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**Berry phase driven Hall effects**

June 22, 2011＠Beijing Naoto Nagaosa Department of Applied Physics The University of Tokyo

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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

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**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

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**Path integral and Aharonov-Bohm effect**

Amplitude from A to B Generalized space Berry Phase

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**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

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**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)

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**Equation of motion Br Bk k-space r-space Luttinger, Blout, Niu e- e-**

Fermi surface e- e- Luttinger, Blout, Niu

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

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**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)]

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**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.

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**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).

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**Spin Chirality due to Spin Wave**

Scalar chirality: Collinear spin structure: Geometric Cancellation Ferromagnet: Antiferromagnet °structure Exact cancellation ∵ 1-magnon term also cancels

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**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

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**Kubo formula for thermal Hall conductivity**

Berry curvature Bose distribution function c.f. Matsumoto- Murakami

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**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).

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**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), …

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**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

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**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

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**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)

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**Target material -Lu2V2O7 Pyrochlore Lattice (111) Plane is Kagome**

Collinear ferromagnet insulator

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**Thermal Hall conductivity for Lu2V2O7**

(=Tc)

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**Temperature dependence, anisotropy**

“spontaneous” component Emergent at Tc Almost isotropic

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**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

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**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

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**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

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**Equation of motion one flux quantum/(nm)2~4000T ! k-space r-space**

Bk Br r-space Fermi surface e- e- Bk induced AHE Br induced AHE Cf. normal HE “dissipationless” nature

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**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

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**Skyrmion and spin Berry phase in real space**

Skyrmion configuration From Senthil et al. Solid angle acts as a fictitious magnetic field for carriers

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**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

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**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.

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**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).

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**Monte Carlo simulation for 2D helimagnet**

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

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

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

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**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

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**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

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**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

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**“Electromagnetic induction”**

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

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**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.

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**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”

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**Hall Effect of Light Photon also has “spin”**

Generalized equation of geometrical optics M.Onoda, S.Murakami, N.N. (PRL2004)

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**Giant X-ray shift in deformed crystal**

PRL2010 Sawada-Murakami-Nagaosa PRL06 Berry curvature in r-k space enhancement

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**Berry phase in r-k space**

D. Xiao et al., PRL (2009) (Real) Space dependent Berry curvature Semiclassical equation of motion

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**Inhomogeneity-induced polarization**

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

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**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

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