1. Berry phase driven Hall effects
June 22, 2011＠Beijing
Naoto Nagaosa
Department of Applied Physics
The University of Tokyo

2. 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

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

4. Path integral and Aharonov-Bohm effect
Amplitude from A to B
Generalized space
Berry Phase

5. Electrons with ”constraint”
doubly degenerate positive energy states.
Dirac electrons
Bloch electrons
Projection onto positive energy state
Spin-orbit interaction
as SU(2) gauge connection
Projection onto each band
Berry phase
of Bloch wavefunction

6. Solid angle by spins acting as a gauge field
|ci＞
|cj＞
gauge flux F Si Sj Sk
conduction
electron
acquire a phase factor
scalar spin chirality
Fictitious flux (in a continuum limit)

7. Equation of motion Br Bk k-space r-space Luttinger, Blout, Niu e- e-
Fermi surface e- e-
Luttinger, Blout, Niu

8. Issues to be discussed
Hall effects of uncharged particles
-- photons and magnons
r-space vs. k-space Berry phase

9. Can neutral particle show Hall effect ?
Hall effect of photon
M. Onoda et al, Phys. Rev. Lett. 93, (2004).
K.Y. Bliokh and Y.P. Bliokh
Phys. Rev. Lett. 96, (2006).
F. D. M. Haldane and S. Raghu,
Phys. Rev. Lett. 100, (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)]

10. Thermal Hall effect in solids
Metals
Wiedemann-Franz law
Righi-Leduc effect
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: )
Thermal Hall effect by phonon：Tb3Ga5O12
Strohm, Rikken, & Wyder, PRL 95 (‘05).
Thermal Hall angle: at 5K.

11. Coupling between spin chirality and magnetic field
Hubbard model with complex hopping ( )
Second-order:
Ring-exchange:
Scalar spin chirality
D. Sen & R. Chitra, PRB (‘95) O.I. Motrunich, PRB (‘06).

12. Spin Chirality due to Spin Wave
Scalar chirality:
Collinear spin structure:
Geometric Cancellation
Ferromagnet:　　　　　　　 　Antiferromagnet 　 °structure
Exact cancellation ∵
1-magnon term also cancels

13. NO-GO Theorem applicable to many cases !
Lattice structure: square(□), triangular (△), kagome, …
Magnetic structure: FM ( ), AFM ( ), 120°, spiral, …
Anisotropy of hopping　→　non-uniform
No-go theorem: FM order with an edge-sharing geometry → ×
Corner sharing geometry, e.g., Kagome !!
classical AFM kagome
q=0 g.s ⇔χ FM
g.s. ⇔χ AFM
Kagome FM
Kagome AFM q=0

14. Kubo formula for thermal Hall conductivity
Berry curvature
Bose distribution function
c.f. Matsumoto- Murakami

15. 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).

16. Quantum spin liquid Candidate materials
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
(constraint: )
Spinon (charge=0, spin 1/2)
(S. Frorens & A. Georges, PRB 70 (‘04))
: gauge field
spin chirality
κ-(BEDT-TTF)Cu2(CN)3, ZnCu3(OH)6Cl2, Na4Ir3O8(3d, strong SO), …

17. 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

18. Spinon v.s. Magnon coupled to （via ring exchange term） Lorentz force
Deconfined spinon ( gauge dependent object)
coupled to 　　　　 （via ring exchange term）
Lorentz force
Magnon (gauge invariant object)
coupled to
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

19. 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 B [T]
M. Yamashita et al., Science 328 (‘10)

20. Target material -Lu2V2O7 Pyrochlore Lattice (111) Plane is Kagome
Collinear ferromagnet
insulator

21. Thermal Hall conductivity for Lu2V2O7
(=Tc)

22. Temperature dependence, anisotropy
“spontaneous” component
Emergent at Tc
Almost isotropic

23. Discussion Origin of thermal Hall conductivity?
Possibility of electronic origin can be ruled out by Wiedemann Franz law.
kxxe<10-5 W/Km below 100K
kxy decreases with H at low T.
Opening of magnon gap
kxy is observed only below TC.
Coherent magnon transport is crucial for the kxy.
External H
kxy is almost proportional to M.
irrelevant

24. Theory of magnon Hall effect based on DM interaction
Katsura & Nagaosa
i site 1 2 3 4
D12
D23
D31
Magnons acquire Berry phase owing to DM interaction.
(isotropic)
D/J=0.32
Cf. D/J=0.19 for CdCr2O4
c.f. Matsumoto
-Murakami

25. 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

26. Equation of motion one flux quantum/(nm)2~4000T ! k-space r-spaceBk Br
r-space
Fermi surface e- e-
Bk induced AHE
Br induced AHE
Cf. normal HE
“dissipationless” nature

27. Pyrochlore Nd2Mo2O7 Y. Taguchi, Y. Oohara, H. Yoshizawa,50
100
150
0.5 1
1.5
Temperature (K)
Resistivity (m Ω
cm) M ( H
= 0.5 T) ( μ B
/Mo) 10 20 30 I 2
/2Nd Mo O 7 ) Nd
(200)
(111) T * TC T* R Mo
Y. Taguchi, Y. Oohara, H. Yoshizawa,
N. Nagoasa, and Y. T., Science 2001

28. Skyrmion and spin Berry phase in real space
Skyrmion configuration
From Senthil et al.
Solid angle acts as a
fictitious magnetic field
for carriers

29. 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

30. Small angle neutron scattering for Skyrmion Xtal
MnSi
S. Mühlbauer et. al., Science (2009)
c.f. early theoretical
prediction by A.N.Bogdanov et al.

31. Skyrmion Crystal Skyrmion Skyrmion crystal
Superposition of three Helix without phase shift
Skyrmion
Skyrmion crystal
3-flod-Q
S. Muhlbauer et al. Science 323, 915 (2009).

32. Monte Carlo simulation for 2D helimagnet
J. H. Park, J. H. Han, S. Onoda and N.N.
anisotropy

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

34. Experiment
Theory
X. Z. Yu, Y. Onose, N. Kanazawa2, J. H. Park, J. H. Han, Y. Matsui, N. N. Y. Tokura
Nature (2010)

35. 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

36. Skyrmion-induced AHE (MnSi)
M. Lee, W. Kang, Y. Onose, Y. Tokura, and N. P. Ong, PRL (2009).
A. Neubauer et al, PRL (2009)
Relation to the magnetic structure??
Finite but quite small 36

37. Fictitious magnetic flux
©Y. Tokura
one flux quantum/(nm)2~4000T !
(double-excahnge model) l
Dryx　∝　F　（Sk density)　
(cal.)
[T]
Dryx(topological) [nWcm]
l(magnetic)
[nm]
FeGe 70 1
indiscernible 28 18 5
MnSi
200
1100
MnGe
3.0
Nd2Mo2O7
(reference)
~0.5
~40000
6000

39. “Electromagnetic induction”
Moving magnetic flux produces
the transverse electric field
Conduction electron number per site
Spin quantum number
c.f.
Topological
Hall effect

40. New dissipative mechanism for spin texture
moving flux  electric field  induced current  dissipation
mean free path
size of Skyrmion
a’ does not require spin-orbit int. and can be as large as ~0.1
But is determined by DM interaction.

41. Skyrmion Hall effect Transverse motion of the Skyrmion as a
back-action to the “electromagnetic induction”
Skyrmion charge determined
by the direction of the external magnetic field
“Hall angle”

42. Hall Effect of Light Photon also has “spin”
Generalized equation of geometrical optics
M.Onoda,
S.Murakami,
N.N. (PRL2004)

43. Giant X-ray shift in deformed crystal
PRL2010
Sawada-Murakami-Nagaosa PRL06
Berry curvature in r-k space
enhancement

44. Berry phase in r-k space
D. Xiao et al., PRL (2009)
(Real) Space dependent
Berry curvature
Semiclassical equation of motion

45. Inhomogeneity-induced polarization
D. Xiao et al., PRL (2009)
Inhomogeneity-induced
topological charge polarization !
(Second Chern form) P r

46. Conclusions Berry phases in r- and k-spaces, and (r,k)-space
Hall effects of uncharged particles
photons and magnons
3. Hall effect and charge pumping in spin textures C