Symposium on Novel Quantum States in Condensed …nqs2017.ws/Slides/Sympo/10th/...Symposium on Novel...

25
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 / )] 1.5 1.0 0.5 0.0 -0.5 κ xy /T (10 -3 W/K 2 m) 5.2 5.0 4.8 4.6 4.4 μ 0 H (T) 0 1/2 9.0 8.5 8.0 7.5 μ 0 H || (T) 4.9 K -3 2 cold hot Majorana fermion Z 2 flux Chiral Majorana edge current Half-integer thermal Hall conductance

Transcript of Symposium on Novel Quantum States in Condensed …nqs2017.ws/Slides/Sympo/10th/...Symposium on Novel...

Page 1: Symposium on Novel Quantum States in Condensed …nqs2017.ws/Slides/Sympo/10th/...Symposium on Novel Quantum States in Condensed Matter 2017 (NQS2017) 8-10 November 2017 YITP, Kyoto

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

hκxy

2D/T [( π/6)(kB 2 /

)]

hκxy

2D/T [(π/6)(kB 2 /

)]h

κxy

2D/T [(π/6)(kB 2 /

)]

hκxy

2D/T [( π/6)(kB 2 /

)]

1.5

1.0

0.5

0

-0.5

κ xy/T

(10

-3 W

/K2 m

)

5.25.04.84.64.4µ0H⊥ (T)

0 1/2

9.08.58.07.5µ0H|| (T)

4.3 K

1.5

1.0

0.5

0.0

-0.5

κ xy/T

(10

-3 W

/K2 m

)

5.25.04.84.64.4µ0H⊥ (T)

0 1/2

9.08.58.07.5µ0H|| (T)

4.9 K2.0

1.5

1.0

0.5

0.0

-0.5

κ xy/T

(10

-3 W

/K2 m

)

6.05.55.04.54.0µ0H⊥ (T)

0 1/2 1

10.09.08.07.0µ0H|| (T)

5.6 K

8.2 K

1.5

1.0

0.5

0

-0.5

κ xy/T

(10

-3 W

/K2 m

)

5.25.04.84.64.4µ0H⊥ (T)

0 1/2

9.08.58.07.5µ0H|| (T)

3.9 K

a b

c d

cold

hot

Majorana fermion

Z2 flux

Chiral Majorana edge currentHalf-integer thermal Hall conductance

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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

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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).

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IntroductionQuantum 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 magnets2D trianglar 2D kagome 3D pyrochlore

Exotic physical properties in QSLsTopological 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

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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

�Jx

Sx

i

Sx

j

�JzSzi S

zj

�JySyi S

yj �JyS

yi S

yj

�Jx

Sx

i

Sx

j

�JzSzi S

zj

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

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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 =iJ

x

4

X

<ij>

x

ci

cj

� iJy

4

X

<ij>

y

ci

cj

� iJz

4

X

<ij>

z

⌘r

ci

cj ⌘r = ic̄ic̄j

Spin 1/2Itinerant Majorana fermion with Dirac cone dispersion

Localized Majorana fermion

Z2 fluxes

Si

ci

c̄iWp = ⌘r⌘r0

Two types of QSLs

Gappless QSL: Majorana metalGapped 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

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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

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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

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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

<ij>

[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

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Possible signatures of Kitaev QSL in α-RuCl3Raman 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

10More direct measurements are required.

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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

= �h̃X

(ijk)

Sx

i

Sy

j

Sz

k

h̃ = �h3 ⇠ h3

�2f

�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

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�2Dxy

= ⌫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 2Dxy

T= q

6

k2B

~2Dxy

T= ⌫

6

k2B

~

q : Central charge

cold

hot cold

hot

electronMajorana fermion

B B

Z2 flux

Chiral edge current of charge neutral Majorana fermions

= # of chiral edge modes

Chiral edge current of electrons

2Dxy

T=

1

2

✓⇡

6

k2B~

◆q = v in IQHE

12

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Thermal Hall effect in insulating magnets✓

q0

◆=

✓xx

xy

�xy

xy

◆✓�rT

x

�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).

spinon

xx

= 2⇡2

3

⇣"F

~ ⌧⌘ k2

B

T

h

1

dspinon

xy

= spinon

xx

(!c

⌧)

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

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

Lu2V2O7

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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

✓q0

◆=

✓xx

xy

�xy

xy

◆✓�rT

x

�rTy

Magnetic susceptibility Specific heat28

26

24

22

20

18

χ (1

0-3 e

mu/

mol

)

20151050

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

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Longitudinal thermal conductivity κxx

15

xx

= spxx

+ phxx

spin phonon

35

30

25

20

15

10

5

0

κ xx

(W/K

m)

706050403020100

T (K)

Sample 10 T12 T

Sample 20 T12 T

TN

12

8

4

0151050

TN

-0.08

-0.04

0.00

Δκ xx(H

)/κxx

(0)

1612840

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 of spin-phonon scattering by spin polarization.

Thermal Hall effectThermal transport is governed by spin excitations.

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Thermal Hall conductivity κxy

Finite κxy ~ 10-2 W/Km at T < JK/kB

-2

0

2

κ xy

(10-3

W/K

m)

-10 0 10µ0H (T)

7 K

12 KT > TN

T < TN

Distinct H-dependence below and above TN • Sign change below TN

• Upward curvature above TN but downward below TNe.g.) κxy < 10-3 W/Km

in volborthite (spin liquid) Tb2Ti2O7 (paramagnet)

16

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

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8

6

4

2

0

-2

-4

κ xy/T

(10

-4 W

/K2 m

)

806040200

T (K)

0.2

0.1

0

-0.1

(κxy

2D/T)/(kB

2/h)

π/12

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

TN

0.2

0.1

0

-0.1

(κxy

2D

/T)/(k B

2 /h

)1.00.50.0

T/(JK/kB)

h*/JK = 0.030.06

π/12

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

Page 18: Symposium on Novel Quantum States in Condensed …nqs2017.ws/Slides/Sympo/10th/...Symposium on Novel Quantum States in Condensed Matter 2017 (NQS2017) 8-10 November 2017 YITP, Kyoto

Comparison with numerical calculations

18Y. 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

2Dxy

= xy

d

interlayer distance d = 5.72 Å

thermal Hall conductance per 2D honeycomb layer

Page 19: Symposium on Novel Quantum States in Condensed …nqs2017.ws/Slides/Sympo/10th/...Symposium on Novel Quantum States in Condensed Matter 2017 (NQS2017) 8-10 November 2017 YITP, Kyoto

J. Nasu, J. Yoshitake & Y. Motome, PRL 119, 127204 (2017).

2Dxy

= xy

d

interlayer distance d = 5.72 Å

thermal Hall conductance per 2D honeycomb layer

Experiments Calculations

Comparison with numerical calculations

19Y. 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

Page 20: Symposium on Novel Quantum States in Condensed …nqs2017.ws/Slides/Sympo/10th/...Symposium on Novel Quantum States in Condensed Matter 2017 (NQS2017) 8-10 November 2017 YITP, Kyoto

Suppression of AFM order by in-plane fields

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

Hc|| ~ 7-8 TM. 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?

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Suppression of AFM order by in-plane fields

T�11 / 1

Texp

✓�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?

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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 scatteringSpin wave

Continuum Continuum

T�11 / 1

Texp

✓�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

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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

Page 24: Symposium on Novel Quantum States in Condensed …nqs2017.ws/Slides/Sympo/10th/...Symposium on Novel Quantum States in Condensed Matter 2017 (NQS2017) 8-10 November 2017 YITP, Kyoto

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

0

κ xx

(W/K

m)

86420

µ0H|| (T)

θ = 60º 7.0 K 6.0 K 5.0 K 2.5 K

10

8

6

4

2

0

κ xx

(W/K

m)

14121086420

T (K)

µ0H|| = 7 T (θ = 90º)

0 Tµ0H⊥ = 12 T(θ = 0º)

µ0H|| = 7 T (θ = 60º) 12

10

8

6

4

2

0

T (K

)

86420µ0H|| (T)

κxx(H) κxx(T) θ = 45° θ = 60°

θ = 90°

M/H (θ = 90°)

θ

Page 25: Symposium on Novel Quantum States in Condensed …nqs2017.ws/Slides/Sympo/10th/...Symposium on Novel Quantum States in Condensed Matter 2017 (NQS2017) 8-10 November 2017 YITP, Kyoto

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