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

Novel Quantum States in Condensed Matter 2017 (NQS2017) YITP, Kyoto University, 10 November, 2017 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 , 127204 (2017).

Contents Introduction Method Thermal transport w/o magnetic fj eld Thermal transport w/ magnetic fj eld Summary 2

Contents Introduction Method Thermal transport w/o magnetic fj eld Thermal transport w/ magnetic fj eld Summary 3

Quantum spin liquid (QSL) QSL Paramagnet T Crossover Quantum fm uctuation disturbs orderings. κ -(BEDT-TTF) 2 Cu 2 (CN) 3 M. Yamashita et al., Nat. Phys. 5 , 44 (2009). S. Yamashita et al., Nat. Phys. 4 , 459 (2008). Quantum spin liquid (QSL): Thermal conductivity Speci fj c heat No singularity in or C v χ No apparent symmetry breakings down to low T Fractional excitations Kagomé volborthite herbertsmithite D. Watanabe et al. , T.-H. Han et al., Nature 492 , 406 (2012). Thermal Hall conductivity Proc. Natl. Acad. Sci. 113 , 8653 (2016). Characterization of QSLs with emergent fermions Neutron scattering Low- T behavior of C v ( T-linear ) Dynamical response ( continuum ) Thermal transport 4

Kitaev model X X X S x i S x S y i S y S z i S z H = − J x j − J y j − J z j < i j > x < i j > y < i j > z A. Kitaev, Annals of Physics 321 , 2 (2006). S =1/2 spin W p Bond-dependent interactions frustration Z 2 fm ux (conserved quantity) W p on each plaquette ground state: quantum spin liquid (Only NN interactions are fj nite) H = � iJ γ X c i c j Assuming W p =+1 { c i } : Majorana fermions 4 emerging from spins h ij i γ J z Free Majorana fermions on a honeycomb lattice; analogous to graphene Dirac cones J y J x 5

Kitaev model X X X S x i S x S y i S y S z i S z H = − J x j − J y j − J z j < i j > x < i j > y < i j > z A. Kitaev, Annals of Physics 321 , 2 (2006). S =1/2 spin W p Bond-dependent interactions frustration Z 2 fm ux (conserved quantity) W p on each plaquette ground state: quantum spin liquid (Only NN interactions are fj nite) Fractional fermionic excitations Emergent fermions may carry heat. J z Majorana Chern insulator 
 by applying magnetic fj eld in gapless phase Gapless QSL Thermal Hall e fg ect J y J x 6

Kitaev spin liquid: fractionalization Itinerant Majorana fermions (IMF) Thermal fractionalization S i Localized Majorana fermions (LMF) JN, M. Udagawa, and Y. Motome, Phys. Rev. Lett. 113 , 197205 (2014). JN, M. Udagawa, and Y. Motome, Phys. Rev. B 92 , 115122 (2015). 0.3 L =8 L =10 0.2 C v L =12 0.1 L =20 T L T H 0 1 S / ln 2 0.5 0 T / J 10 -2 10 -1 10 0 10 1 7 S.-H. Do et al., Nat. Phys. (2017).

Realization of Kitaev QSLs Strong spin-orbit coupling t 2 g 5 j e fg =1/2 localized spin G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102 , 017205 (2009) Kitaev-Heisenberg model X X X X S x i S x S y i S y S z i S z H = − J x j − J z j − J y S i · S j + J H j < i j > x < i j > y < i j > z < ij > Magnetic order Ir 4+ A 2 IrO 3 ( A =Li,Na) Ir 4+ 5 d 5 T c ~10K Y. Singh and P. Gegenwart, Phys. Rev. B 82 , 064412 (2010). Y. Singh et. al., Phys. Rev. Lett. 108 , 127203 (2012). S z i S z R. Comin et. al., Phys. Rev. Lett. 109 , 266406 (2012). j S. K. Choi et. al., Phys. Rev. Lett. 108 , 127204 (2012). S y i S y j α -RuCl 3 Ru 3+ 4 d 5 T c ~10K S x i S x j K. W. Plumb et al., Phys. Rev. B. 90 , 041112 (2014). Kitaev term plays a dominant role. 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). Y. Yamaji et al., Phys. Rev. Lett. 113 , 107201 (2014). M. Majumder et al., Phys. Rev. B 91 , 180401(R) (2015). K. Foyevtsova et al., Phys. Rev. B 88 , 035107 (2013). 8 A. Banerjee et al., Nat. Mater. 15 , 733 (2016).

Dynamical response Raman scattering in RuCl 3 Inelastic neutron scattering in RuCl 3 L. J. Sandilands et al., Phys. Rev. Lett. 114 , 147201 (2015). A. Banerjee et al., Nat. Mater., Nat. Mater. 15 , 733 (2016). Raman scattering in β -, γ -Li 2 IrO 3 A. Glamazda et al., Nat. Commun. 7 , 12286 (2016). Theory J. Knolle et al., Phys. Rev. Lett. 113, 187201 (2014). 9

Comparison of experiment & theory Magnetic order occurs at ~10K in α -RuCl 3 . Good agreement between 
 the present theory and experimental results 0.3 L. J. Sandilands et al., Phys. Rev. Lett. 114 , 147201 (2015). Raman scattering 0.5 JN, J. Knolle, D. L. Kovrizhin, Y. Motome, R. Moessner, Nat. Phys., 12, 912 (2016). 0.4 Experiment Intensity (a.u.) ω f 0.3 Theory i Intensity (a.u.) ω f ω f ε 2 (1- f ) 2 0.2 n +1 n +1 0.2 0.1 [1- f ( ε 1 )][1- f ( ε 2 )] Bosonic background Bosonic background 0.0 Two-fermion excitation ω i ω i 0 50 100 150 200 250 300 T (K) ε 1 0.1 Fermionic T dependence appears around 100K. -J y 0 50 100 150 200 250 300 T (K) -J x Experiment J z S z S z J z spin correlation High-energy features are consistent. Dynamical S.-H. Do et al., Nat. Phys. (2017). J. Yoshitake, JN, and Y. Motome, Phys. Rev. Lett. 117 , 157203 (2016) Theory 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), 10

Thermal transport in α -RuCl 3 I. A. Leahy et al., Phys. Rev. Lett. 118 , 187203 (2017). κ is enhanced in low- T whereas it is suppressed in intermediate- T by applying magnetic fj eld. D. Hirobe, M. Sato, Y. Shiomi, H. Tanaka, and E. Saitoh, Phys. Rev. B 95 , 241112 (2017). Another study for κ in RuCl 3 Longitudinal thermal conductivity κ exhibits a peak R. Hentrich et al., arXiv:1703.08623 (2017). at a peak in speci fj c heat 11

Purpose Candidates of Kitaev materials Magnetic order at low T Cooperation e fg ect of the Kitaev and Heisenberg interactions Two stances: What is the Kitaev QSL? How should it be observed? Our starting point Precursor of Kitaev QSL ( fractionalization ) above T c (~10K) What occurs in the pure Kitaev limit at fj nite temperature? Fractionalization of spins into Majorana fermions Topological nature with magnetic fj eld Heat transport 12

Contents Introduction Method Thermal transport w/o magnetic fj eld Thermal transport w/ magnetic fj eld Summary 13

Jordan-Wigner transformation X X X S x i S x S y i S y S z i S z H = − J x j − J y j − J z j < i j > x < 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 i � 1 i a i − 1 Y i ) † = ( 1 � 2 n i 0 ) a † i = a † S z S + i = ( S � i 2 i 0 = 1 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). Fermions : a i , a † H.-D. Chen and Z. Nussinov, J. Phys. A Math. Theor. 41, 075001 (2008). i Introducing Majorana fermions c i c j � iJ y H = iJ x c i c j + J z c i = a i + a † X X X c i ¯ ¯ c j c i c j c i ¯ ¯ c i c j c i c j c j c i c j i 4 4 4 c i = ( a i − a † i ) / i ¯ h ij i x h ij i y h ij i z c j , H ] = 0 [ ¯ c i ¯ c i c j � iJ y H = iJ x c i c j � iJ z X X X : local conserved quantity η r ≡ i ¯ c i ¯ c j η r c i c j c i c j c i c j η r c i c j 4 4 4 h ij i x h ij i y h ij i z 14

Method Quantum spin model X X X S x i S x S y i S y S z i S z H = − J x j − J y j − J z j < i j > x < i j > y < i j > z Jordan-Wigner transformation Itinerant fermion model c i c j � iJ y H = iJ x c i c j � iJ z X X X η r c i c j c i c j c i c j η r c i c j 4 4 4 h ij i x h ij i y h ij i z η r = i ¯ c i ¯ c j : Itinerant Majorana c i S i ¯ : Localized Majorana c i Free Majorana fermion system with thermally fm uctuating fm uxes W p = η r η r 0 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 fm ipping Ising valuables η r . J x = J y = J z = J 15

Speci fj c heat and entropy 0.3 L =8 0.2 L =10 C v Double peak structure L =12 0.1 L =20 T L T H 0 1 Release of a half of entropy 0.5 at each crossover S (Entropy) 0 j = − i 0.15 S x i S x NN x bond 4 c i c j (NN correlation) 〈 S ix S jx 〉 0.1 : itinerant Majorana c i 0.05 (matter Majorana) 0 1 Local conserved quantity (local conserved quantity) Y 〈 W p 〉 η r = i ¯ c i ¯ c j W p = η r r ∈ p 0 ¯ c i : localized Majorana 10 -2 10 -1 10 0 10 1 ( fm ux Majorana) T / J c i : itinerant Majorana Entropy release at T ** development of spin correlation S i ¯ coherent develop of W p : localized Majorana Entropy release at T * c i 16

Contents Introduction Method Thermal transport w/o magnetic fj eld Thermal transport w/ magnetic fj eld Summary 17