Berry phase driven Hall effects
June 22, 2011＠Beijing
Naoto Nagaosa
Department of Applied Physics
The University of Tokyo
Collaborators
Theory
H. Katsura, J. H. Han, J. Zang, J. H. Park, K. Nomura,
M. Mostovoy, B.J.Yang
Experiment
X. Z. Yu, Y. Onose, N. Kanazawa, Y. Matsui,
Y. Shiomi, Y. Tokura
Berry phase
M.V.Berry, Proc. R.Soc. Lond. A392, 45(1984)
En ( X (t ))
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
t
Path integral and Aharonov-Bohm effect
x1

aN
CN
x1
a2
C2
a1
C1
a2 a1
C2 C1
x0
Amplitude from A to B 
N
aj
j 1
r  r, k , X1 , X 2 ,  , X n
x0
a1a2 |  a1a2 |0 eie / c
Generalized space
Berry Phase
Electrons with ”constraint”
E
E
doubly
degenerate
positive energy
states.
k
k
Dirac electrons
Projection onto positive energy state
Spin-orbit interaction
as SU(2) gauge connection
Bloch electrons
Projection onto each band
Berry phase
of Bloch wavefunction
Solid angle by spins acting as a gauge field
|ci＞
gauge flux 
Sk
Si
|cj＞
conduction
electron
Sj
acquire a phase factor
scalar spin chirality
Fictitious flux (in a continuum limit)
Equation of motion
lB  l
c  1
lB  l
c  1
dx  dk

  Bk
dt k dt
k-space
Fermi surface
Bk
e-
Luttinger, Blout, Niu
dk
  dr

 e 
  Br 
dt
 r dt

r-space
Br
e-
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 ?
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).
Thermal Hall effect by phonon：Tb3Ga5O12
Strohm, Rikken, & Wyder, PRL 95 (‘05).
Thermal Hall angle:
at 5K.
Hall effect of magnons in insulating magnets ?
 Yes ! [ H.Katsura-N.N.-P.A.Lee (PRL09)]
Thermal Hall effect in solids
Metals
Righi-Leduc effect
Wiedemann-Franz law
F.D.M. Haldane, PRL 93 (‘04).
applicable to AHE also
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.
Coupling between spin chirality and magnetic field
• Hubbard model with complex hopping (
Second-order:
e
e
)
iij
iij
Ring-exchange:
Scalar spin chirality
D. Sen & R. Chitra, PRB (‘95)
O.I. Motrunich, PRB (‘06).
Spin Chirality due to Spin Wave
Scalar chirality:
Collinear spin structure:
• Geometric Cancellation
Ferromagnet:
Antiferromagnet
120°structure
i
i
j
k
i
Exact cancellation
∵
j
k
i
1-magnon term
also cancels
NO-GO Theorem applicable to many cases !
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 → ×
Corner sharing geometry, e.g., Kagome !!
classical AFM kagome
i)
ii)
Kagome FM
Kagome AFM q=0
q=0 g.s.
⇔χ FM
g.s. ⇔χ AFM
Kubo formula for thermal Hall conductivity
Bose distribution
function
c.f. Matsumoto- Murakami
Berry curvature
Thermal Hall effect in Kagome ferromagnet
Spin Wave Hamiltonian
Magnon dispersion
Around k=0
TKNN-like formula:
T-linear & B-linear!
Skew scattering ?
Small in the scattering of low energy limit (s-wave).
Quantum spin liquid
RVB（resonating valence bond）state,
P. W. Anderson(‘87)
quantum liquid of singlets
Mean field theory of RVB state
U(1) (internal) gauge-field
(constraint:
Spinon (charge=0, spin 1/2)
)
(S. Frorens & A. Georges, PRB 70 (‘04))
aij
: gauge field
spin chirality
Candidate materials
κ-(BEDT-TTF)Cu2(CN)3, ZnCu3(OH)6Cl2, Na4Ir3O8(3d, strong SO), …
RVB theory under magnetic field
Lee and Lee,
O. I. Motrunich
• spin model(△)：
Scalar chirality
Slave rotor rep.:
Ring exchange term
κ-(BEDT-TTF)Cu2(CN)3
Ioffe-Larkin, Nagaosa-Lee
Spinon v.s. Magnon
Deconfined spinon ( gauge dependent object)
coupled to
A
（via ring exchange term）
Lorentz force
Magnon (gauge invariant object)
coupled to
B   A
intrinsic Hall effect
Thermal Hall effect due to spinons
spinon metal ・Fermi surface（gapless spinon picture）
spinon current conductivity：
Wiedemann-Franz law
Thermal Hall angle
Thermal conductivity in κ-(BEDT-TTF)Cu2(CN)3
M. Yamashita et al., Nature Phys. 5 (‘09)
0.02 W/Km ⇔
@0.3 K
T-linear
Spinon lifetime
Spinon effective mass
Thermal Hall angle @ B [T]
M. Yamashita et al., Science 328 (‘10)
Target material -Lu2V2O7
1
Lu2V2O7
H || [111]
H=0.1T
M(B/V))
0.8
0.6
0.4
0.2
Resistivity(cm)
0
insulator
xx(W/Km)
Collinear ferromagnet
103
102
101
100
Pyrochlore Lattice
(111) Plane is Kagome
104
1.5
1
0.5
0
0
50
100
T(K)
150
Thermal Hall conductivity for Lu2V2O7
Lu2V2O7 H||[100]
2 80K
70K (=Tc)
60K
50K
30K
20K
10K
1
xy (10-3 W/Km)
0
-1
-2
2 40K
1
0
-1
-2
-5
0
5
-5
0
5
-5 0
Magnetic Field (T)
5
-5
0
5
Temperature dependence, anisotropy
 (10-3 W/Km)
2
T=50K
1
0
-1
-10
0
10
0H (T)
“spontaneous” component
Emergent at Tc
Almost isotropic
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.
 0 H = 0.1 T
0H=7T
1
1
0.8
0.6
0.4
0.5
M ( B / V4+)
xy (10-3 W / Km)
1.5
Coherent magnon transport is
crucial for the xy.
0.2
0
50
100
0
T (K)
xy is almost proportional to M.
irrelevant
External H
23
Theory of magnon Hall effect based on DM interaction
| i 
2
D31
D23
~ i
Je ij  J  i Dij  n
Magnons acquire Berry phase owing to
DM interaction.
4
3
xy (10-3 W/Km)
1
 ( H , T )  
C
kB2T 
gB H 
2



 3 / 2a 
2 JS 
(isotropic)
0
-1
-10
Katsura & Nagaosa
    
~
i
 j|  JSi  S j  Dij  (Si  S j )|i  (J / 2 )e ij
D12
1
i site
xy calc(H)
D/J=0.32
H||[100]
T=20K
0
Magnetic Field (T)
Cf. D/J=0.19 for CdCr2O4
10
c.f. Matsumoto
-Murakami
2
  g H 
kBT
Li5/2 exp  B ,
2 JS
  kBT 
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
Equation of motion
dx  dk

  Bk
dt k dt
k-space
Fermi surface
Bk
dk
  dr

 e 
  Br 
dt
 r dt

one flux quantum/(nm)2~4000T !
r-space
e-
Bk induced AHE
 xy   0 , xy   2
“dissipationless” nature
Br
e-
Br induced AHE
 xy   2 , xy   0
Cf. normal HE
 xy  B / ne,  xy   xx2 B / ne
Pyrochlore Nd2Mo2O7
Mo
TC
Nd Mo O
2
2 7
30
I(200)
I (111)
20
1.5
1
1
0.5
0.5
0
0
T *50
100
Temperature (K)
Y. Taguchi, Y. Oohara, H. Yoshizawa,
N. Nagoasa, and Y. T., Science 2001
0
150
μ
0
B /Mo)
10
M ( H = 0.5 T) (
R
Resistivity (m
Ω cm)
2
I ( μ B /2Nd 2 Mo 2 O 7)
T*
Skyrmion and spin Berry phase in real space
Skyrmion configuration
From Senthil et al.
Solid angle acts as a
fictitious magnetic field
for carriers
S i  (S j  S k )    a
Quantum Phase Transition in MnSi
Pfleiderer, Rosch, Lonzarich et al
DM magnet
Spin fluctuation on a sphere
in momentum space
Non-Fermi liquid charge transport
Small angle neutron scattering for Skyrmion Xtal
MnSi
S. Mühlbauer et. al.,
Science 323 915
(2009)
c.f. early theoretical
prediction by A.N.Bogdanov et al.
Skyrmion Crystal
Superposition of three Helix without phase shift
Q1  Q2 Q3  0
Skyrmion
Skyrmion crystal
3-flod-Q
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.
 xy  S i  (S j  S k )
anisotropy
Lorentz TEM observation of Skyrmion crystal in (Fe,Co)Si
Experiment
Theory
X. Z. Yu, Y. Onose, N. Kanazawa2, J. H. Park, J. H. Han, Y. Matsui, N. N. Y. Tokura
Nature (2010)
Coupled dynamics of conduction electrons and SkX
J.D.Zang, J.H. Han, M.Mostovoy, and N.N.
Effective EMF due to spin texture
acting on conduction electrons
Coupling term
Lorentz force
Boltzmann equation
LLG equation
Skyrmion-induced AHE (MnSi)
A. Neubauer et al, PRL 102 186602 (2009)
M. Lee, W. Kang, Y. Onose, Y. Tokura,
and N. P. Ong, PRL (2009).
Finite but quite small
Relation to the magnetic structure??
Fictitious magnetic flux
©Y. Tokura
one flux quantum/(nm)2~4000T !
(double-excahnge model)
l
Dyx ∝  （Sk
FeGe
MnSi
MnGe
Nd2Mo2O7
(reference)
density)
l(magnetic)
[nm]
(cal.)
[T]
Dyx(topological)
[ncm]
70
1
indiscernible
18
28
5
3.0
1100
200
~0.5
~40000
6000
“Electromagnetic induction”
V
Moving magnetic flux produces
the transverse electric field
j
x
Conduction electron number per site
S
Spin quantum number
 xytop e  hz  

c.f.

mc
V||
 xy
Topological
Hall effect

j
j
jc
j
2S
2S  x
New dissipative mechanism for spin texture
moving flux  electric field  induced current  dissipation
 (kF l )(a /  )2
mean free path
l  
size of Skyrmion
’ does not require spin-orbit int. and can be as large as ~0.1
But  is determined by DM interaction.
Skyrmion Hall effect
V
Transverse motion of the Skyrmion as a
back-action to the “electromagnetic induction”
V  Q(   ' )(V||  ez )
Q  1
Skyrmion charge determined
by the direction of the external magnetic field
“Hall angle”
tan H     '
j
Hall Effect of Light
Photon also has “spin”
Generalized equation of geometrical optics



 kc 
velocity： rc  v(rc )  kc  ( zc |  k | zc )
c
kc

 
force： kc  [v(rc )]kc
 
polarization： | zc )  ikc   k | zc )
c
M.Onoda,
S.Murakami,
N.N. (PRL2004)
Giant X-ray shift in deformed crystal
Sawada-Murakami-Nagaosa PRL06
Berry curvature in r-k space
 106
enhancement
PRL2010
Berry phase in r-k space
D. Xiao et al., PRL (2009)
(Real) Space dependent
Berry curvature
• Semiclassical equation of motion
Inhomogeneity-induced polarization
D. Xiao et al., PRL (2009)
Inhomogeneity-induced
topological charge polarization !
(Second Chern form)
r
P
Conclusions
1. Berry phases in r- and k-spaces,
and (r,k)-space
C
2. Hall effects of uncharged particles
photons and magnons
3. Hall effect and charge pumping in spin textures