Thermal transport in the Kitaev model Joji Nasu Department of - - PowerPoint PPT Presentation

YITP, Kyoto University, 10 November, 2017 Novel Quantum States in Condensed Matter 2017 (NQS2017)

Joji Nasu Department of Physics, Tokyo Institute of Technology

Thermal transport in the Kitaev model

• J. Nasu, J. Yoshitake, and Y. Motome, Phys. Rev. Lett. 119, 127204 (2017).

Collaborators: Yukitoshi Motome, Junki Yoshitake (UTokyo)

Contents

2

Introduction Method Thermal transport w/o magnetic fjeld Thermal transport w/ magnetic fjeld Summary

Contents

3

Introduction Method Thermal transport w/o magnetic fjeld Thermal transport w/ magnetic fjeld Summary

Quantum spin liquid (QSL)

4

Quantum fmuctuation disturbs orderings.

Quantum spin liquid (QSL): No singularity in or

Cv χ

No apparent symmetry breakings down to low T

QSL Paramagnet Crossover T

Fractional excitations Characterization of QSLs with emergent fermions Low-T behavior of Cv (T-linear) Dynamical response (continuum) Thermal transport

• S. Yamashita et al., Nat. Phys. 4, 459 (2008).

κ-(BEDT-TTF)2Cu2(CN)3

• M. Yamashita et al., Nat. Phys. 5, 44 (2009).

Specifjc heat Thermal conductivity Thermal Hall conductivity

• D. Watanabe et al.,
• Proc. Natl. Acad. Sci. 113, 8653 (2016).

Kagomé volborthite herbertsmithite

T.-H. Han et al., Nature 492, 406 (2012).

Neutron scattering

Kitaev model

5

H =−Jx X

<i j>x

Sx

i Sx j −Jy

X

<i j> y

Sy

i Sy j −Jz

X

<i j>z

Sz

i Sz j

• A. Kitaev, Annals of Physics 321, 2 (2006).

frustration Z2 fmux (conserved quantity) Wp on each plaquette

S=1/2 spin

Wp

ground state: quantum spin liquid (Only NN interactions are fjnite) Bond-dependent interactions Free Majorana fermions on a honeycomb lattice; analogous to graphene

Jz Jx

Jy

Dirac cones Assuming Wp=+1

H = iJγ 4 X

hijiγ

cicj

{ci} : Majorana fermions

emerging from spins

Kitaev model

6

H =−Jx X

<i j>x

Sx

i Sx j −Jy

X

<i j> y

Sy

i Sy j −Jz

X

<i j>z

Sz

i Sz j

• A. Kitaev, Annals of Physics 321, 2 (2006).

Fractional fermionic excitations frustration Majorana Chern insulator
 by applying magnetic fjeld in gapless phase Thermal Hall efgect Z2 fmux (conserved quantity) Wp on each plaquette

S=1/2 spin

Wp

Emergent fermions may carry heat. Bond-dependent interactions

Jz Jx

Jy

Gapless QSL ground state: quantum spin liquid (Only NN interactions are fjnite)

Kitaev spin liquid: fractionalization

7

Itinerant Majorana fermions (IMF) Localized Majorana fermions (LMF)

Si

TL TH

0.1 0.2 0.3 0.5 1 10-2 10-1 100 101

L=8 L=10 L=12 L=20

Cv S / ln 2 Thermal fractionalization T/J

JN, M. Udagawa, and Y. Motome, Phys. Rev. Lett. 113, 197205 (2014). JN, M. Udagawa, and Y. Motome, Phys. Rev. B 92, 115122 (2015). S.-H. Do et al., Nat. Phys. (2017).

Realization of Kitaev QSLs

8

t2g5

jefg=1/2 localized spin

• G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009)

Strong spin-orbit coupling

H =−Jx X

<i j>x

Sx

i Sx j −Jy

X

<i j> y

Sy

i Sy j −Jz

X

<i j>z

Sz

i Sz j

+JH X

<ij>

Si · Sj

Kitaev-Heisenberg model

A2IrO3 (A=Li,Na)

• Y. Singh and P. Gegenwart, Phys. Rev. B 82, 064412 (2010).
• Y. Singh et. al., Phys. Rev. Lett. 108, 127203 (2012).
• R. Comin et. al., Phys. Rev. Lett. 109, 266406 (2012).
• S. K. Choi et. al., Phys. Rev. Lett. 108, 127204 (2012).
• K. W. Plumb et al., Phys. Rev. B. 90, 041112 (2014).
• Y. Kubota et al., Phys. Rev. B 91, 094422 (2015).
• L. J. Sandilands et al., Phys. Rev. Lett. 114, 147201 (2015).
• J. A. Sears, M. Songvilay et al., Phys. Rev. B 91, 144420 (2015).
• M. Majumder et al., Phys. Rev. B 91, 180401(R) (2015).

α-RuCl3

Sx

i Sx j

Sy

i Sy j

Sz

i Sz j

Ir4+

Tc~10K Tc~10K

Magnetic order

Ir4+ 5d5 Ru3+ 4d5

Kitaev term plays a dominant role.

• A. Banerjee et al., Nat. Mater. 15, 733 (2016).
• Y. Yamaji et al., Phys. Rev. Lett. 113, 107201 (2014).
• K. Foyevtsova et al., Phys. Rev. B 88, 035107 (2013).

Dynamical response

9

• L. J. Sandilands et al., Phys. Rev. Lett. 114, 147201 (2015).

Raman scattering in RuCl3 Inelastic neutron scattering in RuCl3

• A. Banerjee et al., Nat. Mater., Nat. Mater. 15, 733 (2016).

Raman scattering in β-,γ-Li2IrO3

• A. Glamazda et al., Nat. Commun. 7, 12286 (2016).
• J. Knolle et al., Phys. Rev. Lett. 113, 187201 (2014).

Theory

Comparison of experiment & theory

10

Bosonic background

n+1

0.1 0.2 0.3 50 100 150 200 250 300 Intensity (a.u.) T (K) Experiment Theory (1-f )2

0.0 0.1 0.2 0.3 0.4 0.5 50 100 150 200 250 300 Intensity (a.u.) T (K)

Bosonic background

n+1

Theory Experiment

Raman scattering Dynamical spin correlation

S.-H. Do et al., Nat. Phys. (2017).

• J. Yoshitake, JN, and Y. Motome, Phys. Rev. Lett. 117, 157203 (2016)
• J. Yoshitake, JN, Y. Kato, and Y. Motome, Phys. Rev. B 96, 024438 (2017)
• J. Yoshitake, JN, and Y. Motome, Phys. Rev. B 96, 064433 (2017),
• L. J. Sandilands et al., Phys. Rev. Lett. 114, 147201 (2015).

JN, J. Knolle, D. L. Kovrizhin, Y. Motome, R. Moessner, Nat. Phys., 12, 912 (2016).

JzS zS z

• Jx
• Jy

J

ωf

ωf ωi ωf ωi ε1 ε2 [1-f(ε1)][1-f(ε2)]

Two-fermion excitation

Good agreement between
 the present theory and experimental results Magnetic order occurs at ~10K in α-RuCl3. Fermionic T dependence appears around 100K. High-energy features are consistent.

Thermal transport in α-RuCl3

11

• D. Hirobe, M. Sato, Y. Shiomi, H. Tanaka, and E. Saitoh, Phys. Rev. B 95, 241112 (2017).
• I. A. Leahy et al., Phys. Rev. Lett. 118, 187203 (2017).

Longitudinal thermal conductivity κ exhibits a peak at a peak in specifjc heat κ is enhanced in low-T whereas it is suppressed in intermediate-T by applying magnetic fjeld.

• R. Hentrich et al., arXiv:1703.08623 (2017).

Another study for κ in RuCl3

Purpose

12

Magnetic order at low T Precursor of Kitaev QSL (fractionalization) above Tc (~10K) Candidates of Kitaev materials Two stances:

Cooperation efgect of the Kitaev and Heisenberg interactions What is the Kitaev QSL? How should it be observed? Our starting point

Heat transport

Fractionalization of spins into Majorana fermions

What occurs in the pure Kitaev limit at fjnite temperature?

Topological nature with magnetic fjeld

Contents

13

Introduction Method Thermal transport w/o magnetic fjeld Thermal transport w/ magnetic fjeld Summary

Jordan-Wigner transformation

14

Honeycomb lattice: a zigzag xy chain connected by z-bonds Jordan-Wigner transformation

H.-D. Chen and J. Hu, Phys. Rev. B 76, 193101 (2007).

• X. Y. Feng, G.-M. Zhang, and T. Xiang, Phys. Rev. Lett. 98, 087204 (2007).

H.-D. Chen and Z. Nussinov, J. Phys. A Math. Theor. 41, 075001 (2008).

regarding the honeycomb lattice as one open chain

H =−Jx X

<i j>x

Sx

i Sx j −Jy

X

<i j> y

Sy

i Sy j −Jz

X

<i j>z

Sz

i Sz j

Introducing Majorana fermions

Sz

i = a† i ai − 1

2 S+

i = (S i )† = i1

Y

i0=1

(1 2ni0)a†

i

H = iJx 4 X

hijix

cicj iJy 4 X

hijiy

cicj + Jz 4 X

hijiz

¯ ci ¯ cjcicj cicj ¯ ci ¯ cj cicj cicj

¯ ci = (ai − a†

i )/i

ci = ai + a†

i

Fermions: ai, a†

i

: local conserved quantity

ηr ≡ i ¯ ci ¯ cj [ ¯ ci ¯ cj, H] = 0

H = iJx 4 X

hijix

cicj iJy 4 X

hijiy

cicj iJz 4 X

hijiz

ηrcicj cicj

cicj cicj ηr

Method

15

H =−Jx X

<i j>x

Sx

i Sx j −Jy

X

<i j> y

Sy

i Sy j −Jz

X

<i j>z

Sz

i Sz j

H = iJx 4 X

hijix

cicj iJy 4 X

hijiy

cicj iJz 4 X

hijiz

ηrcicj cicj cicj cicj ηr ηr = i ¯ ci ¯ cj

Quantum spin model

: Itinerant Majorana : Localized Majorana

¯ ci

ci

Itinerant fermion model

Si

Jordan-Wigner transformation

Free Majorana fermion system with thermally fmuctuating fmuxes Wp = ηrηr0

Sign problem-free “Quantum” Monte Carlo simulations

Jx = Jy = Jz = J

Simulations are classical and done for fmipping Ising valuables ηr.

Quantum nature of S=1/2 spins is fully taken into account!

TL TH

Specifjc heat and entropy

16

0.1 0.2 0.3 0.5 1 0.05 0.1 0.15 1 10-2 10-1 100 101

L=8 L=10 L=12 L=20

Cv 〈Six Sjx〉

(NN correlation)

〈Wp〉

(local conserved quantity)

S T/J

Si

: itinerant Majorana : localized Majorana

¯ ci ci

Entropy release at T** Entropy release at T* development of spin correlation coherent develop of Wp (Entropy)

Sx

i Sx j = − i

4cicj

NN x bond Wp = Y

r ∈p

ηr Local conserved quantity

: itinerant Majorana : localized Majorana

¯ ci ci

(matter Majorana) (fmux Majorana)

ηr = i ¯ ci ¯ cj Double peak structure Release of a half of entropy at each crossover

Contents

17

Introduction Method Thermal transport w/o magnetic fjeld Thermal transport w/ magnetic fjeld Summary

Thermal conductivity

18

H = iJx 4 X

hijix

cicj iJy 4 X

hijiy

cicj iJz 4 X

hijiz

ηrcicj cicj cicj cicj ηr ηr = i ¯ ci ¯ cj

: Itinerant Majorana : Localized Majorana

¯ ci

ci

Itinerant fermion model

Si

κxx = κyy Jx = Jy = Jz = J

Isotropic case Itinerant Majoranas carry thermal current: Jγ

Q = κγγ0∂γ0T

Energy polarization: Energy current: which is written by itinerant Majoranas

• K. Nomura, S. Ryu, A. Furusaki, and N. Nagaosa, Phys. Rev. Lett. 108, 26802 (2012), H. Sumiyoshi and S. Fujimoto, JPSJ 82, 023602 (2013).

Kubo formula + “gravitomagnetic energy magnetization” JQ = JE

Thermal current:

(zero chemical potential for Majorana fermions)

AC thermal conductivity

19

0.1 0.2 0.3

• 2
• 1

1

L=8 L=10 L=12 L=20

Cv

1

• 2
• 1

1

〈Wp〉

Free fmux Random fmux

ω/J T/J

10-2 10-1 100 101 0.0 0.5 1.0 1.5 2.0 0.00 0.04 0.08 0.12 0.16

κxx(ω)/J

AC part of thermal conductivity 
 vanishes in the free-fmux case.

Wp = +1

Fluctuation of fmuxes yields
 fjnite value of κxx(ω). Finite-ω contribution detects fmux fmuctuations.

TL TH

Low-energy contribution increases around TH High-energy contribution develops around TL [H, JQ] = 0

Wp = +1

with

AC thermal conductivity

20

0.00 0.02 0.04 0.06 0.08 0.10 0.0 0.5 1.0 1.5 2.0 2.5 T/J=0.009 T/J=0.0375 T/J=0.75 T/J=1.97 0.00 0.02 0.04 0.06 0.08 0.10 0.0 0.5 1.0 1.5 2.0 2.5

ω/J κxx(ω)/J

AC component grows with increasing T. The peak shifts to low-T side and decreases.

0.000 0.002 0.004 0.006 0.00 0.01 0.02 0.03 0.04 0.05 0.06 L=10 L=12 L=20 0.000 0.002 0.004 0.006 0.00 0.01 0.02 0.03 0.04 0.05 0.06

ω/J κxx(ω→0)

Low-energy part shows size dependence. DC component is obtained by the extrapolation. Dip becomes smaller with increase of size. AC part of thermal conductivity 
 vanishes in the free-fmux case.

Wp = +1

[H, JQ] = 0

Wp = +1

with

Thermal conductivity

21

0.1 0.2 0.3

L=8 L=10 L=12 L=20

Cv

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

• 2
• 1

1

T/J κxx/J

DC component is obtained by extrapolation. κxx takes a peak around TH. Transport is governed by itinerant Majoranas.

TL TH Iκxx/J2

0.1 0.2 10-2 10-1 100 101

Integrated AC thermal conductivity:

Ixx

κ

= Z ∞ κxx(ω)dω

Fluctuation of fmuxes yields
 fjnite value of .

Ixx

κ

Ixx

κ detects fmux fmuctuations.

Contents

22

Introduction Method Thermal transport w/o magnetic fjeld Thermal transport w/ magnetic fjeld Summary

Introduction of magnetic fjeld

23

:low-energy efgective model

−Jx X

<i j>x

Sx

i Sx j −Jy

X

<i j> y

Sy

i Sy j −Jz

X

<i j>z

Sz

i Sz j

Model Hamiltonian: H = HK + H eff

h

HK =

i j k

Magnetic fjeld for the direction perpendicular to the honeycomb plane

˜ h = λh3 ∼ h3 ∆2

Efgective magnetic fjeld:

Hh = −h X

i

⇣ Sx

i + Sy i + Sz i

⌘ H eff

h

= −˜ h X

(ijk)

Sx

i Sy j Sz k

• A. Kitaev, Annals of Physics 321, 2 (2006).

Majorana Chern insulator by applying the Magnetic fjeld Chiral edge mode with x y h

∆/J ∼ 0.1

Longitudinal thermal conductivity

24

T/J Cv κxx/J κxx/T L=12 TL TH

κxx takes a peak around TH. κxx is insensitive to h.

~

κxx/T at low T limit vanishes due to the gap opening.

0.0 0.1 0.2 0.3 0.4 0.00 0.02 0.04 0.06 0.08 0.0 0.1 0.2 0.3 0.4 10-2 10-1 100 101

h/J=0.012 h/J=0.024 h/J=0.036 h/J=0.048 h/J=0.0

~ ~ ~ ~

Transverse thermal conductivity

25

T/J Cv κxy/J κxy/T L=12

π/12 Quantization at low T κxy takes a peak around TH. κxy increases with increasing h.

~

contrasting behavior to κxx Deviation from π/12 around TL Nonmonotonic behavior

h/J=0.012 h/J=0.024 h/J=0.036 h/J=0.048 h/J=0.0

~ ~ ~ ~

0.0 0.1 0.2 0.3 0.4 0.00 0.01 0.02 0.0 0.1 0.2 0.3 10-2 10-1 100 101

TL TH

Efgect of fmux excitation

26

Wp

: Z2 fmux

Wp = ±1 Wp ≡ +1: ground state

(free fmux)

Wp = −1: fmux excitation

High-T peak is well accounted for by random fmux excitation. MC result deviates from zero-fmux one around low-T crossover. TL TH κxy/J κxy/T T/J

h/J=0.048

~

π/12

0.01 0.02 MC random flux zero flux 0.05 0.1 0.15 0.2 0.25 0.3 10-2 10-1 100 101

free flux

Deviation from Plateau

27

Arrhenius plot for κxy/T Magnetic fjeld dependence of gaps

Majorana gap Flux gap

Flux gap is almost independent of h. Majorana gap linearly depend on h.

~ ~

Deviation of κxy/T from the quantized value is determined by the fmux gap.

Magnetic fjeld dependence

28

h/J κxy/T κxx/T

~

Longitudinal component Transverse component

almost unchanged by h Enhancement by h

~ ~

Almost zero at low T Linear dependence for h

~

at fjnite T κxy is proportional to h3.

Contrasting magnetic-fjeld dependence between κxx and κxy.

˜ h ∼ h3 ∆2

0.0 0.1 0.2 0.3 0.4 0.5 0.6

• 0.3
• 0.2
• 0.1

0.0 0.1 0.2 0.3

• 0.06
• 0.04
• 0.02

0.00 0.02 0.04 0.06 T/J=0.00675 T/J=0.03 T/J=0.099 T/J=1.0

Contents

29

Introduction Method Thermal transport w/o magnetic fjeld Thermal transport w/ magnetic fjeld Summary

Summary

30

Kitaev model on a honeycomb lattice

“Quantum” Monte Carlo simulation in Majorana representation Magnetic fjeld is introduced as an efgective model.

With magnetic fjeld

Peculiar T dependence of κxy/T due to the fmux excitations. Thermal Hall conductivity is proportional to h3.

Without magnetic fjeld

Longitudinal thermal conductivity exhibits a peak at high-T crossover Dynamical component detects fmux fmuctuation. attributed to the itinerant Majorana fermions