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  • Symposium on Novel Quantum States in Condensed Matter 2017 (NQS2017)

    8-10 November 2017 YITP, Kyoto University

    Half-integer thermal Hall conductance in a Kitaev spin liquid

    − Evidence for chiral Majorana edge current −

    Yuichi KASAHARA Department of Physics, Kyoto University

    h κ xy

    2D/T [( π/6)(k B 2 /

    )]

    h κ xy

    2D/T [(π/6)(k B 2 /

    )] h

    κ xy

    2D/T [(π/6)(k B 2 /

    )]

    h κ xy

    2D/T [( π/6)(k B 2 /

    )]

    1.5

    1.0

    0.5

    0

    -0.5

    κ x y/T

    ( 10

    -3 W

    /K 2 m

    )

    5.25.04.84.64.4 µ0H⊥ (T)

    0 1/2

    9.08.58.07.5 µ0H|| (T)

    4.3 K

    1.5

    1.0

    0.5

    0.0

    -0.5

    κ x y/T

    ( 10

    -3 W

    /K 2 m

    )

    5.25.04.84.64.4 µ0H⊥ (T)

    0 1/2

    9.08.58.07.5 µ0H|| (T)

    4.9 K 2.0

    1.5

    1.0

    0.5

    0.0

    -0.5

    κ x y/T

    ( 10

    -3 W

    /K 2 m

    )

    6.05.55.04.54.0 µ0H⊥ (T)

    0 1/2 1

    10.09.08.07.0 µ0H|| (T)

    5.6 K

    8.2 K

    1.5

    1.0

    0.5

    0

    -0.5

    κ x y/T

    ( 10

    -3 W

    /K 2 m

    )

    5.25.04.84.64.4 µ0H⊥ (T)

    0 1/2

    9.08.58.07.5 µ0H|| (T)

    3.9 K

    a b

    c d

    cold

    hot

    Majorana fermion

    Z2 flux

    Chiral Majorana edge currentHalf-integer thermal Hall conductance

  • Collaborators

    Kaori Sugii, Masaaki Shimozawa, Minoru Yamashita Institute for Solid State Physics, The University of Tokyo

    Taka Shibauchi Department of Advanced Materials Science, The University of Tokyo

    Takafumi Onishi, Yuji Matsuda Department of Physics, Kyoto University

    Nobuyuki Kurita, Hidekazu Tanaka Department of Physics, Tokyo Institute of Technology

    Joji Nasu Department of Physics, Tokyo Institute of Technology

    Yukitoshi Motome Department of Applied Physics, The University of Tokyo

    Matsuda

    Shibauchi

    Tanaka

    Nasu

    Yamashita

    Shimozawa

    Sugii

    Motome

  • Outline

    1. Introduction: Kitaev quantum spin liquid

    2. A candidate of Kitaev magnet α-RuCl3

    3. Thermal Hall effect in perpendicular fields

    4. Thermal Hall effect in tilted fields Observation of half-integer thermal Hall conductance

    5. Summary

    Y. Kasahara et al., arXiv:1709.10286 (2017).

  • Introduction Quantum spin liquid (QSL)

    Quantum fluctuations melt the long-range magnetic order even at T = 0.

    1D: S = 1/2 XXZ chain

    2D & 3D: Geometrically frustrated magnets 2D trianglar 2D kagome 3D pyrochlore

    Exotic physical properties in QSLs Topological phases Gauge fluctuations Fractionalized excitations

    Spin liquid are states which do not break any simple symmetry: Neither spin-rotational symmetry nor lattice translational symmetry.

    Platforms of QSL

    The ground state with massive entanglement of local spins.

    4

  • H = �J x

    X

    x

    Sx i

    Sx j

    � J y

    X

    y

    Sy i

    Sy j

    � J z

    X

    z

    Sz i

    Sz j

    Kitaev model

    Kitaev Interaction Bond-dependent Ising-like interaction

    Honeycomb lattice (2D) A. Kitaev, Ann. Phys. 321, 2 (2006).

    Hyper-honeycomb lattice (3D) S. Mandal & N. Surendran, PRB 79, 024426 (2009).

    Exchange frustration

    �J x

    Sx i

    Sx j

    �JzSzi Szj �JySyi S

    y j �JySyi S

    y j

    �J x

    Sx i

    Sx j

    �JzSzi Szj

    S = 1/2 spins on tri-coordinate lattices A. Kitaev, Ann. Phys. 321, 2 (2006).

  • HK = �Jx X

    x

    Sx i

    Sx j

    � J y

    X

    y

    Sy i

    Sy j

    � J z

    X

    z

    Sz i

    Sz j

    Kitaev model

    Fractionalization of quantum spins

    Wp = +1

    HK = iJx 4

    X

    x

    c i

    c j

    � iJy 4

    X

    y

    c i

    c j

    � iJz 4

    X

    z

    ⌘ r

    c i

    c j ⌘r = ic̄ic̄j

    Spin 1/2 Itinerant Majorana fermion with Dirac cone dispersion

    Localized Majorana fermion Z2 fluxes

    Si ci

    c̄i Wp = ⌘r⌘r0

    Two types of QSLs

    Gappless QSL: Majorana metal Gapped QSL: Toric code

    A. Kitaev, Ann. Phys. 321, 2 (2006).Wp = �1

    Z2 flux Itinerant Majorana fermion

    6

    Spin fractionalization occurs below ~JK/kB. (proximate spin liquid state)

    Jordan-Wigner transformation

    Majorana representation

    Free Majorana fermions on a honeycomb lattice

  • Candidate materials

    G. Jackeli & G. Khaliullin, PRL 102, 017205 (2006).

    Spin-orbit assisted Mott insulator with j = 1/2

    90° bond formed by edge-shared octahedra

    7

    4d5 or 5d5

  • Candidate materials

    β-Li2IrO3

    2D honeycomb lattice 3D hyper-honeycomb lattice

    Y. Singh & P. Gegenwart, PRB 82, 064412 (2010).

    T. Takayama et al., PRL 114, 077202 (2015).

    K. W. Plumb et al., PRB 90, 041112 (2014).

    α-RuCl3

    Na2IrO3

    8

  • J = �1.7meV K = �6.7meV � = +6.6meV

    Layered honeycomb magnet α-RuCl3

    •AFM order with zigzag spin structure at TN ~ 7.5 K

    •Transition at 14 K appears due to stacking faults.

    J. A. Sears et al., PRB 91, 144420 (2015).

    S. M. Winter et al., PRB 93, 214431 (2016).

    Presence of non-Kitaev interaction

    Dominant Kitaev term JK/kB ~ 100 K

    9

    H = X

    [J ~Si · ~Sj + JKS�i S�j + �(S↵i S�j + S�i S↵j )] Heisenberg Kitaev off-diagonal exchange

    Kx = −6.7 meV, Ky = −6.7 meV, Kz = −5.0 meV

  • Possible signatures of Kitaev QSL in α-RuCl3 Raman scattering

    Fermionic excitations

    L. J. Sandilands et al., PRL 114, 147201 (2015).

    J. Nasu et al., Nat. Phys. 12, 912 (2016).

    Broad magnetic continuum at high energy

    Inelastic neutron scattering

    Broad magnetic continuum appears below ~ JK/kB

    S.-H. Do et al., Nat. Phys. http://doi.org/10.1038/nphys4298. A. Banerjee et al., Nat. Mater. 15, 733 (2016). A. Banerjee et al., Science 356, 1055 (2017).

    Experiment

    Theory

    Possible signature of spin fractionalization

    10 More direct measurements are required.

  • What gives direct signature of Majorana fermions?

    Topological system characterized by Chern insulator under H

    Chiral edge current of Majorana fermions

    H = 0 H ≠ 0

    H = HK +He↵h

    He↵ h

    = �h̃ X

    (ijk)

    Sx i

    Sy j

    Sz k

    h̃ = �h3 ⇠ h 3

    �2f �f ⇠ 0.06JK

    HK = �Jx X

    x

    Sx i

    Sx j

    � J y

    X

    y

    Sy i

    Sy j

    � J z

    X

    z

    Sz i

    Sz j

    Effect of magnetic field

    Massless Dirac cone

    Massive Dirac cone

    (h || [111]) A. Kitaev, Ann. Phys. 321, 2 (2006).

    Flux gap

    11

    Non Abelian phase

  • �2D xy

    = ⌫ e2

    h

    What gives direct signature of Majorana fermions? Kitaev QSLInteger QHE

    q = ⌫

    2

    Half-integer thermal Hall conductance in a Kitaev QSL

    ν : Chern number 2D xy

    T = q

    6

    k2 B

    ~ 2D xy

    T = ⌫

    6

    k2 B

    ~

    q : Central charge

    cold

    hot cold

    hot

    electron Majorana fermion

    B B

    Z2 flux

    Chiral edge current of charge neutral Majorana fermions

    = # of chiral edge modes

    Chiral edge current of electrons

    2D xy

    T =

    1

    2

    ✓ ⇡

    6

    k2B ~

    ◆ q = v in IQHE

    12

  • Thermal Hall effect in insulating magnets ✓

    q 0

    ◆ =

    ✓  xx

     xy

    � xy

     xy

    ◆✓ �rT

    x

    �rT y

    Ferromagnetically ordered state

    Paramagnetic state

    Spin liquid state

    D. Watanabe, PNAS 113, 8653 (2016). Cu3V2O7(OH)2・2H2O

    Lu2V2O7 Ho2V2O7, In2Mn2O7, BiMnO3 Cu(1-3,bdc)

    Magnon Hall effect arising from Berry phase

    Tb2Ti2O7

    Spinon Hall effect in QSL state with spinon Fermi surface

    M. Hirschberger et al., Science 348, 106 (2015).

    Y. Onose et al, Science 329, 297 (2010). Ideue et al., PRB 85, 134411 (2012). M. Hirschberger et al., PRL 115, 106603 (2015).

    spinon xx

    = 2 ⇡2

    3

    ⇣" F

    ~ ⌧ ⌘ k2

    B

    T

    h

    1

    d spinon xy

    = spinon xx

    (! c

    ⌧)

    H. Katsura et al., PRL 104, 066403 (2010).

    R. Matsumoto et al., PRB 89, 054420 (2014).

    Lu2V2O7

  • Thermal transport measurements in α-RuCl3

    14

    • Clear anomaly at TN ~ 7.5 K • No discernible anomaly at

    ~ 14 K due to stacking faults

    High quality single crystal