Orbital Dynamics coupled with Jahn-Teller phonons in Strongly Correlated Electron System

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1 The 5 th Scienceweb GCOE International Symposium 1 Orbital Dynamics coupled with Jahn-Teller phonons in Strongly Correlated Electron System Department of Physics, Tohoku University Joji Nasu In collaboration with Sumio Ishihara

2 Strongly correlated electron system Internal degrees of freedom of electron Charge Spin Orbital Interplay Lattice Exotic phenomena High-Tc superconductivity Colossal magneto-resistance effect Multi-ferroics Quantum spin liquid etc. 2

3 3 Orbital Degree of Freedom Crystalline field with high symmetry (e.g. Perovskite Manganites) Orbital degeneracy Orbital degree of freedom 3d e g orbitals t 2g orbitals Anisotropy of electronic distribution Exchange interaction of electronic spins Anisotropy of electric conductance Coupling with lattice distortion

4 Orbital Excitation -Orbiton- Spin Degree of freedom Spin order up down Spin wave(magnon) e g orbital degree of freedom Orbital order Orbital wave (orbiton) up down Orbital Pseudo spin 4

5 Example for materials with orbital degree LaMnO 3 : End material of colossal magnetoresistance manganites KCuF 3 : the material with one-dimensional magnetism e g Mn 3+ e g Cu 2+ hole t 2g t 2g staggered orbital order Superexchange interaction 5

6 Observation of Orbiton E. Saitoh, et. al., Nature (2001) In LaMnO 3, Raman scattering K. Ishii, et.al., PRB83,241101(R) (2011) In KCuF 3, Resonant Inelastic X-ray Scattering The collective excitation from orbital order Orbiton has been observed. 6

7 Difference between Magnon and Orbiton Orbital degree of freedom Anisotropy of electronic distribution 1) Anisotropy of orbital interaction Orbiton dispersion is anisotropic. In LaMnO 3 S. Ishihara and S. Maekawa, PRB62, 2338(2000) 2) Strong coupling with lattice distortion Jahn-Teller effect In particular, Dynamical aspect of Jahn-Teller effect might affect orbital dynamics. 7

8 Jahn-Teller Effect Lattice potential Metal ion Ligand Q Energy loss Q 2 Electron-lattice coupling Energy gain Q Lattice must be distorted! Jahn-Teller effect = Coupling of orbital and lattice distortion Dynamics of lattice Kinetic energy 2 / Q 2 Orbital dynamics 8

9 Jahn-Teller Effect Jahn-Teller Theorem Any nonlinear molecule with a spatially degenerate electronic ground state will undergo a geometrical distortion that removes that degeneracy Orbital essentially couples with lattice Adiabatic potentials Vibronic Hamiltonian Kinetic energy of lattice Lattice potential Electron-lattice coupling Low-energy dynamics along rotational direction is expected. 9

10 Weak coupling approach Vibronic Hamiltonian J. van den Brink, PRL87, 19(2001) Neglected Orbiton-phonon hybridization Repulsion between orbiton and phonon modes Orbiton Phonon Energy of orbiton-phonon mode deceases. Assumed ground state is unstable. Hybridization picture is not correct in strong coupling case. 10

11 Purpose Orbital Dynamics coupled with Jahn-Teller phonons in Strongly Correlated Electron System On-site dynamical Jahn-Teller effect originating from Kinetic energy of lattice vibration Inter-site superexchange interaction Unique excitation due to Jahn-Teller effect 11

12 12 Model Exchange interaction Jahn-Teller coupling is local. e g orbital degree of freedom + 2 kinds of phonons :Orbital pseudo-spin operator

13 13 Symmetry of Hamiltonian Exchange interaction Symmetry for infinitesimal rotation If, the Hamiltonian is invariant under and Existence of Goldstone mode

14 Our approach Generalized spin wave approximation with mean-field approximation Hamiltonian On site part Exchange part: Mean-field approximation under Ferro orbital order :Off-site part :On-site part : Coordination number Off-site On-site 14

15 Solution of on-site Hamiltonian Off-site On-site Solve self-consistently Eigenstates in with Exact evaluation of Jahn-Teller coupling Exact diagonalization by using Householder method ~3000 dimensions ( ) 15

16 16 Inter-site Hamiltonian Eigenstates of is expanded by using basis of :Projection operator Introduction of N~3000 kinds of boson Approximation for orbital operator The states involving ground state are taken into account. Approximation by bosons Generalization of spin-wave approximation F. P. Onufrieva, Sov. Phys. JETP 89, 2270 (1985). N. Papanicolaou, Nucl. Phys. B 305, 367 (1988).

17 Diagonalization of Hamiltonian Bosonic Hamiltonian Diagonalization for system consisting of kinds of bosons J. Colpa, Physica A93,327 (1978) Generalized Bogoliubov transformation :Bogoliubov boson 17

18 18 Condition Ferro orbital order On one dimensional chain : There should be Goldstone mode We can easily generalize its dimension and assumed order. Dynamical susceptibility :Bogoliubov vacuum

19 Static properties In all parameter region, the ferro-orbital order is stabilized. Orbital order is the same regardless of JT coupling By introducing Jahn-Teller coupling, dynamical JT effect reduces orbital moment. (Ham s reduction effect) In strong coupling limit, dynamical JT effect is proportional to 1/g 2. Non-monotonic behavior 19

20 Weak coupling case Gapless excitation: consistent with Goldstone theorem White lines: Conventional spin-wave approximation where orbiton is hybridized with phonon J. van den Brink, PRL87, 19(2001) Gapped excitation Violation of Goldstone theorem Orbital excitation mode by our method agrees with that by conventional spin-wave approximation except for gapless feature. 20

21 Excited states Strong coupling case Local picture Adiabatic planes Frank-Condon excitations Gound state Phonon sidebands appear around JT energy The flat dispersion Local excitation 21

22 Dispersion of Phonon sidebands Phonon sidebands Excitations between adiabatic planes Large J Dispersive sidebands Propagation of local excitations between adiabatic planes Amplitude is determined by exchange interaction J. Extent is determined by JT coupling g. 22

23 Low-energy excitation Adiabatic plane Frank-Condon excitation Orbital excitation involving lattice vibration within the lowest adiabatic plane Frank-Condon excitation:orbital excitation with phonon sidebands Low-energy excitation:collective excitation involving lattice vibration 23

24 Low-energy effective model Effective model on the lowest adiabatic plane in strong coupling limit (Born-Oppenheimer approximation) Kinetic energy of rotational mode Born-Oppenheimer approx.: Lattice dynamics is strongly coupled with orbital degree of freedom Electronic Vibration wave-function wave-function 24

25 Low-energy effective model Original model Effective model Good agreement Low-energy mode corresponds to collective mode of orbital-lattice coupled excitation 25

26 Origin of low-energy excitation Adiabatic plane Adiabatic potential Degeneracy along rotational direction Frank-Condon excitation Orbital excitation involving lattice vibration Degeneracy along rotational direction Orbital excitation involving lattice vibration Characteristic of Exe Jahn-Teller system with two kinds of phonons 26

27 Degree of freedom of phonon 2 kinds of phonons, Exe JT: 1 kind of phonon, Exb 1 JT: Degree of freedom of phonon Low-energy excitation 27

28 28 Conclusion Orbital Dynamics coupled with Jahn-Teller phonons in Strongly Correlated Electron System There are two kinds of excitations Frank-Condon excitation with phonon sidebands Dispersive due to superexchange interaction Orbital excitation involving lattice vibration Orbital and lattice are strongly coupled. This excitation originates from degeneracy of adiabatic plane.