥
𝑬0𝑩
4𝜋ℏ 𝜋

+ + + +- -++ + + +- -+- - -Surface QH states

𝑗@ = 𝜎"#𝜖 @BC𝐹BC

B E

+- +-

𝜌 = 𝜎"# 𝐵>
𝒋 = 𝜎"#𝑬×𝒛;

Response to ElectroMagnetic fields* 𝜃𝑒
𝐸HI = − J 𝑑 L 𝑥
𝑬0𝑩
4𝜋ℏ 𝜋

+

B

+ + + +
+ + + + +
𝜃=𝜋
- - -

-

E

𝜃=𝜋
j
-

𝛿𝐸HI
𝑒* 𝜃
𝑷=−
= 𝜌 =𝑩𝜎"# 𝐵>
4𝜋ℏ 𝜋
𝑗@ = 𝜎"#𝜖 @BC𝐹𝛿𝑬
𝛿𝐸BCHI𝑒 *𝒋 =
𝜃 𝜎 𝑬×𝒛;
𝑴=−
=
𝑬 "#
𝛿𝑩
4𝜋ℏ 𝜋

Surface QH states

Qi, Hughes, Zhang ’08

Axion term (q term)
𝑆XYZ[\]]= − J 𝑑𝑡𝑑 L 𝑥

𝑗𝜇𝐴𝜇 +

1(𝑩2 − 𝑬2)8𝜋*𝑒𝑆h = J 𝑑𝑡𝑑 L 𝑥
𝜃𝑬 0 𝑩2𝜋ℎ𝜃 = 𝜃 𝒙, 𝑡

Peccei, Quinn 1977
Wilczek 1987

𝛿𝑆
=0
𝛿𝐴𝜇

𝑒2
𝜃
𝜵 0 𝑬 = 4𝜋 𝜌 +
𝜵
0𝑩
2ℎ
𝜋
𝜕𝑬
𝑒2
𝜃
𝑒2 𝜃̇
𝜵×𝑩 −
= 4𝜋 𝒋 +
𝜵
×𝑬 +
𝑩
𝜕𝑡
2ℎ
𝜋
2ℎ 𝜋

Axion term (q term)+B+ + + +
+ + + + +
𝜃=𝜋
- - -

𝜃=0

-

E

𝜃=𝜋
j
-

𝜃=0
Qi, Hughes, Zhang 2008𝑒2
𝜃
𝜵 0 𝑬 = 4𝜋 𝜌 +
𝜵
0𝑩
2ℎ
𝜋
𝜕𝑬
𝑒2
𝜃
𝑒2 𝜃̇
𝜵×𝑩 −
= 4𝜋 𝒋 +
𝜵
×𝑬 +
𝑩
𝜕𝑡
2ℎ
𝜋
2ℎ 𝜋

Topological states- Quantum Hall effect
Laughlin (1980)
TKNN (1982)
Kohmoto (1985)

Z = {..,-1, 0, 1, 2, …}
B- Quantum spin Hall effect(topological insulators)
2d Kane-Mele (2005), BHZ(2006)
3d Fu et al., Moore-Balents, Roy- Chiral superconductors
Read-Green (2000)
Kitaev (2000)- Topological SC (3d)Schnyder et al. (2008)
Fu-Berg, Sato (2010)

Z2 = { 0, 1 }even or odd

Topological states- Quantum Hall effect
Laughlin (1980)
TKNN (1982)
Kohmoto (1985)- Quantum spin Hall effect(topological insulators)
2d Kane-Mele (2005), BHZ(2006)
3d Fu et al., Moore-Balents, Roy- Chiral superconductors
Read-Green (2000)
Kitaev (2000)- Topological SC (3d)Schnyder et al. (2008)
Fu-Berg, Sato (2010)dsymmetry

QHE and chiral SC
2d chiral SC

QHE
2DEG

3 He-A, Sr RuO , …
2
4

B

E = vpqlz = -1 pair

Bulk: gapped (Landau gap)

Bulk: gapped (SC gap)

Edge: chiral Dirac modes(edge states)

Edge: chiral Majorana modes(Andreev bound states)

QHE andand chiralchiral
SC
QHE
QHE
and
chiralSC
SC
2d chiralchiral SC
SC
2d

QHE
QHE
2DEG(
2DEG(
2DEG


B

333HeMA,(SrHeMA,(SrRuO444,(…
,(…
He-A, Sr222RuO
, …

=






E
vp
q
1pair
l
z



=-

(gauge(symmetry(preserved(((((((((((((((((((((((((((((((gauge(symmetry(broken%(gauge(symmetry(preserved(((((((((((((((((((((((((((((((gauge(symmetry(broken%
gauge symmetry preserved
gauge symmetry broken(total(charge(is(conserved
(total(charge(is(conserved
total charge is conserved(but(total(energy(is(sCll(conserved
(but(total(energy(is(sCll(conserved
but total energy is still conserved

Charge(transport(((((((((((((((((((((((((((((((((((((((((((((((((((((((((Thermal(transportCharge(transport(((((((((((((((((((((((((((((((((((((((((((((((((((((((((Thermal(transportCharge transport
Thermal transport

QuantumHall effect
QHE
QHE
QHE
2DEG
2DEG(

B

Charge current
𝒋 = 𝜎j 𝑬×𝒛;
𝒋 = 𝜵×𝑴 =Streda formula :Streda(formula(:Cross-correlated
CrossMcorrelated(
responses:((((((((((responses:

Mzz== ((σsHH/ec)/ec)δµdµδdMdN == ((σsHH/ec)/ec)δdBBzz
δΝ

𝜕𝑴
×𝜵𝜇
𝜕𝜇

Streda formula: sH a kH
We consider the BdG theory of SCs
Energy current
E = vpq

𝒋I = −𝜅j 𝜵𝑇×𝒛;

lz = -1 pair𝒋I = 𝜵×𝑴I =

𝜕𝑴I
×𝜵𝑇
𝜕𝑇

Qin et al. PRL (2011), Nomura et al. PRL (2012)

Streda-like formula :

𝜕𝑀I>
𝜅j =
𝜕𝑇

Streda formula: sH a kH
Hamiltonian𝜀"𝑝
𝑇 @B = #𝑝𝑝>

Energy current
𝑗I" 𝑗I# 𝐽I>

Energy current
𝒋I = −𝜅j 𝜵𝑇×𝒛;
𝒋I = 𝜵×𝑴I =Momentum𝐿rs = 𝑥r 𝑝s − 𝑥s 𝑝r

𝜕𝑴I
×𝜵𝑇
𝜕𝑇

Qin et al. PRL (2011), Nomura et al. PRL (2012)

Orbital angular momentum L
rs
𝑀I =

s
𝑥 r 𝑗I − 𝑥 s 𝑗Ir

𝒋I = 𝜵×𝑴I

𝜕𝑀I>
𝜅j =
𝜕𝑇

sHH aκHkH
Stredaformula:Stredaformula: σHamiltonian
Hamiltonian𝜀"𝑝
𝑇 @B = #𝑝𝑝>

Energy current 
𝑗I" 𝑗I# 𝐽I>

In (pseudo) Lorentz invariant cases
In((pseudo)(Lorentz(invariant(cases
(e.g. BdG Hamiltonian)(e.g.(BdG(Hamiltonian)







Momentum
MomentumME ∝ L

𝐿Lrs= 𝑥r 𝑝s − 𝑥s 𝑝r

Nomura et al. PRL (2012)

Orbital angular momentum L
rs

s

r
s r
𝑀
MI = 𝑥 𝑗I − 𝑥 𝑗I

𝒋I = 𝜵×𝑴M
I E

𝜕𝑀I> 𝑣 * 𝜕𝐿>
𝜅j =
=
𝜕𝑇
2 𝜕𝑇

Temperature
gradient== gravity
Temperaturegradientgradient=
gravity
Temperaturegravity
Equivalence+Principle:
Equivalence+Principle:
Equivalence
Principle:
The(laws(of(physics(in(a(gravita>onal%ﬁeld(are(idenCcal(
The
laws of physics in a gravitational field are identicalThe(laws(of(physics(in(a(gravita>onal%ﬁeld(are(idenCcal((to(those(in(a(local(accelera>ng%frame(and(in(a(temperature%gradient.
tothose in a local accelerating frame and in a temperature gradient. (to(those(in(a(local(accelera>ng%frame(and(in(a(temperature%gradient.
TolmanMEhrenfest(,(Lujnger(
Tolman-Ehrenfest, Luttinger
TolmanMEhrenfest(,(Lujnger(

gravitaConal(ﬁeld
gravitational
field
gravitaConal(ﬁeld

TT1T11
TT2T22

φφg ==gNewlyfound
formulae
Newlyfound
foundformulae
formulae
Newly
Streda (1982)
Present work
Streda((1982)((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Present(work
Streda((1982)((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Present(work

φφgg

zz
BB


φφ


BBzgzgdfg = T-1dT , Bzg=(2/v2)W z
2)2Ωz%d%dφφg g=%T=%TD1D1dT%,%%B
dT%,%%Bzg=(2/v)Ωz z % g=(2/v%'

F = e (E + v v x B)

F = m (Eg + 2v x Bg)
FF==mm(E(Egg++2v
2vBBgg))

Lorentz force
Coriolis force
Lorentz(force((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Coriolis(force
Lorentz(force((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Coriolis(force
Electromagnetism-gravity analogy
ElectromagneCsmMgravity(analogy
ElectromagneCsmMgravity(analogyResponsesof
2d-TSC
Responses
of
2d-TSC
Newlyfound
foundformulae
formulae
Newly
Streda (1982)
Present work
Streda((1982)((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Present(work
Streda((1982)((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Present(work
Streda((1982)((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Present(work

φg
φφgg

Bz
zz
BB


φ
φφ


Bzg
BBzgzg

B
2)2Ω
z
%d%dφφg g=%T=%TD1D1dT%,%%B
dT%,%%Bzg=(2/v)Ωz z % g=(2/v%

FF==mm(E(Egg++2v
2vBBgg))

Thermal energy changes as
Charge density changes as
Thermal(energy(changes(as(Charge(density(changes(as(Lorentz(force((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Coriolis(force
Lorentz(force((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Coriolis(forceΔQ =(2κHHT/v
T/v22)ΔΩ)ΔΩzz%((((((ΔQ%=(2κElectromagneCsmMgravity(analogy
ElectromagneCsmMgravity(analogyNewlyfound
formulae
Newly
found
formulae
Newlyfound
foundformulae
formulaeNewly
2 dimensions
2(dimensionsQHE
2d chiral SC
QHE((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((2d(chiral(SCStreda((1982)((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Present(work
Streda((1982)((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Present(work


φφφ


φφφggg

z
BB
zz
B


z
B
z
BBggzgPrevious section
Previous(secCon
3 dimensions
3(dimensions2)2Ω
z%d%dφφg g=%T=%TD1D1dT%,%%B
dT%,%%Bzg=(2/v)Ωz z % g=(2/v%

3d-TI
3d-TSC
3dMTI(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((3dMTSC

FF==mm(E(Egg++2v
2vBBgg))
E j

Lorentz(force((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Coriolis(force
Lorentz(force((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Coriolis(force
Egj
Bgj
B j
ElectromagneCsmMgravity(analogy
ElectromagneCsmMgravity(analogy

This(secConElectric/Thermal responses
Quantum Hall effectat the surface
3d TI

Quantum thermal Hall effectat the surface
3d TSC𝜵𝑇j : electric charge current
Magnetization is generated electrically
Qi, Hughes, Zhang (2008)jE : thermal energy current
Present workElectric/Thermal responses
Quantum Hall effectat the surface
3d TI

Quantum thermal Hall effectat the surface
3d TSC
MajoranaTsurface

Dirac surface
E

E
k

k

j : electric charge current
Magnetization is generated electrically
Qi, Hughes, Zhang (2008)jE : thermal energy current
Present work

QTHE
surface Majorana
Majorana fermions
QTHE
ofofsurface
fermions
Majorana fermions at the surface Quantum(thermal(Hall(eﬀect(
Quantum
thermal Hall effect((((((((((((((((((((((((((((((at(the(surface(at the surface

Majorana(
MajoranaT(surfacesurface%%%EE (φg: gravitational potential [Luttinger (1964)]

k k

j%jE E:%:(thermal(energy(current(
thermal energy current

Present(work(
Present
work

QTHE ofof surface
surface Majorana
Majorana fermionsfermions
QTHE
QTHE of surface Majorana fermions
Majoranafermionsfermionsat
atthe
thesurface
surface
Majorana

Quantum(thermal(Hall(eﬀect(jE(/v2(=Pφ :(momentum(((((((((((((((((((((((((((((((at(the(surface(
Orbital(angular(momentum((((((((((((((((((((((((((((((per(unit(volume
Majorana((surface

gravitationalpotential
potential[Luttinger[Luttinger(1964)]
(1964)]
(φ(φg:g:gravitational%%%E

coupling(energyk

L . Ω

%jE%:(thermal(energy(current(
Present(work(

Thermalresponses
responses in
Thermal
in3d
3dTSC
TSC
Topological insulator
Topological insulator
Bi22SeSe3(,(Bi2Te3(,…(Bi
2 33(, Bi22Te33 ,…
(((QHE((of(QHE of((((Dirac(fermions
Dirac fermions

Topological superconductor
Topological superconductor
3HeMB,(Cu Bi Se ((?)
33He-B,
CuxxxBi222Se333 (?) (((QTHE((of(QTHE of(((Majorana(fermions
Majorana fermionsT(((((((((((((((((((Qi,Qi,(Hughes,(Zhang((2008)
Hughes, Zhang (2008)

Thermalresponses
responses in
Thermal
in 3d
3dTSC
TSC
Topologicalinsulator
insulator
Topological
Bi22SeSe33(,,(BiTe33,…
(,…(Bi
Bi22TeTopologicalsuperconductor
superconductor
Topological
33He-B,
HeMB,(Cu
Se33(?)((?)
CuxBi
xBi22Se(((QTHE((of(QTHE of
Majorana fermions(((Majorana(fermions(((QHE((of(QHE ofDirac fermions((((Dirac(fermionsT

M == 2ehc EEM
e2
B
PP == 2hc B
2(((((((((((((((((((Qi,Qi,(Hughes,(Zhang((2008)
Hughes, Zhang (2008)

L =
PE =

Eg
Ω

Present(work(

Summary
Summary
Topologicalproperties of insulators and superconductors are characterized by
Topological(properCes(of(insulators(and(superconductors(are(characterized(by((Cross-correlated
responses
CrossMcorrelated(responses
EM(responses(((((((((((((((((((((((((((((((((((((((((((((((((Thermal(responses
EM
responses
Thermal responses

2D(

3D(

δMz= (σH/ec)δµδLz= (2κH/ v2) δΤ

δN%%= (σH/ec)δBz

δQ = (2κHΤ/v2)δΩz

M =
P =

E
B

L =
PE =

Eg
Ω

Response toto thermal
thermal gradient
Response
gradient
Cross-correlation between
between ElectricElectric field
field EE andand Magnetic
Magnetic field
field B
B
Cross-correlation

M == aαmmEEM
Cross-correlation between
between Thermal
Thermal gradientgradient andand Rotational
Rotational motion
motionCross-correlation

L
TT

T

(angular(momentum
angularmomentum

T

T+ΔT
T+DT

Related works
Hidaka, Hirono, Kimura, Minami
5
ShitadeStreda formula and (b) momentum pumping/transport
betweenthe edges states using a Laughlin gauge arguKimura
ment. Let us discuss these in a bit more detail.
One can write the Streda formula for the Hall viscosShiozaki,
Fujimoto
ity in terms of bulk quantities following analogy to theelectromagnetic response. Let us motivate the idea by
Stone
first
reviewing the electromagnetic response. The Streda
formula for the Hall conductance is✓◆Goswami,= Roy
@M
e
@µ
13
Goswami,Tewariwhere e is the elementary charge, M is the bulk magnetization, µ is the chemical potential, T is temperature,
Gromov,AbanovandB denotes the magnetic
field strength. The conjugate quantities to the magnetization and the chemical
potential
are magnetic
field and particle number density,
Bradlyn,Readrespectively. By means of thermodynamic conjugacy relations we obtain
Nakai, Ryu,✓Nomura
◆
@n
=e
.
@B
….
ber in the periodic y direction. A generalized Laughlin
y

51

x

H

z

T,B

FIG. 2. The generalized Laughlin experiment for the Hallviscosity. A viscoelastic flux is threaded through the cylinder,
which one should think of as threading a dislocation through
0.1 Electrons encircling the cylindrical hole will be
the cylinder.
translated by the Burgers’ vector of the dislocation flux. For
the case illustrated here the Burgers’ vector is in the periodic y 0.05
direction. If the system has a Hall viscosity there will
be a momentum current flowing in the x-direction carrying
momentum pointing in the y-direction.

ζH

z

0

H

T,µ

Intuitively, this expression states that inserting a magnetic flux binds charge to that flux. In a lattice model,one
P can look at the2 change in the particle density n(x) =⌫2occ. |hx|⌫(B)i| as a function of the magnetic field(B) where the single particle states for a fixed B are denoted by |⌫(B)i. For small amplitudes of B the density
varies linearly with B, and the Hall conductance can be−0.05
experiment8 can be done to measure the Hall viscosity.
Threading a vicoelastic magnetic flux G along the axis
−0.1 changes the transverse momentum argumentof cylinder
−4
−2
0
2
4
k2 of the Hamiltonian h(k1 , k2 ) into k2
G sin(k2 a)/a.
m
Thus, as G is increased, the spectrum of the edge states
is modified and there is a net momentum transfer from
FIG. 9. Comparison
44 of the Hall viscosity using the gauge
one end to. TheandHalltheviscosity
is givenbyformalism
in another
Eq. (4.3) (solid)
momentumpolarization

熱場の量子論とその応用
8/22/2016 RIKEN

トポロジカル物質における電磁結合
Electromagnetic coupling in topological matters

Kentaro Nomura (IMR, Tohoku)
outline• Introduction to topological insulators• Electromagnetic and thermal responses• Charge-spin coupling in Weyl semimetals

Dirac semimetal
E

Normalinsulator

Topologicalinsulator
_

+

EF

_

+

weak SOC

strong SOC

Dirac semimetalE(p) = ±v F px2 + py2 + pz2

Relativistic quantum mechanics• Dirac theory• Weyl theory

𝐻 = 𝑝" 𝛼' + 𝑝# 𝛼* + 𝑝> 𝛼L + 𝑚z 𝛼{

ai: 4x4 Dirac matrices

𝐻 = 𝑝" 𝜎' + 𝑝# 𝜎* + 𝑝> 𝜎L

si: 2x2 Pauli matrices
massless

degeneratepx

non-degenerate E

E

py

px

py

Dirac-Weyl semimetals• Dirac semimetals
𝐻 = 𝑝" 𝛼' + 𝑝# 𝛼* + 𝑝> 𝛼L + 𝑚z 𝛼{• Weyl semimetals
𝐻 = 𝑝" 𝜎' + 𝑝# 𝜎* + 𝑝> 𝜎L

ai: 4x4 Dirac matrices

si: 2x2 Pauli matrices

m0=0 (massless)degeneratemassless

non-degeneratek = p + K0

Dirac-Weyl semimetals• Dirac semimetals• Weyl semimetalspz

degeneratepy

non-degenerateDirac-Weyl semimetals• Dirac semimetals• Weyl semimetals

H = pxα1 + pyα 2 + pzα 3 + m0α 4 + bSz
Spin terma breaks time-reversal symmetry

degenerate

b=0

non-degenerateb≠0Dirac-Weyl semimetals• Dirac semimetals• Weyl semimetals

H = pxα1 + pyα 2 + pzα 3 + m0α 4 + JMSz

Hexc
= J∑
sd

Nimp
i=1

high T
degenerate

M=0paramagnetic

S(Ri ) ⋅ Σ
low T

non-degenerateM≠0ferromagnetic

Dirac-Weyl semimetals
Kurebayashi, KN. JPSJ 83, 063709 (2014)70high T

60

Dirac semimetal

Temperarture [K]

50

40

30

Weyl semimetal

20

low T

10

0
-0.3

degenerate

M=0paramagnetic

-0.2

-0.1

0
M0 [eV]

0.1

0.2

0.3

non-degenerateM≠0ferromagnetic

Magnetically doped TI
Bi2-yCry(Sex Te1-x)3Cr
Science 339, 1582 (2013)

Anomalous Hall Effectρ xy = R0B z + Rs Mz

ρ xy ∝ M (B → 0)
occurs in a system with• Spin-orbit coupling• Ferromagnetic orderAnomalous Hall Effect_ _ _ _ _ρ xy = R0B z + Rs Mz

jx+ + + + +Mz >0

ρ xy ∝ M (B → 0)
occurs in a system with• Spin-orbit coupling• Ferromagnetic order

E
E

Chiral anomaly
1D Weyl fermions

(

)

(

)

H1D = ∫ dx ψR+ −i∂ x + eAx ψR − ψL+ −i∂ x + eAx ψL

EnergyL

Energy

k

R

k

dk
= −eE
dt

Chiral anomaly
1D Weyl fermions

(

)

(

)

H1D = ∫ dx ψR+ −i∂ x + eAx ψR − ψL+ −i∂ x + eAx ψL
dNL−e= − ∫ dx E
dt
2π

dNR
−e
= + ∫ dx E
dt
2π

EnergyL

Energy

k

R

k

dk
= −eE
dt

Chiral anomaly
1D Weyl fermions

(

)

(

)

H1D = ∫ dx ψR+ −i∂ x + eAx ψR − ψL+ −i∂ x + eAx ψL
dNL−e= − ∫ dx Edt
2πd(NR + NL )
=
dt

dNR
−e
= + ∫ dx E
dt
2π

0

d(NR − NL )−e
= ∫ dx E
dt
πtotal charge

axial charge

Chiral anomaly
3D Weyl fermions

!
!
!
+!
H3D = ∫ d x ψ σ ⋅ −i∇ + eA + JM ψR − ψL σ ⋅ −i∇ + eA − JM ψL
3

+
R

(

(

)

)

B=0
kz

d(NR + NL )=dt

0d(NR − NL )−e
= ∫ dx E
dt
πtotal charge

axial chargefor 1D

Chiral anomaly

12.2. 経路積分形式による定式化

287

掛けた

3D Weyl fermionsd(NR − NL ) =
dt

!

d3 x

−e2
E·B
πhc

(12.31)!
!
!
+!
H3D = ∫ d x ψ σ ⋅ −i∇ + eA + JM ψR − ψL σ ⋅ −i∇ + eA − JM ψL
3(+が成立することが示唆される。
R

(

)

)

En(kz)

B≠0
kz

0

L

R

NLL =

図 12.3: 磁場中の３次元 Weyl 半金属のエネルギー分散。Landau 量子化d(N + N )=磁場極限での低エネルギー励起はE = 0 と交差するゼロ Landau 準位のdt

R 準位のスペクトルはLによって各 Landau
0 kz の関数として与えられる。強total charge
中に限られる。

d(NR − NL )−eaxial charge= ∫ dx E
２つの Weyl ノード間の散乱があるときはdt
π NR − NL は一定値に緩和す

1Dる。緩和時間を τ とすると運動方程式 k̇z = −eEz for− ∆kz /τ から波数の変化分 ∆kz = −τ eEz が得られる。この強磁場条件で発生する電流は

BLx Lyhc / eChiral anomaly

12.2. 経路積分形式による定式化

287

掛けた

3D Weyl fermionsd(NR − NL ) =
dt

!

d3 x

−e2
E·B
πhc

(12.31)!
!
!
+!
H3D = ∫ d x ψ σ ⋅ −i∇ + eA + JM ψR − ψL σ ⋅ −i∇ + eA − JM ψL
3(+が成立することが示唆される。
R

(

)

)

En(kz)

B≠0
kz

0

L

R

NLL =

図 12.3: 磁場中の３次元 Weyl 半金属のエネルギー分散。Landau 量子化d(N + N )=磁場極限での低エネルギー励起はE = 0 と交差するゼロ Landau 準位のdt

R 準位のスペクトルはLによって各 Landau
0 kz の関数として与えられる。強total charge
中に限られる。

d(NR − NL )2e 2
3= ∫d xE ⋅ B axial charge
2
２つの Weyl ノード間の散乱があるときはdt(2NπR)− NL は一定値に緩和する。緩和時間を τ とすると運動方程式 k̇z = −eEz − ∆kz /τ から波数の変化分 ∆kz = −τ eEz が得られる。この強磁場条件で発生する電流は

BLx Lyhc / eChiral anomaly

12.2. 経路積分形式による定式化

287

掛けた

3D Weyl fermionsd(NR − NL ) =
dt

!

d3 x

−e2
E·B
πhc

(12.31)!
!
!
+!
H3D = ∫ d x ψ σ ⋅ −i∇ + eA + JM ψR − ψL σ ⋅ −i∇ + eA − JM ψL
3(+が成立することが示唆される。
R

(

)

)

En(kz)

a = JM“chiral vector potential”
kz

0

L

R

図 12.3: 磁場中の３次元 Weyl 半金属のエネルギー分散。Landau 量子化d(N + N )=磁場極限での低エネルギー励起はE = 0 と交差するゼロ Landau 準位のdt

R 準位のスペクトルはLによって各 Landau
0 kz の関数として与えられる。強total charge
中に限られる。

d(NR − NL )2e 2
3= ∫d xE ⋅ B axial charge
2
２つの Weyl ノード間の散乱があるときはdt(2NπR)− NL は一定値に緩和する。緩和時間を τ とすると運動方程式 k̇z = −eEz − ∆kz /τ から波数の変化分 ∆kz = −τ eEz が得られる。この強磁場条件で発生する電流は

Chiral anomaly
3D Weyl fermions

!
!
!
+!
H3D = ∫ d x ψ σ ⋅ −i∇ + eA + JM ψR − ψL σ ⋅ −i∇ + eA − JM ψL
3

+
R

(

)

b = ∇ × a = ∇ × JM!
e = −a! = − JM()

a = JM“chiral vector potential”

Zyusin & Burkov (2012), Liu, Ye, Qi (2013)

d(NR + NL )
=
dt

2e 2
∫ d x (2π )2 (E ⋅ b + e ⋅ B)

d(NR − NL )
=
dt

2
2e
3
d
∫ x (2π )2 (E ⋅ B + e ⋅ b)3

Chiral anomaly
2e∂µ j µ = − 2 (E ⋅ b + e ⋅ B)
2π

b = ∇ × a = ∇ × JM!
e = −a! = − JM
Zyusin & Burkov (2012), Liu, Ye, Qi (2013)

d(NR + NL )
=
dt

2e 2
∫ d x (2π )2 (E ⋅ b + e ⋅ B)

d(NR − NL )
=
dt

2
2e
3
d
∫ x (2π )2 (E ⋅ B + e ⋅ b)3

Chiral anomaly
2e∂µ j µ = − 2 (E ⋅ b + e ⋅ B)
2π

b = ∇ × a = ∇ × JM!
e = −a! = − JMµ∂µ janomaly

Zyusin & Burkov (2012), Liu, Ye, Qi (2013)

d(NR + NL )
=
dt

2e 2
∫ d x (2π )2 (E ⋅ b + e ⋅ B)

d(NR − NL )
=
dt

2
2e
3
d
∫ x (2π )2 (E ⋅ B + e ⋅ b)3

Chiral anomaly
2e∂µ j µ = − 2 (E ⋅ b + e ⋅ B)
2π

b = ∇ × a = ∇ × JM!
e = −a! = − JM
E

Mz ≠ 0janomalyEEµ∂µ janomaly

Zyusin & Burkov (2012), Liu, Ye, Qi (2013)e2
janomaly =JM × E22πe2ρanomaly =JM ⋅ B22π

Weyl fermions in B fieldnth Landau level :En (k z ) = ±!v F

2(k + τ M̂ ) +
z

xJS
!v F

z

z

2eB z
!c

|n|

(n=±1, ±2, ±3,…)
0th Landau level :charge density

ρ AHE = σ AHE M̂ ⋅ B

E0 (k z ) = −τ z !v F k z − xJSM̂z(not resistivity)
NLL =En(kz)

M0

𝑛 ≠ 0 LL𝑛 = 0 LL

En(kz)

M
kz

0

kz

BLx Lyhc / eGate-tuned magnetization dynamics
(a) z
x

(b)
y

WSM

V(t)

Insulator
Metal

(a)

d0 B

Metal

d

WSM

V(t)

Insulator

charge density

B

ρ AHE = σ AHE M̂ ⋅ B

Metal(not resistivity)

FIG. 1. (Color online) Schematic illustration of the multilayer structure of a Weyl semimetal, an insulator, and a metal. The applied
voltage between the Weyl semimetal layer and the bottom metal
Zeemandynamics.
Anisotropy
layer induces magnetization
The thickness of the Weyl
semimetal and the insulator are denoted as
d0 and d, respectively.
*~~
𝐸 𝑴pulses= −𝑔𝜇𝑴magnetization.
0 𝑩 − 𝐾𝑀# (b) Oscillating voltage€ 𝑆the(a) Voltage
switch*' and generates
induces magnetization precession
spin
~
~ The
+ *‚ 𝜎𝐴𝐻𝐸𝑴 0 𝑩 a +𝑉 current.
𝜎𝐴𝐻𝐸𝑴
0𝑩
generated spin current is injected into an adjacent metal layer.
Charging

B

Gate voltage

FIG. 2. (Coloduced by therection determ
Zeeman energ
plied voltage.
the sign of the

field as B =Gate-tuned magnetization dynamics
(a) z
x

(b)
y

WSM

V(t)

Insulator
Metal

(a)

d0 B

Metal

d

WSM

V(t)

Insulator

charge density

B

ρ AHE = σ AHE M̂ ⋅ B

Metal(not resistivity)

FIG. 1. (Color online) Schematic illustration of the multilayer structure of a Weyl semimetal, an insulator, and a metal. The applied
voltage between the Weyl semimetal layer and the bottom metal
Zeemandynamics.
Anisotropy
layer induces magnetization
The thickness of the Weyl
semimetal and the insulator are denoted as
d0 and d, respectively.
*~~
𝐸 𝑴pulses= −𝑔𝜇𝑴magnetization.
0 𝑩 − 𝐾𝑀# (b) Oscillating voltage€ 𝑆the(a) Voltage
switch*' and generates
induces magnetization precession
spin
~
~ The
+ *‚ 𝜎𝐴𝐻𝐸𝑴 0 𝑩 a +𝑉 current.
𝜎𝐴𝐻𝐸𝑴
0𝑩
generated spin current is injected into an adjacent metal layer.
Charging
Gate voltage
1 *
𝜌
2𝐶

B

𝑉𝜌

FIG. 2. (Coloduced by therection determ
Zeeman energ
plied voltage.
the sign of the

field as B =Gate-tuned magnetization dynamics
(b)

(a) z

By
x d0

Metal

d

WSM

V(t)

WSM

V(t)

Insulator
Metal

(a)

(b)

Insulator
Metal

B

d0 B

Metal

d

WSM

V(t)

(b)

B

ZeemanZeemanB(a)

B

Charging
Charging

Insulator
Metalnline) Schematicof themultilayer
struc- of the multilayer strucFIG. 1. illustration(Color online)
Schematic
illustration
emimetal,ture
an ofinsulator,
a metal.an The
appliedand a FIG.
a Weyl andsemimetal,
insulator,
metal.2. The
applied(Coloronline) Schematicillu
FIG. 2. (Colothe Weylvoltage
semimetal
layer
and
the
bottommetal
between the Weyl semimetal layer andthe bybottommetal voltage.
duced
the electric
(a)
The
duced
by
the
agnetization
dynamics.
The thicknessof the Anisotropy
Weyl
Zeemandynamics.
layer
induces magnetization
The thickness
of
the
Weyl
rection determined by thebalanceofrectiondetermhe insulator
are denoted
as dinsulator
respectively.
0 and d, aresemimetal
and thedenoted as
d0 Zeemanand d, respectively.
energy. (b),(c) Magnetization*
Zeeman energ
~
~
𝐸
𝑴
=
−𝑔𝜇
𝑆
𝑴
0
𝑩
−
𝐾𝑀
s switch themagnetization.
(b)
Oscillating
voltage€ the magnetization.
# (b)plied(a) Voltage pulses switch
Oscillating
voltagevoltage. The directionof voltage.
the spinplied*'zation precession
and generates aprecession
spin current.
The
induces magnetizationand generates
a+
spin
The
~ 0 𝑩 the~voltage.
sign
the𝑴+ *‚ 𝜎𝑴
𝑉 current.
𝜎of𝐴𝐻𝐸0𝑩
the sign of the𝐴𝐻𝐸urrent is injected
intospin
an current
adjacentismetal
layer.
generatedinjected
into an adjacent metal layer.
Charging
1 *𝜌
2𝐶

nd a metal. An electric voltage is applied be-

Gate voltage

field as B =𝑉𝜌 Bẑ. Asfield
shown
in F
as B
=

Gate-tuned magnetization dynamics
(b)

(a) z

By
x d0

Metal

d

WSM

V(t)

WSM

V(t)

Insulator
Metal

(a)

(b)

Insulator
Metal

B

d0 B

Metal

d

WSM

V(t)

(b)

B

ZeemanZeemanB(a)

B

Charging
Charging

Insulator
Metalnline) Schematicof themultilayer
struc- of the multilayer strucFIG. 1. illustration(Color online)
Schematic
illustration
emimetal,ture
an ofinsulator,
a metal.an The
appliedand a FIG.
a Weyl andsemimetal,
insulator,
metal.2. The
applied(Coloronline) Schematicillu
FIG. 2. (Colothe Weylvoltage
semimetal
layer
and
the
bottommetal
between the Weyl semimetal layer andthe bybottommetal voltage.
duced
the electric
(a)
The
duced
by
the
agnetization
dynamics.
The thicknessof the Anisotropy
Weyl
Zeemandynamics.
layer
induces magnetization
The thickness
of
the
Weyl
rection determined by thebalanceofrectiondetermhe insulator
are denoted
as dinsulator
respectively.
0 and d, aresemimetal
and thedenoted as
d0 Zeemanand d, respectively.
energy. (b),(c) Magnetization*
Zeeman energ
~
~
𝐸
𝑴
=
−𝑔𝜇
𝑆
𝑴
0
𝑩
−
𝐾𝑀
s switch themagnetization.
(b)
Oscillating
voltage€ the magnetization.
# (b)plied(a) Voltage pulses switch
Oscillating
voltagevoltage. The directionof voltage.
the spinplied*'zation precession
and generates aprecession
spin current.
The
induces magnetizationand generates
a+
spin
The
~ 0 𝑩 the~voltage.
sign
the𝑴+ *‚ 𝜎𝑴
𝑉 current.
𝜎of𝐴𝐻𝐸0𝑩
the sign of the𝐴𝐻𝐸urrent is injected
intospin
an current
adjacentismetal
layer.
generatedinjected
into an adjacent metal layer.
Charging
1 *𝜌
2𝐶

nd a metal. An electric voltage is applied be-

Gate voltage

field as B =𝑉𝜌 Bẑ. Asfield
shown
in F
as B
=

Gate-tuned magnetization dynamics
2
(a)

B

(a) z
B
x

(b)
y

WSM

V(t)

Insulator
Metalruclied
etal
Weyl
vely.
tage
The

be-

ZeemanCharging

(c)

(b)

V< 0

d0 B

Metal

d

WSM

V(t)

B

(a)

B

Insulator
Metal• Charge-induced
spin torque
FIG. 1. (Color online) Schematic illustration of the multilayer
strucFIG.
2. a(Color
Schematic
illustration
the spinThetorquein- KN, Kurebaashi, 2015
ture of
Weyl online)
semimetal,
an insulator,and ofa metal.
applied
FIG. 2. (Colo
ducedthe electric(a) The equilibrium
divoltagebybetween
thevoltage.
Weyl semimetal
layer and magnetization
the bottom metal
duced by therectiondetermined
byZeemanthe balanceofAnisotropy
theThecharging
energyandWeylthelayer induces
magnetization
dynamics.
thicknessof therection determ
Zeemanenergy.
(b),(c)Magnetization
precession
drivenbytheapsemimetal and the insulator are denoted as
d0 and d, respectively.
*
Zeeman energ~~
𝐸 𝑴pulses= −𝑔𝜇𝑴
0 𝑩spin− 𝐾𝑀
plied
voltage.
The
directionof magnetization.
thetorquedependingon€ 𝑆the# changes(a) Voltage
switch(b) Oscillating
voltageplied voltage.
*
'
the
sign
of
the
voltage.
induces magnetization precession
and generates
a+
spin
~
~ The
+ *‚ 𝜎𝑉 current.
𝜎𝐴𝐻𝐸𝑴
0𝑩
the sign of the
𝐴𝐻𝐸 𝑴 0 𝑩
generated spin current is injected into an adjacent metal layer.
Charging
Gate voltage

1 *2 (a) the equilibrium
field as B = Bẑ. As shown in FIG.
𝑉𝜌
𝜌
2𝐶
magnetization direction is obtained by minimizing EC + EZ +field as B =Gate-tuned magnetization dynamics

Metal

x

WSM

nsulator

Metal

(a)

(a) z
y

B

WSM

V(t)

Insulator
Metal

(b)

(b)

V> 0

B
d0 B

(c)

Zeeman
Metal

B

(a)

B

WSM
ChargingdV(t)Insulator
Metal

on of the multilayer struc• Charge-induced
spin torque
FIG. 1. (Color online) Schematic illustration of the multilayer
strucand a metal. The applied
FIG. 2.
online)illustrationof Kurebaashi,
the spin 2015
torqueture of a Weyl semimetal,an (Color
insulator,
and Schematica metal. The
appliedKN,
FIG. 2. (Coloyer and the bottom metal
duced semimetalby the electric
The equilibrium
magnetization
voltage between the Weyl
layervoltage.
and the(a)bottom
metal
duced by the
The thickness of the Weyl
rection dynamics.
determinedby thebalanceofofthethecharging energy and
ZeemanAnisotropy
layer induces magnetization
The
thickness
Weyl
rection determ
as d0 and d, respectively.
Zeemanenergy.
(b),(c)Magnetization
precession
drivenby thesemimetal and the insulator are denoted as
d0 and d, respectively.
*
Zeemanenerg~~
on. (b) Oscillating
𝐸 voltage𝑴pulses= −𝑔𝜇𝑴
0 𝑩 −The
𝐾𝑀direction
plied
voltage.
of the spin voltage
torque changes depending€ 𝑆the# (b) Oscillating(a) Voltage
switchmagnetization.
plied voltage.
nerates a spin current. The
*
'
the
sign
of
the
voltage.
induces magnetization precession
and generates
a+
spin
~
~ The
+ *‚ 𝜎𝑉 current.
𝜎𝐴𝐻𝐸𝑴
0𝑩
the sign of thedjacent metal layer.
𝐴𝐻𝐸 𝑴 0 𝑩
generated spin current is injected into an adjacent metal layer.
Charging
Gate voltage

c voltage is applied be-

1 * As shown in FIG. 2 (a) the equilibri
field as B = Bẑ.
𝑉𝜌𝜌
field as B =2𝐶magnetization direction is obtained by minimizing EC + E

Gate-tuned magnetization dynamics

Metal

x

WSM

nsulator

Metal

(a)

(a) z
y

B

WSM

V(t)

Insulator
Metal

(b)

(b)

V> 0

B
d0 B

(c)

Zeeman
Metal

B

(a)

B

WSM
ChargingdV(t)Insulator
Metal

on of the multilayer strucFIG. 1. (Color online) Schematic illustration of the multilayer strucand a metal. The applied
FIG. 2.
online)illustration
spin
ture of a Weyl semimetal,an (Color
insulator,
and Schematica metal. The
applied of the~2.torque
𝛿𝐸
FIG.𝑴
(Colo
yer and the bottom metal
duced semimetalby the electric
The equilibrium
𝑩𝑒𝑓𝑓 = magnetization
voltage between the Weyl
layervoltage.
and the(a)bottom
metal
duced
the
~ by andThe thickness of the Weyl
rection dynamics.
determinedby thebalanceofofthethecharging 𝛿𝑴energyZeemanAnisotropy
layer induces magnetization
The
thickness
Weyl
rection determ
as d0 and d, respectively.
Zeemanenergy.
(b),(c)Magnetization
precession
drivenby thesemimetal and the insulator are denoted as
d0 and d, respectively.
*
Zeemanenerg~~
on. (b) Oscillating
𝐸 voltage𝑴pulses= −𝑔𝜇𝑴
0 𝑩 −The
𝐾𝑀direction
plied
voltage.
of the spin voltage
torque changes depending€ 𝑆the# (b) Oscillating(a) Voltage
switchmagnetization.
plied voltage.
nerates a spin current. The
*
'
the
sign
of
the
voltage.
induces magnetization precession
and generates
a+
spin
~
~ The
+ *‚ 𝜎𝑉 current.
𝜎𝐴𝐻𝐸𝑴
0𝑩
the sign of thedjacent metal layer.
𝐴𝐻𝐸 𝑴 0 𝑩
generated spin current is injected into an adjacent metal layer.
Charging
Gate voltage

c voltage is applied be-

1 * As shown in FIG. 2 (a) the equilibri
field as B = Bẑ.
𝑉𝜌𝜌
field as B =2𝐶magnetization direction is obtained by minimizing EC + E

Gate-tuned magnetization dynamics

Metal

x

WSM

nsulator

Metal

(a)

(a) z
y

B

WSM

V(t)

Insulator
Metal

(b)

(b)

V> 0

B
d0 B

(c)

Zeeman
Metal

B

(a)

B

WSM
ChargingdV(t)Insulator
Metal

on of the multilayer strucFIG. 1. (Color online) Schematic illustration of the multilayer strucand a metal. The applied
FIG. 2.
online)illustration
spin
ture of a Weyl semimetal,an (Color
insulator,
and Schematica metal. The
applied of the~2.torque
𝛿𝐸
FIG.𝑴
(Colo
yer and the bottom metal
duced semimetalby the electric
The equilibrium
𝑩𝑒𝑓𝑓 = magnetization
voltage between the Weyl
layervoltage.
and the(a)bottom
metal
duced
the
~ by andThe thickness of the Weyl
rection dynamics.
determinedTheby thebalanceofofthethecharging 𝛿𝑴energylayer induces magnetization
thickness
Weyl
rection determ
as d0 and d, respectively.
Zeemanenergy.
(b),(c)Magnetization
precession
drivenby thesemimetal and the insulator are denoted as d0 and d, respectively.
Zeemanenerg
on. (b) Oscillating voltage
pliedthevoltage.
The direction
of
the
spin
torquechangesdepending(a) Voltage pulses switch
magnetization.
(b)
Oscillating
voltage~~plied𝑑𝑴
voltage.
𝑑𝑴
nerates a spin current. The
the
sign
of
the
voltage.
~ ×𝑩𝑒𝑓𝑓~the×sign of theinduces magnetization precession and generates a=spincurrent.
The+ 𝛼𝑴−𝛾𝑴
djacent metal layer.
𝑑𝑡 metal layer.
𝑑𝑡generated spin current is injected into an adjacentc voltage is applied be-field as B = Bẑ. As shown in FIG. 2 (a) the equilibri
field as B =magnetization direction is obtained by minimizing EC + E

Gate-tuned magnetization dynamics
(a) z

(b)
y

x

WSM

V(t)

(a)

d0 B

Metal

d

WSM

V(t)

Insulator
Metal

B

B

Insulator
Metal
4

Input voltage [ V ]Input voltage [ V ]FIG.
1. (Color online) Schematic illustrationof the multilayer struc(a)(b)
ture 0.6of a Weyl semimetal, an insulator,
The applied
0.015and a metal.
~2. (Colo
𝛿𝐸
𝑴
FIG.
0.01
𝑩𝑒𝑓𝑓 =
0.4
voltage
between the Weyl semimetal 0.005
layer
and the bottom metal0
duced~ by the𝛿𝑴-0.01The thickness of the Weyl
layer0.20 induces magnetization dynamics.-0.005-0.015
rection determ
1.5
semimetal
and the insulator are denoted
as d0 and d, respectively.
Zeeman energ
0.01
1
0.008
(a) Voltage
pulsesswitchthemagnetization.
(b)
Oscillating
voltage
~
~
0.5
0.006
plied𝑑𝑴
voltage.
𝑑𝑴
0.004
0~ ×𝑩𝑒𝑓𝑓~the×sign of theinduces magnetization precession and 0.002
generates a=spincurrent.
The+ 𝛼𝑴−𝛾𝑴-0.50𝑑𝑡 metal layer.
𝑑𝑡generatedspin current is injected into an
adjacent-0.002-1-1.5
0-0.00450

100
Time [nsec]

150

200

0

0.5

1
Time [nsec]

1.5

2

FIG. 4. (Color online) (a) The time evolution of the y component of the magnetization with the pulsed voltage. The numerical calculation

field as B =Gate-tuned spin pumping
spin current
(b)

(a)

jspin

⎛ g↑↓ ⎞dM̂= ⎜⎜ r ⎟⎟ M̂ ×dt
⎝ 4π ⎠
(a)

B

(b)

(b)
B

prising a
0.5
0.5
In this st
B
Metal
d B
Zeeman00of a cha
WSM
d
-0.5
-0.5
electric
ChargingV(t)-1-1
Insulator
-1
-1
-0.5
-0.5
to produ
-1
-1
0
0
-0.5
-0.5
0
0
0.5
0.5
Metal
0.5
0.5
ing with
11
11
4
nonosca(b)
hematic illustrationof thestruc-(a) Magnetization trajectory with the pulsed
FIG.multilayer
3. (Color online)from a c
0.015~
𝛿𝐸 𝑴there is t
0.01 and a metal.
voltage. ItTheshowsthat the magnetization changes the sign from +y to, an insulator,
applied
0.005
FIG.
2.
(Coloronline)Schematic
illustratio
𝑩
=
𝑒𝑓𝑓
0
−y.
(b)
Magnetization
trajectory
drivenbytheoscillating
voltage.
It~yl semimetal-0.005
layer and the bottom metal
charge
i
𝛿𝑴
duced
by
the
electric
voltage.
(a)Theequili
-0.01
shows
that
the
magnetization
precessesaroundtheequilibrium
axis.
-0.015quires thon dynamics.
The thickness of the Weyl
rection determined by the balance of
the c
0.01
which cor are denoted
as d0 and d, respectively.
0.008
Zeemanenergy.
(b),(c)Magnetization
prece
~
~
0.006
𝑑𝑴
𝑑𝑴
with
the
the magnetization.
(b) Oscillating voltage
0.004
~
~
plied
voltage.
The
direction
of
the
spin
torqu
Then spin torque associated with the
charging
poten= −𝛾𝑴×𝑩and𝛼𝑴 × practical𝑒𝑓𝑓 +ecession and0.002
generatesa spin current. The0𝑑𝑡
𝑑𝑡
the sign
tial contributions are obtained
as ofTCthe=voltage.
!γBC × M̂ and-0.002
Next w
njected into-0.004
an adjacentT metal
layer.
0200
00.5
V =1 !γB1.5
V × M̂.2 The anisotropy energy gives a contricillating
Time [nsec]
bution which is independent of the magnetic field as BA =tion of te y component of the magnetization with the pulsed∂E
voltage.
The numerical
K calculation
A
Input voltage [ V ]

0

1

B

1

Summary• A Weyl semimetal is predicted to realized
in a magnetically doped topological insulators~ ×𝑩
𝑻 = 𝜎ˆjI 𝑉(𝑡)𝑴• Voltage-induced spin torque• We demonstrated voltage-induced
magnetization switching and spin pumping
in Weyl semimetals
(a) z
x

(b)
y

WSM

V(t)

Insulator
Metal

(a)

d0 B

Metal

d

WSM

V(t)

Insulator
Metal

B

B