Half-integer thermal Hall conductance in a Kitaev spin liquid - - PowerPoint PPT Presentation

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

κxy

2D/T [(π/6)(kB 2 / )]

1.5 1.0 0.5 0.0

• 0.5

κxy/T (10

• 3 W/K

2m)

5.2 5.0 4.8 4.6 4.4 µ0H⊥ (T) 1/2 9.0 8.5 8.0 7.5 µ0H|| (T)

4.9 K

• 3

2

c d

cold hot Majorana fermion Z2 flux

Chiral Majorana edge current Half-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 = −Jx X

<ij>x

Sx

i Sx j − Jy

X

<ij>y

Sy

i Sy j − Jz

X

<ij>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

−JxSx

i Sx j

−JzSz

i Sz j

−JySy

i Sy j

−JySy

i Sy j

−JxSx

i Sx j

−JzSz

i Sz j

S = 1/2 spins on tri-coordinate lattices

• A. Kitaev, Ann. Phys. 321, 2 (2006).

HK = −Jx X

<ij>x

Sx

i Sx j − Jy

X

<ij>y

Sy

i Sy j − Jz

X

<ij>z

Sz

i Sz j

Kitaev model

Fractionalization of quantum spins

Wp = +1

HK = iJx 4 X

<ij>x

cicj − iJy 4 X

<ij>y

cicj − iJz 4 X

<ij>z

ηrcicj

ηr = i¯ ci ¯ cj

Spin 1/2 Itinerant Majorana fermion with Dirac cone dispersion Localized Majorana fermion Z2 fluxes Si

ci

¯ ci

Wp = ηrηr0

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

• n 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 = −1.7 meV K = −6.7 meV Γ = +6.6 meV

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

<ij>

[J ~ Si · ~ Sj + JKSγ

i Sγ j + Γ(Sα i Sβ j + Sβ i Sα j )]

Heisenberg Kitaev

• ff-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 + Heff

h

Heff

h = −˜

h X

(ijk)

Sx

i Sy j Sz k

˜ h = λh3 ∼ h3 ∆2

f

∆f ∼ 0.06JK

HK = −Jx X

<ij>x

Sx

i Sx j − Jy

X

<ij>y

Sy

i Sy j − Jz

X

<ij>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

σ2D

xy = ν e2

h

What gives direct signature of Majorana fermions?

Kitaev QSL Integer QHE

q = ν 2

Half-integer thermal Hall conductance in a Kitaev QSL

ν : Chern number

κ2D

xy

T = q π 6 k2

B

~

κ2D

xy

T = ν π 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

κ2D

xy

T = 1 2 ✓π 6 k2

B

~ ◆

q = v in IQHE

12

Thermal Hall effect in insulating magnets

✓ q ◆ = ✓ κxx κxy κxy κxy ◆ ✓ rTx rTy ◆

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).

κspinon

xx

= 2π2 3 ⇣εF ~ τ ⌘ k2

BT

h 1 d κspinon

xy

= κspinon

xx

(ωcτ)

• 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

✓ q ◆ = ✓ κxx κxy κxy κxy ◆ ✓ rTx rTy ◆

Magnetic susceptibility Specific heat

28 26 24 22 20 18 χ (10

• 3 emu/mol)

20 15 10 5 T (K)

Sample 1 Sample 2

Thermal Hall effect

ex.) Spin liquid state: Kagome volborthite Cu3V2O7(OH)2･2H2O Magnetically ordered state: Kagome Cu-(1-3,bdc) Pyrochlore Lu2V2O7

0 T

spinon magnon

cf.) Phonon thermal Hall effect κxy/T ~ 10-5 W/K2m κxy/T ~ 10-6 W/K2m κxy/T ~ 10-5 - 10-4 W/K2m

α-RuCl3 single crystals

Longitudinal thermal conductivity κxx

15

κxx = κsp

xx + κph xx

spin phonon

35 30 25 20 15 10 5 κxx (W/Km) 70 60 50 40 30 20 10 T (K)

Sample 1 0 T 12 T Sample 2 0 T 12 T

TN 12 8 4 15 10 5 TN

• 0.08
• 0.04

0.00 Δκxx(H)/κxx(0) 16 12 8 4 µ0H (T)

50 K 20 K 12 K 35 K 27 K

Sample 2

However, it is difficult to separate spin & phonon contributions.

• Clear anomaly in κxx at TN
• Suppression of κxx by magnetic field

κxxph is usually enhanced due to suppression

• f spin-phonon scattering by spin polarization.

Thermal Hall effect

Thermal transport is governed by spin excitations.

Thermal Hall conductivity κxy

Finite κxy ~ 10-2 W/Km at T < JK/kB

• 2

2 κxy (10

• 3 W/Km)
• 10

10 µ0H (T)

7 K 12 K

T > TN T < TN

Distinct H-dependence below and above TN

• Sign change below TN
• Upward curvature above TN

but downward below TN

e.g.) κxy < 10-3 W/Km in volborthite (spin liquid) Tb2Ti2O7 (paramagnet)

16

Thermal Hall effect below and above TN is different in origin.

8 6 4 2

• 2
• 4

κxy/T (10

• 4 W/K

2m)

80 60 40 20 T (K) 0.2 0.1

• 0.1

(κxy

2D/T)/(kB 2/h)

π/12

Sample 1 (6 T) Sample 1 (12 T) Sample 2 (15 T) TN

Thermal Hall conductivity κxy

17

• Spin liquid with spinon Fermi surface
• Enhancement of κxy with

positive sign below JK/kB ~ 80 K

• Broad peak at ~ 20 K
• A. V. Inyushkin & . N. Taldenkov,

JETP Lett. 86, 379 (2007).

• Phonons
• Magnons

Small DM interaction D/kB ~ 5 K << J/kB ~ 80 K κxy/T ~ 10-6 W/K2m Finite κxy/T usually appears in the ordered state.

• Exotic quasiparticle excitations

inherent to the spin-liquid state of α-RuCl3.

• D. Watanabe, PNAS 113, 8653 (2006).

In volborthite, Hall signal is negative. κxy/T ~ 10-5 W/K2m

• S. M. Winter et al.,

PRB 93, 214431 (2016).

Different T-dependence

Comparison with numerical calculations

18

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

T-dependence is consistent with numerical calculations for the 2D pure Kitaev model.

• J. Nasu, J. Yoshitake & Y. Motome,

PRL 119, 127204 (2017).

• Enhancement of κxy with positive sign below T < JK/kB
• Broad peak at T ~ 0.1JK/kB
• κxy/T reaches close to half of the quantization value.

Experiments Calculations

Possible signature of Majorana fermion excitations

κ2D

xy = κxyd

interlayer distance d = 5.72 Å thermal Hall conductance per 2D honeycomb layer

• J. Nasu, J. Yoshitake & Y. Motome,

PRL 119, 127204 (2017).

κ2D

xy = κxyd

interlayer distance d = 5.72 Å thermal Hall conductance per 2D honeycomb layer

Experiments Calculations

Comparison with numerical calculations

19

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

However, quantization of κxy2D/T is not attained due to the magnetic order.

T-dependence is consistent with numerical calculations for the 2D pure Kitaev model.

• Enhancement of κxy with positive sign below T < JK/kB
• Broad peak at T ~ 0.1JK/kB
• κxy/T reaches close to half of the quantization value.

Possible signature of Majorana fermion excitations

Suppression of AFM order by in-plane fields

• A. Banerjee et al., arXiv:1706.0703 (2017).

Hc|| ~ 7-8 T

• M. Majumder et al., PRB 91, 180401(R) (2015).

AFM order is little influenced by out-of-plane fields, but it is easily suppressed by in-plane fields.

Low-temperature properties are masked by the magnetic order.

Key questions: Is the magnetic order suppressed by tuning parameters?

Whether Kitaev QSL survives when suppressing the order?

20

Suppression of AFM order by in-plane fields

T −1

1

∝ 1 T exp ✓ −n∆ T ◆

NMR: Unusual spin gap

• N. Jansa et al., arXiv:1706.08455 (2017).

NMR

• A. Banerjee et al., arXiv:1706.0703 (2017).

Hc|| ~ 7-8 T Low-temperature properties are masked by the magnetic order.

21

Key questions: Is the magnetic order suppressed by tuning parameters?

Whether Kitaev QSL survives when suppressing the order?

NMR: Unusual spin gap Neutron scattering: Magnetic continuum at high energy above Hc||

Suppression of AFM order by in-plane fields

• A. Banerjee et al., arXiv:1706.0703 (2017).

Inelastic neutron scattering

Spin wave Continuum Continuum

T −1

1

∝ 1 T exp ✓ −n∆ T ◆

Low-temperature properties are masked by the magnetic order.

Key question: Kitaev QSL survives when suppressing the magnetic order? Kitaev QSL emerges under in-plane magnetic fields.

22

What gives direct signature of Majorana fermions?

Phase transition is tuned by H|| = Hsinθ. Thermal Hall response is determined by H⊥ = Hcosθ. Measurements of thermal Hall effect in tilted fields

Thermal Hall effect in a Kitaev QSL state

Low-temperature properties are masked by the magnetic order.

Key questions: The magnetic order can be suppressed by tuning parameters?

Kitaev QSL survives when suppressing the magnetic order?

23

Longitudinal thermal conductivity κxx

24

Strongly anisotropic response: Quasi 2D nature of magnetic properties. Suppression of the AFM order by in-plane field component.

6 5 4 3 2 1 κxx (W/Km) 8 6 4 2 µ0H|| (T)

θ = 60º 7.0 K 6.0 K 5.0 K 2.5 K

10 8 6 4 2 κxx (W/Km) 14 12 10 8 6 4 2 T (K)

µ0H|| = 7 T (θ = 90º) 0 T µ0H⊥ = 12 T (θ = 0º) µ0H|| = 7 T (θ = 60º)

12 10 8 6 4 2 T (K) 8 6 4 2 µ0H|| (T)

κxx(H) κxx(T) θ = 45° θ = 60° θ = 90° M/H (θ = 90°)

θ

Summary

25

Measurements of thermal Hall effect in a Kitaev magnet candidate α-RuCl3 Perpendicular fields Striking enhancement of κxy/T with positive sign below T ~ JK/kB A broad peak at T ~ 0.1JK/kB Tilted fields Observation of half-integer thermal Hall conductance for the first time.

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

Signature of Majorana fermion excitations Sudden disappearance of the quantum Hall plateau at high field Evidence for chiral Majorana edge current Topological phase transition