Quantum order-by-disorder in Kitaev model on a triangular lattice

Transcription

1 Quantum order-by-disorder in Kitaev model on a triangular lattice George Jackeli Max-Planck Institute & University of Stuttgart, Germany Andronikashvili Institute of Physics, Tbilisi, Georgia GJ & Avella, arxiv 15 May 23,

2 Transition metal oxides: Plethora of Challenging Phenomena Metall to Insulator transitions Spin Orbital Lattice Charge High-Tc Superconductivity Colossal Magnetoresistance Unexpected variety of phases and transitions between them 2

3 Relativistic in origin, Spin-orbit coupling Coulomb force Orbital Spin Charge Lattice Enhance interplay by going to heavy TM elements 3

4 correlations spin-orbit 4

5 Kramers doublet of Ir 4+ A. Abragam and B. Bleaney, EPR of Transition Ions Kramers pair! S=1/2 10 t 2g 5d 5 (10-3 emu/mol) Sr 2 IrO 4 B y~ A y z x~ φ A 3λ/2~0.6 x y~ ev x~ B (K)

6 between: Iridates Zoo of Iridate compounds In betwee observed large intensity modulation along the l direction strong SOC typically results in anisotropicreflects magnetic cou- magnetic structure factor. The magnetic the bilayer plings that deviate from the pure Heisenberg-like spin at each l, and the corresponding intenpeaks were refined interaction in the weak SOC limit. A compelling outcome sities obtained from integrating rocking curves are plotted Fig. The intensity modulation has a periodicity set is that a novel Heisenberg antiferromagnet in can be2(b). realized by the ratio between the lattice parameter c and the bilayer (b) Sr2IrO4 Sr2IrO4 Sr3Ir2O7 O SrIrO3 A2BIrO6 rgy scales comparable U ~ W ~ λ Ir Introduction Resistivity (polycrystalline samples) ro4 Sr3Ir2O7 107 Na2IrO3 Ln2Ir2O7 106 Jeff=1/2 moments Heisenberg-like Ho Dy 4 Ising-like Semi-Metal, Topological? 5! (m" cm) 3 4 Ho Tb! 2012 American Gd 103 Eu netic structure. The up and down magnetic moments correlate with counterclockwise and clockwise rotations of the IrO6 octahedra,society respectively. Physical A2IrO3 Sm 102 Nd 101 R2Ir2O T(K) 3d-5d Interplay Eu A2Ir2O7 102 Nd Pr R2Ir2O Metal Insu (Ln=Nd, Sm, E pyrochlo Metal Insulator Transition (Ln=Nd, Sm, Eu, Gd, Tb, Dy, Ho) pyrochlores Kitaev-like Sr Ln O! 300 K. Matsuhira et al. : J. Phys. Soc. Jpn. 76 (2007) (Ln=Nd, Sm, Eu) 1 B IrO6 Na4Ir3AO Sr2IrO4 FIG. 2 (color online). (a) l scan measured in!-" polarization channel showing peaks. (b)electrons Integrated intenir4+magnetic : 5d5 Bragg Conduction sities at each peak obtained from rocking curves (red dots). Red Ir[t2g]+O[2p] conduction solid (green dashed) line is bilayer structural factor band expected for Itinerant electron antiferromagnetic (ferromagnetic) alignmentsystem of two adjacent 2 2!d expressed by cos (sin2 2!d IrO2 planes in a bilayer on the pyrochlore lattice c c ). (c) Temperature dependence of (0 1 19) peak. A4Ir O8 Ln2Ir2O7 Pr Resistivity (pol pyrochlore oxides Ln3+: (4f)n Localized moment Magnetic frustration FIG. 1 (color online). (a) Because of a10 staggered in-layer FIG. 1 (color online). (a) Crystal structure of Sr3 Ir2 O7 as Tb srotation of oxygen 10octahedra, Sr2 IrO4 has fourreported in Ref.in[17]. Every neighboring IrO6 octahedra are IrO layers 2 es A2IrO3 (A=Na, Li) 10 rotated unit in opposite sense about the c axis by 12$. the unit cell [9], which coincides with the magnetic cell. nd 60 (b) Magnetic 300order has a c-axis collinear G-type antiferromag10 (b) Jeff ¼ 1=2 moments lie and are canteddyin the IrO2 plane [8]. 3-1 All energy scales c Sr! (m" cm) (a) QSL? K. Matsuhira et al. : J. Phy (Ln=Nd T-induced MIT, Quantum critical metal 6

7 Trianglar lattice Ba 3 IrTi2O 9 (a) crystal structure (b) single Iridium layer (c) Ir-O-O-Ir exchange path ẑ ˆx ŷ ẑ ŷ z y x ẑ ŷ ˆx view along (111) ˆx view along (311) Ir Ti O Ba from Becker et al PRB 15

8 Frustration from anisotropy x x KxS S KyS y S y KzS z S z Impossible to satisfy simultaneously every pairwise interactions Infinitely many classical ground states

9 and ground-state energies G. Khaliullin, PTPS 05 CMC simulations on Classical model: Rousochatzakis et al, arxiv 14 DMRG and ED on Quantum model: Becker et al, PRB 15 dual 120 order dual Z 2 vortex crystal nematic phase Fig. 4 have been determined by nergy for clusters with N = 6 dic boundary conditions FM as well Z 2 vortex crystal 120 order Z 6 FM 4-leg ladder dual Z 6 FM dual O(3) FM

10 Model on Triangular Lattice: Symmetry (S x,s y,s z ) ->(-S x,s y,-s z ) x x KxS S KyS y S y KzS z S z Kz -> - Kz We can thus focus on the case all couplings being FM

11 Classical Ground State Manifold Kx=Ky=Kz>0 x x KxS S KyS y S y KzS z S z In FM state Classical energy E=-(M x M x +M y M y +M z M z )=-M 2 Global moment M can be freely rotated: accidental symmetry

12 Classical Ground State Manifold Kx=Ky=Kz>0 x x KxS S KyS y S y KzS z S z Coupling between NN chains E12=-(M1 x M2 x +M1 y M2 y )

13 Classical Ground State Manifold Kx=Ky=Kz>0 x x KxS S KyS y S y KzS z S z Coupling between NN chains E12=-(M1 x M2 x +M1 y M2 y ) M z of each chain can be individually flipped

14 Classical Ground State Manifold Accidental degeneracies - not related to symmetry: can be lifted by fluctuations 1 2 ~! ~! 2 For magnets we need to calculate SW spectra for each Classical state and compare X ~!n (k) 1! 2! Not always possible!

15 Classical Ground State Manifold Accidental degeneracies - not related to symmetry: can be lifted by fluctuations 1 2 ~! ~! 2 Linked cluster expansion: calculate corrections from short wave-length fluctuations

16 Selection of quantum easy axes z y x

17 Selection of quantum easy axes z y x E (2) (m) = X T 2 ' X m 4!

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19 Linked cluster expansion: calculate corrections from short wave-length fluctuations H = D(S z 1S z 3)S z 2S z 4 Gives the coupling between NNN chains, forming two sub-lattices decoupled from each other

20 from Becker et al PRB 15

21 Symmetry protected degeneracy A B A x x KxS S KyS y S y D C D KzS z S z B C A D B C Canonical transformation: A: (x,y,z) B: (-x,-y,z) C:(x,-y,-z) D:(-x,y,-z) Hamiltonian remains unchanged, but z-comp. every 2nd chain gets flipped. No correlations of z-comp between NN chains