Berry phase driven Hall effects

Naoto Nagaosa

Department of Applied PhysicsThe University of Tokyo

June 22, 2011 ＠ Beijing

Collaborators

TheoryH. Katsura, J. H. Han, J. Zang, J. H. Park, K. Nomura, M. Mostovoy, B.J.Yang

ExperimentX. Z. Yu, Y. Onose, N. Kanazawa, Y. Matsui,

Y. Shiomi, Y. Tokura

Berry phase 　 M.V.Berry, Proc. R.Soc. Lond. A392, 45(1984)

t

))(( tXEn

Transitions between eigen-states are forbidden during the adiabatic change

Projection to the sub-space of Hilbert space and constrained quantum system

Connection of the wave-function in sub-space of Hilbert space

Berry phase, gauge connection

Path integral and Aharonov-Bohm effect

1C

2C

1C2C

NC

1a2a

Na

Amplitude from A to B

N

jja

1

1a

2a

cieeaaaa /02121 ||

0x 0x

1x 1x

nXXXkrr ,,,,, 21 Generalized space

Berry Phase

Electrons with ”constraint”

Projection onto positive energy stateSpin-orbit interaction

as SU(2) gauge connection

Dirac electrons

doublydegenerate

positive energy states.

E

k

Bloch electrons

Projection onto each bandBerry phase

of Bloch wavefunction

k

E

Solid angle by spins acting as a gauge field

gauge flux

Si

Sj

Sk

|ci ＞ |cj ＞

Fictitious flux (in a continuum limit)

conductionelectron

acquire a phase factor

scalar spin chiralityscalar spin chirality

k-space

e-

Fermi surface

r-space

e-

kBk

x

dt

dk

dt

d

rdt

dre

dt

dB

r

k

Equation of motion

BrBk

1c1c

llB llB

Luttinger, Blout, Niu

Issues to be discussed

1. Hall effects of uncharged particles -- photons and magnons

2. r-space vs. k-space Berry phase

Can neutral particle show Hall effect ?

Thermal Hall effect by phonon ： Tb3Ga5O12

Strohm, Rikken, & Wyder, PRL 95 (‘05).

Thermal Hall angle: at 5K.

Hall effect of photon

M. Onoda et al, Phys. Rev. Lett. 93, 083901 (2004).K.Y. Bliokh and Y.P. Bliokh Phys. Rev. Lett. 96, 073903 (2006).F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008) O. Hosten, P. Kwiat, Science 319, 787 (2008).

Hall effect of magnons in insulating magnets ? Yes ! [ H.Katsura-N.N.-P.A.Lee (PRL09)]

Metals

Wiedemann-Franz law

Righi-Leduc effect

F.D.M. Haldane, PRL 93 (‘04).

How about Mott insulators ?

Spins can carry thermal Hall current ?

cf.) Magnon spin Hall effect (S. Fujimoto, arXiv: 0811.2263)

Thermal Hall effect by phonon ： Tb3Ga5O12

Strohm, Rikken, & Wyder, PRL 95 (‘05).

Thermal Hall angle: at 5K.

Thermal Hall effect in solids

applicable to AHE also

• Hubbard model with complex hopping ( )

Ring-exchange:

Scalar spin chirality

D. Sen & R. Chitra, PRB (‘95) O.I. Motrunich, PRB (‘06).

Coupling between spin chirality and magnetic field

Second-order: ijie

ijie

Spin Chirality due to Spin Wave

Ferromagnet: 　　　　　　　 　 Antiferromagnet 　 120°structure

Scalar chirality:

Collinear spin structure:

i

j k

i

• Geometric Cancellation

Exact cancellation

∵

i

j k

i

1-magnon term also cancels

i) Lattice structure: square(□), triangular ( ), kagome, …△

ii) Magnetic structure: FM ( ), AFM ( ), 120°, spiral, …

iii) Anisotropy of hopping 　→　 non-uniform

• No-go theorem: FM order with an edge-sharing geometry → ×

Kagome FM Kagome AFM q=0

Corner sharing geometry, e.g., Kagome !!

classical AFM kagome

i) q=0 g.s. ⇔χ FM

ii) g.s. ⇔χ AFM

NO-GO Theorem applicable to many cases !

Berry curvatureBose distribution function

Kubo formula for thermal Hall conductivity

c.f. Matsumoto- Murakami

Magnon dispersion

Around k=0

Spin Wave Hamiltonian

TKNN-like formula:

T-linear & B-linear!

Thermal Hall effect in Kagome ferromagnet

Skew scattering ? Small in the scattering of low energy limit (s-wave).

(constraint: )

RVB （ resonating valence bond ） state, P. W. Anderson(‘87)

quantum liquid of singlets

　

Mean field theory of RVB state

U(1) (internal) gauge-field

Candidate materials κ-(BEDT-TTF)Cu2(CN)3, ZnCu3(OH)6Cl2, Na4Ir3O8(3d, strong SO), …

Spinon (charge=0, spin 1/2)(S. Frorens & A. Georges, PRB 70

(‘04))

Quantum spin liquid

ija : gauge field spin chirality

RVB theory under magnetic field

• spin model( )△ ：

Slave rotor rep.:

κ-(BEDT-TTF)Cu2(CN)3

Ring exchange term

Scalar chirality

Lee and Lee, O. I. Motrunich

Ioffe-Larkin, Nagaosa-Lee

Deconfined spinon ( gauge dependent object)

coupled to 　　　　 （ via ring exchange term ）

Lorentz force

Magnon (gauge invariant object)

coupled to

intrinsic Hall effectThermal Hall effect due to spinons spinon metal ・ Fermi surface （ gapless spinon picture ）

　 spinon current conductivity ：Wiedemann-Franz law

Spinon v.s. Magnon

AB

A

Thermal Hall angle

Thermal conductivity in κ-(BEDT-TTF)Cu2(CN)3 M. Yamashita et al., Nature Phys. 5 (‘09)

T-linear

0.02 W/Km ⇔@0.3 K

Spinon lifetime

Spinon effective mass

Thermal Hall angle @ B [T]

M. Yamashita et al., Science 328 (‘10)

Target material -Lu2V2O7

　

Pyrochlore Lattice

(111) Plane is Kagome

Collinear ferromagnet

insulator

0

0.2

0.4

0.6

0.8

1Lu2V2O7

H || [111]H=0.1T

M(

B/V

))

100

101

102

103

104

Res

isti

vity

(cm

)

0 50 100 1500

0.5

1

1.5 x

x(W

/Km

)

T(K)

Thermal Hall conductivity for Lu2V2O7

(=Tc)

-5 0 5

20K

Magnetic Field (T)-5 0 5

30K

-5 0 5

-2

-1

0

1

2 40K

50K

-5 0 5

10K

60K 70K

-2

-1

0

1

2 80K x

y (1

0-3 W

/Km

)Lu2V2O7 H||[100]

-10 0 10

-1

0

1

2T=50K

(1

0-3 W

/Km

)

0H (T)

“spontaneous” component

Emergent at Tc

Almost isotropic

Temperature dependence, anisotropy

Discussion

Origin of thermal Hall conductivity?Possibility of electronic origin can be ruled out by Wiedemann Franz law.

xxe<10-5 W/Km below 100K

xy decreases with H at low T. Opening of magnon gap

xy is observed only below TC.

Coherent magnon transport is crucial for the xy.

xy is almost proportional to M.External Hirrelevant

23

50 100

0.5

1

1.5

0

0.2

0.4

0.6

0.8

1

0

T (K)

xy

(10

-3 W

/ K

m)

M (

B / V4

+)

0 H = 0.1 T 0 H = 7 T

1

2

3

4

D12

D23D31

Theory of magnon Hall effect based on DM interaction Katsura & Nagaosa

ijijiijji )e/J()|iSS(DSSJj| 2

~

nDiJeJ iji ij ~

Magnons acquire Berry phase owing to DM interaction.

(isotropic)

,expLi22

2),(B

B5/2

B

2

B2/3

2B

Tk

Hg

JS

Tk

JS

Hg

a

TkTH

-10 0 10-1

0

1

T=20K

Magnetic Field (T)

xy

(10-3

W/K

m)

H||[100]

xycalc(H)

C

D/J=0.32Cf. D/J=0.19 for CdCr2O4

i|i site

c.f. Matsumoto -Murakami

Gauge field of spin textures in insulating magnets M.Mostovoy, K.Nomura and N.N. PRL2011

Spin dynamics in the intermediate virtual states of the exchange int.Coupling between gauge field e and E Multi-orbital Mott insulator

Finite even without inversion asymmetry or spin-orbit interaction

k-space

e-

Fermi surface

r-space

e-

kBk

x

dt

dk

dt

d

rdt

dre

dt

dB

r

k

Equation of motion

20 , xyxy

Bk induced AHE

“dissipationless” nature

02 , xyxy

Br induced AHE

Cf. normal HEneBneB xxxyxy /,/ 2

BrBk

one flux quantum/(nm)2~4000T !

0 50 100 1500

0.5

1

0

0.5

1

1.5

Temperature (K)

Resi

stivi

ty (m

Ωcm

)

M( H

= 0.

5 T)

( μ

B/M

o)

0

10

20

30

I (

μB2

/2N

d2

Mo 2

O7) Nd 2Mo 2O 7

I (200) I(111)

T*

TCT*

RMo

Pyrochlore Nd2Mo2O7

Y. Taguchi, Y. Oohara, H. Yoshizawa, N. Nagoasa, and Y. T., Science 2001

Skyrmion configuration

From Senthil et al.

Skyrmion and spin Berry phase in real space

Solid angle acts as a fictitious magnetic field for carriers

aSSS kji )(

Pfleiderer, Rosch, Lonzarich et al

Quantum Phase Transition in MnSi

Non-Fermi liquid charge transport

Spin fluctuation on a sphere in momentum space

DM magnet

MnSiMnSi

S. Mühlbauer et. al., Science 323 915 (2009)

Small angle neutron scattering for Skyrmion Xtal

c.f. early theoretical prediction by A.N.Bogdanov et al.

Skyrmion Crystal

3-flod-Q

Superposition of three Helix without phase shift

Skyrmion Skyrmion crystal

0321 QQQ

S. Muhlbauer et al. Science 323, 915 (2009).

Monte Carlo simulation for 2D helimagnet

J. H. Park, J. H. Han, S. Onoda and N.N.

)( kjixy SSS

anisotropy

Lorentz TEM observation of Skyrmion crystal in (Fe,Co)Si

Nature (2010)

X. Z. Yu, Y. Onose, N. Kanazawa2, J. H. Park, J. H. Han, Y. Matsui, N. N. Y. Tokura

TheoryExperiment

Coupled dynamics of conduction electrons and SkX

Effective EMF due to spin texture acting on conduction electrons

Coupling term

Boltzmann equation

LLG equation

Lorentz force

J.D.Zang, J.H. Han, M.Mostovoy, and N.N.

Skyrmion-induced AHE (MnSi)

Finite but quite small

A. Neubauer et al, PRL 102 186602 (2009)

Relation to the magnetic structure??

M. Lee, W. Kang, Y. Onose, Y. Tokura, and N. P. Ong, PRL (2009).

Fictitious magnetic flux 　

one flux quantum/(nm)2~4000T !(double-excahnge model)

MnGe

Nd2Mo2O7

(reference)

FeGe

MnSi

(magnetic) [nm]

yx(topological) [ncm]

(cal.)[T]

70

18

3.0

~0.5

1

1100

~40000

28 5

200

6000

indiscernible

yx 　 ∝　　 （ Sk density) 　

©Y. Tokura

V

j

mc

he ztopxy

Moving magnetic flux produces the transverse electric field

c.f.

xConduction electron number per site

SSpin quantum number

||V

jcj

xy

jcj

xS

S

2

2

Topological Hall effect

“Electromagnetic induction”

New dissipative mechanism for spin texture

moving flux electric field induced current dissipation

2)/)(( alkF

lmean free path size of Skyrmion

’　　does not require spin-orbit int. and can be as large as ~0.1But is determined by DM interaction.

V

j

))('( || zeVQV

Transverse motion of the Skyrmion as a back-action to the “electromagnetic induction”

Skyrmion Hall effect

1Q Skyrmion charge determined by the direction of the external magnetic field

“Hall angle” 'tan H

Hall Effect of Light

)|)|

)]([

)||()(

ckcc

ccc

ckccc

ccc

zkizonpolarizati

krvkforce

zzkk

krvrvelocity

c

c

：

：

：

M.Onoda,S.Murakami,N.N. (PRL2004)

Generalized equation of geometrical optics

Photon also has “spin”

Giant X-ray shift in deformed crystal

Sawada-Murakami-Nagaosa PRL06

Berry curvature in r-k space

610 enhancement

PRL2010

Berry phase in r-k space

(Real) Space dependent Berry curvature

• Semiclassical equation of motion

D. Xiao et al., PRL (2009)

Inhomogeneity-induced polarization

Inhomogeneity-induced topological charge polarization !

(Second Chern form)P

r

D. Xiao et al., PRL (2009)

1. Berry phases in r- and k-spaces,

and (r,k)-space

2. Hall effects of uncharged particles

photons and magnons

3. Hall effect and charge pumping in spin textures

C

Conclusions