Novel Quantum States in Condensed Matter 217 (NQS217) YITP, Kyoto University, 1 November, 217 Thermal transport in the Kitaev model Joji Nasu Department of Physics, Tokyo Institute of Technology Collaborators: Yukitoshi Motome, Junki Yoshitake (UTokyo) J. Nasu, J. Yoshitake, and Y. Motome, Phys. Rev. Lett. 119, 12724 (217).
Contents Introduction Method Thermal transport w/o magnetic field Thermal transport w/ magnetic field Summary 2
Contents Introduction Method Thermal transport w/o magnetic field Thermal transport w/ magnetic field Summary 3
Quantum spin liquid (QSL) QSL Paramagnet Crossover Quantum fluctuation disturbs orderings. T κ-(bedt-ttf)2cu2(cn)3 No apparent symmetry breakings down to low T Fractional excitations Thermal conductivity No singularity in Cv or Specific heat S. Yamashita et al., Nat. Phys. 4, 459 (28). M. Yamashita et al., Nat. Phys. 5, 44 (29). Quantum spin liquid (QSL): Low-T behavior of Cv (T-linear) Dynamical response (continuum) Thermal transport herbertsmithite D. Watanabe et al., Proc. Natl. Acad. Sci. 113, 8653 (216). T.-H. Han et al., Nature 492, 46 (212). Neutron scattering Characterization of QSLs with emergent fermions Thermal Hall conductivity Kagomé volborthite 4
H = Jx Kitaev model Six S jx Jy <i j > x y y Si S j <i j > y Jz Siz S zj <i j > z A. Kitaev, Annals of Physics 321, 2 (26). S=1/2 spin Wp Bond-dependent interactions frustration Z2 flux (conserved quantity) Wp on each plaquette Assuming Wp=+1 Jz Jx H = ij ci c j 4 hi ji ground state: quantum spin liquid (Only NN interactions are finite) {ci } : Majorana fermions emerging from spins Free Majorana fermions on a honeycomb lattice; analogous to graphene Jy Dirac cones 5
H = Jx Kitaev model Six S jx <i j > x Jy <i j > y y y Si S j Jz Siz S zj <i j > z A. Kitaev, Annals of Physics 321, 2 (26). S=1/2 spin Wp Bond-dependent interactions frustration Z2 flux (conserved quantity) Wp on each plaquette ground state: quantum spin liquid (Only NN interactions are finite) Fractional fermionic excitations Emergent fermions may carry heat. Jz Gapless QSL Jx Jy Majorana Chern insulator by applying magnetic field in gapless phase Thermal Hall effect 6
Kitaev spin liquid: fractionalization Thermal fractionalization JN, M. Udagawa, and Y. Motome, Phys. Rev. Lett. 113, 19725 (214). JN, M. Udagawa, and Y. Motome, Phys. Rev. B 92, 115122 (215). Cv Localized Majorana fermions (LMF).3.3 S / ln 2 Si Itinerant Majorana fermions (IMF) L=8 L=1 L=12 L=2.2.1 1 TL TH.2 1-2 1-1 1 11-1 1-2 1 111 1.1.5 1-2 1 1-1 1 T/J 7 S.-H. Do et al., Nat. Phys. (217).
Realization of Kitaev QSLs Strong spin-orbit coupling t2g5 jeff=1/2 localized spin G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 12, 1725 (29) Kitaev-Heisenberg model y y x x H = Jx Si S j J y Si S j <i j > x Jz <i j > y Siz S zj <i j > z +JH <i j> Si S j Magnetic order A2IrO3 (A=Li,Na) Ir4+ 5d5 Tc~1K Y. Singh and P. Gegenwart, Phys. Rev. B 82, 64412 (21). Y. Singh et. al., Phys. Rev. Lett. 18, 12723 (212). R. Comin et. al., Phys. Rev. Lett. 19, 26646 (212). S. K. Choi et. al., Phys. Rev. Lett. 18, 12724 (212). α-rucl3 Ru3+ 4d5 K. W. Plumb et al., Phys. Rev. B. 9, 41112 (214). Y. Kubota et al., Phys. Rev. B 91, 94422 (215). L. J. Sandilands et al., Phys. Rev. Lett. 114, 14721 (215). J. A. Sears, M. Songvilay et al., Phys. Rev. B 91, 14442 (215). M. Majumder et al., Phys. Rev. B 91, 1841(R) (215). Ir4+ Siz Sjz Siy Sjy Tc~1K Six Sjx Kitaev term plays a dominant role. Y. Yamaji et al., Phys. Rev. Lett. 113, 1721 (214). K. Foyevtsova et al., Phys. Rev. B 88, 3517 (213). A. Banerjee et al., Nat. Mater. 15, 733 (216). 8
Dynamical response Raman scattering in RuCl3 L. J. Sandilands et al., Phys. Rev. Lett. 114, 14721 (215). Inelastic neutron scattering in RuCl3 A. Banerjee et al., Nat. Mater., Nat. Mater. 15, 733 (216). Raman scattering in β-,γ-li2iro3 A. Glamazda et al., Nat. Commun. 7, 12286 (216). Theory J. Knolle et al., Phys. Rev. Lett. 113, 18721 (214). 9
Comparison of experiment & theory Magnetic order occurs at ~1K in α-rucl3. Experiment a Theory ωi (1-f )2.2 Intensity (a.u.).3 Intensity (a.u.) Raman scattering Good agreement between the present theory and experimental results.5 L. J. Sandilands et al., Phys. Rev. Lett. 114, 14721 (215)..4 JN, J. Knolle, D. L. Kovrizhin, Y. Motome, R. Moessner, Nat. Phys., 12, 912 (216). ωf.3.2 b ωf n+1 ε2 ωf.1 [1-f(ε1)][1-f(ε2)] Bosonic background. 5 1 15 2 25 3 T (K).1 5 1 15 T (K) 2 25 -JySiySjy 3 ωi ε1 ωi ω Fermionic T dependence appears around 1K. ω c ε f f 2 Experiment -JxSixSjx -JJzSizSjz f(ε1)[1-f(ε2)] ωi -ε1 ωi High-energy features are consistent. S.-H. Do et al., Nat. Phys. (217). J. Yoshitake, JN, and Y. Motome, Phys. Rev. Lett. 117, 15723 (216) Theory Dynamical spin correlation Two-fermion excitation J. Yoshitake, JN, Y. Kato, and Y. Motome, Phys. Rev. B 96, 24438 (217) J. Yoshitake, JN, and Y. Motome, Phys. Rev. B 96, 64433 (217), 1
Thermal transport in α-rucl 3 I. A. Leahy et al., Phys. Rev. Lett. 118, 18723 (217). κ is enhanced in low-t whereas it is suppressed in intermediate-t by applying magnetic field. D. Hirobe, M. Sato, Y. Shiomi, H. Tanaka, and E. Saitoh, Phys. Rev. B 95, 241112 (217). Longitudinal thermal conductivity κ exhibits a peak at a peak in specific heat Another study for κ in RuCl 3 R. Hentrich et al., ariv:173.8623 (217). 11
Purpose Candidates of Kitaev materials Magnetic order at low T Two stances: Cooperation effect of the Kitaev and Heisenberg interactions What is the Kitaev QSL? How should it be observed? Our starting point Precursor of Kitaev QSL (fractionalization) above T c (~1K) What occurs in the pure Kitaev limit at finite temperature? Fractionalization of spins into Majorana fermions Topological nature with magnetic field Heat transport 12
Contents Introduction Method Thermal transport w/o magnetic field Thermal transport w/ magnetic field Summary 13
Jordan-Wigner transformation H = Jx Six S jx <i j > x y Jy y Si S j Siz S zj Jz <i j > y <i j > z Honeycomb lattice: a zigzag xy chain connected by z-bonds Jordan-Wigner transformation regarding the honeycomb lattice as one open chain Si+ = (Si ) = i 1 Y 2ni )ai (1 i =1 Fermions: ai, ai ci = ai + c i = (ai ai )/i [c i c j, H ] = r i c i c j : local conserved quantity 1 2 H.-D. Chen and J. Hu, Phys. Rev. B 76, 19311 (27).. Y. Feng, G.-M. Zhang, and T. iang, Phys. Rev. Lett. 98, 8724 (27). H.-D. Chen and Z. Nussinov, J. Phys. A Math. Theor. 41, 751 (28). ai Introducing Majorana fermions Siz = ai ai i Jx H = ciic j 4 hi ji x i Jx H = ci c j 4 hi ji x i Jy Jz ciic jj + c c iic c jjcciiccjj 4 hi ji 4 hi ji y z i Jy ci c j 4 hi ji y i Jz r ci c j 4 hi ji z 14
Method Quantum spin model H = Jx Six S jx Jy <i j > x y y Si S j <i j > y Jz Siz S zj <i j > z Jordan-Wigner transformation Itinerant fermion model i Jx H = ci c j 4 hi ji x Si ci c i i Jy ci c j 4 hi ji y : Itinerant Majorana i Jz r ci c j 4 hi ji z r = i c i c j : Localized Majorana Free Majorana fermion system with thermally fluctuating fluxes W p = r r Sign problem-free Quantum Monte Carlo simulations Quantum nature of S=1/2 spins is fully taken into account! Simulations are classical and done for flipping Ising valuables ηr. Jx = J y = Jz = J 15
Specific heat and entropy Cv.3.3 L=8 L=1 L=12 L=2.2.1.2 TL 1 1-2 TH 1-1 1 Double peak structure 11 Release of a half of entropy at each crossover.1 S.5 (Entropy) Wp Six Sjx.15 1-2 1-2 1-11-1.1 11 1 x x 1 1 (NN correlation) 1 NN x bond Si S j = ci.5 1 1-2 1 : itinerant Majorana (matter Majorana) 11 Local conserved quantity (local conserved quantity) Wp = Y r r = i c i c j r 2p 1 Si 1-1 i ci c j 4 ci c i -2 1-1 T/J 1 1 1 c i : localized Majorana (flux Majorana) : itinerant Majorana Entropy release at T** development of spin correlation : localized Majorana Entropy release at T* coherent develop of Wp 16
Contents Introduction Method Thermal transport w/o magnetic field Thermal transport w/ magnetic field Summary 17
Thermal conductivity Itinerant fermion model i Jx H = ci c j 4 hi ji x Si ci c i i Jy ci c j 4 hi ji y i Jz r ci c j 4 hi ji : Itinerant Majorana z r = i c i c j : Localized Majorana Itinerant Majoranas carry thermal current: JQ = @ T Energy polarization: Energy current: Thermal current: JQ = JE which is written by itinerant Majoranas (zero chemical potential for Majorana fermions) Kubo formula + gravitomagnetic energy magnetization K. Nomura, S. Ryu, A. Furusaki, and N. Nagaosa, Phys. Rev. Lett. 18, 2682 (212), H. Sumiyoshi and S. Fujimoto, JPSJ 82, 2362 (213). Isotropic case Jx = Jy = Jz = J xx = y y 18
AC thermal conductivity.3.3 L=8 L=1 L=12 L=2 Wp Cv.2.1.2 TL 1 1-2 TH 1-1 1 AC part of thermal conductivity vanishes in the free-flux case. W p = +1 11 Free.1 flux [H, JQ ] = with W p = +1 Fluctuation of fluxes yields finite value of κxx(ω). 1 1 11.16 Random flux 1-2 1-2 1-11-1 11 2. ω/j 1..8.5.4.. 1-2 High-energy contribution develops around TL 1-1 1 T/J T/J 1 κxx(ω)/j.12 1.5 Finite-ω contribution detects flux fluctuations. 1 Low-energy contribution increases around TH 19
AC thermal conductivity κ xx (ω)/j.6.4.1.8.6.4.2 T/J=.9 T/J=.375 T/J=.75 T/J=1.97...5 1. 1.5 2. 2.5 κ xx (ω ) ω/j AC part of thermal conductivity vanishes in the free-flux case. W p =+1 [H, J Q ] = W p =+1 with AC component grows with increasing T. The peak shifts to low-t side and decreases. Low-energy part shows size dependence..2 L=1 L=12. L=2..1.2.3.4.5.6 ω/j Dip becomes smaller with increase of size. DC component is obtained by the extrapolation. 2
Thermal conductivity Cv κ xx /J κ xx /J Iκ xx /J 2.3.2.1.8.7.6.5.4.3.2.1.2.1 T L L=8 L=1 L=12 L=2 1-2 1-1 1 1 1 T/J T H 1-2 1-1 1 1 1 DC component is obtained by extrapolation. κ xx takes a peak around T H. Transport is governed by itinerant Majoranas. Integrated AC thermal conductivity: I xx apple = Z 1 apple xx (!)d! Fluctuation of fluxes yields finite value of. Iapple xx Iapple xx detects flux fluctuations. 21
Contents Introduction Method Thermal transport w/o magnetic field Thermal transport w/ magnetic field Summary 22
Introduction of magnetic field HK = Jx Six S jx Jy <i j > x y y Si S j <i j > y Jz Siz S zj <i j > z A. Kitaev, Annals of Physics 321, 2 (26). Magnetic field for the direction perpendicular to the honeycomb plane y z x Hh = h Si + Si + Si i Hheff = h (i jk) y Six S j Skz :low-energy effective model h Model Hamiltonian: H = HK + Hheff Effective magnetic field: h = 3 h h3 2 k with /J.1 Majorana Chern insulator by applying the Magnetic field Chiral edge mode i y j x 23
Longitudinal thermal conductivity Cv κ xx /J.4.3.2.1..8.6.4 T L.4.3.2.1 T H L=12 ~ h/j=.12 ~ h/j=.24 ~ h/j=.36 ~ h/j=.48 h/j=. κ xx takes a peak around T H. κ xx ~ is insensitive to h. κ xx /T.2..4.3.2. 1-2 1-1 1 1 1 κ xx /T at low T limit vanishes due to the gap opening..1. 1-2 1-1 1 1 1 T/J 24
Transverse thermal conductivity Cv.4.3.2.1..2 T L T H L=12 ~ h/j=.12 ~ h/j=.24 ~ h/j=.36 ~ h/j=.48 h/j=. κ xy takes a peak around T H. κ xy /J.1 κ xy ~ increases with increasing h. contrasting behavior to κ xx..3 1-2 1-1 1 1 1 π/12 κ xy /T.2.1 Quantization at low T Deviation from π/12 around T L. 1-2 1-1 1 1 1 T/J Nonmonotonic behavior 25
Effect of flux excitation κ xy /J.2.1 T L MC random flux zero free flux T H ~ h/j=.48 W p = ±1 : Z 2 flux W p κ xy /T.3.25.2.15.1.5 π/12 1-2 1-1 1 1 1 T/J W p +1: ground state (free flux) W p = MC result deviates from zero-flux one around low-t crossover. High-T peak is well accounted for by random flux excitation. 1: flux excitation 26
Deviation from Plateau Magnetic field dependence of gaps Arrhenius plot for κ xy /T Flux gap Majorana gap Flux gap is almost independent of h. Majorana gap linearly depend on h. ~ ~ Deviation of κ xy /T from the quantized value is determined by the flux gap. 27
Magnetic field dependence κ xy /T κ xx /T.6.5.4.3.2.1..3.2.1. -.1 -.2 -.3 -.6 -.4 -.2..2.4.6 ~ h/j T/J=.675 T/J=.3 T/J=.99 T/J=1. Longitudinal component Almost zero at low T almost unchanged by h Transverse component ~ Enhancement by h Linear dependence for h ~ at finite T κ xy is proportional to h 3. h h3 2 ~ Contrasting magnetic-field dependence between κ xx and κ xy. 28
Contents Introduction Method Thermal transport w/o magnetic field Thermal transport w/ magnetic field Summary 29
Summary Kitaev model on a honeycomb lattice Quantum Monte Carlo simulation in Majorana representation Magnetic field is introduced as an effective model. Without magnetic field Longitudinal thermal conductivity exhibits a peak at high-t crossover attributed to the itinerant Majorana fermions Dynamical component detects flux fluctuation. With magnetic field Thermal Hall conductivity is proportional to h 3. Peculiar T dependence of κ xy /T due to the flux excitations. 3