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SPIN STRUCTURE FACTOR OF THE FRUSTRATED QUANTUM MAGNET Cs 2 CuCl 4

SPIN STRUCTURE FACTOR OF THE FRUSTRATED QUANTUM MAGNET Cs 2 CuCl 4. Rastko Sknepnek. Department of Physics and Astronomy McMaster University. In collaboration with:. Denis Dalidovich John Berlinsky Junhua Zhang Catherine Kallin. 1/30. Duke University. March 9, 2006. Outline.

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SPIN STRUCTURE FACTOR OF THE FRUSTRATED QUANTUM MAGNET Cs 2 CuCl 4

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  1. SPIN STRUCTURE FACTOR OF THE FRUSTRATED QUANTUM MAGNET Cs2CuCl4 Rastko Sknepnek Department of Physics and Astronomy McMaster University In collaboration with: • Denis Dalidovich • John Berlinsky • Junhua Zhang • Catherine Kallin 1/30 Duke University March 9, 2006

  2. Outline • Motivation • Spin waves vs. spinons • Experiments on Cs2CuCl4 • Nonlinear spin wave theory for Cs2CuCl4 • Summary Duke University March 9, 2006 2/30

  3. Motivation Neutron scattering measurements on quantum magnet Cs2CuCl4. extended scattering continuum. Signature of deconfined, fractionalized spin-1/2 (spinon) excitations? (R. Coldea, et al., PRB 68, 134424 (2003)) Can this broad scattering continuum be explained within a conventional 1/S expansion? (Complementary work: M. Y. Veillette, et al., PRB (2005)) 3/30 Duke University March 9, 2006

  4. Spin Waves Heisenberg Hamiltonian: • J<0 – ferromagnetic ground state • J>0 – antiferromagnet (Néel ground state) • Spin waves are excitations of the (anti)-ferromagnetically ordered state. • Exciting a spin wave means creating a quasi-particle called magnon. • Magnons are S=1 bosons. Dispersion relations (k0): (ferromagnet) (antiferromagnet) Duke University March 9, 2006 4/30

  5. Ground state of an antiferromagnet Antiferromagnetic Heisenberg Hamiltonian: (J>0) State can not be the ground state - it is not an eigenstate of the Hamiltonian. • Antiparallel alignment gains energy only from the z-z part of the Hamiltonian. • True ground state - the spins fluctuate so the system gains energy from the spin-flip terms. Ground state of the Heisenberg antiferromagnet shows quantum fluctuations. How important is the quantum nature of the spin? quantum correction 1 ~ classicalenergy S Reduction of the staggered magnetization due to quantum fluctuations: 5/30 Duke University March 9, 2006

  6. Spin Liquid and Fractionalization in 1d (half-integer spin) • In D=1 quantum fluctuations destroy long range order. • Spin-spin correlation falls off as a power law. • Ground state is a singlet with total spin Stot=0 (exactly found using Bethe ansatz). Excitations are not spin-1 magnons but pairs of fractionalized spin-1/2 spinons. Main feature of a fractionalized state  broad scattering continua. Prototypical system KCuF3. (D.A.Tennant, et al., PRL (1993)) Fractionalization: Excitations have quantum numbers that are fractions of quantum numbers of the local degrees of freedom. 6/30 Duke University March 9, 2006

  7. Geometrical Frustration in 2d • Ising-like ground state is possible only on bipartite lattices. • Non-bipartite lattices (e.g., triangular) exhibit geometrical • frustration. • On an isotropic triangular lattice the ground state is a three sub-lattice Néel state. 7/30 Duke University March 9, 2006

  8. Resonating valence bond (RVB) (P.W. Anderson, Mater. Res. Bull. (1973)) • Ground state – linear superposition of disordered valence bond configurations. • Each bond is formed by a pair of spins in a singlet state. RVB state has the following properties: • spin rotation SU(2) symmetry is not broken. • spin-spin, dimer-dimer, etc. correlations are exponentially decaying – no LRO. • excitations are gapped spin-1/2 deconfined spinons. RVB state is an example of a two dimensional spin liquid. 8/30 Duke University March 9, 2006

  9. Is there any experimental realization of two dimensional spin liquid? 9/30 Duke University March 9, 2006

  10. Cs2CuCl4 - a spin-1/2 frustrated quantum magnet. Crystalline structure: • Orthorhombic (Pnma) structure. • Lattice parameters (at T=0.3K) • a = 9.65Å • b = 7.48Å • c = 12.26Å. • CuCl42- tetrahedra arranged in layers. • (bc plane) separated along a by Cs+ ions. Cs2CuCl4 is an insulator with each Cu2+ carrying a spin 1/2. Crystal field quenches the orbital angular momentum resulting in near-isotropic Heisenberg spin on each Cu2+. • Spins interact via antiferromagnetic superexchange • coupling. • Superexchange route is mediated by two nonmagnetic • Cl- ions. • Superexchange is mainly restricted to the bc planes 10/30 Duke University March 9, 2006

  11. Coupling constants J J’ Measurements in high magnetic field (12T): J’’ J = 0.374(5) meV J’ = 0.128(5) meV J’’= 0.017(2) meV High magnetic field experiment also observe small splitting into two magnon branches. Indication of a weak Dzyaloshinskii-Moriya (DM) interaction. DM interaction creates an easy plane anisotropy. D = 0.020(2) meV D Below TN=0.62K the interlayer coupling J’’ stabilizes long range order. The order is an incommensurate spin spiral in the (bc) plane. 17.7o e0=0.030(2) 11/30 Duke University March 9, 2006

  12. The Hamiltonian Relatively large ratio J’/J≈1/3 and considerable dispersion along both b and c directions indicate two dimensional nature of the system. Effective Hamiltonian: 12/30 Duke University March 9, 2006

  13. A few remarks... • A strong scattering continuum does not automatically • entail a spin liquid phase. • Magnon-magnon interaction can cause a broad scattering • continuum in a conventional magnetically ordered phase. In Cs2CuCl4 strong scattering continuum is expected because: • low (S=1/2) spin and the frustration lead to a small ordered moment and strong • quantum fluctuations • the magnon interaction in non-collinear spin structures induces coupling between • transverse and longitudinal spin fluctuations  additional damping of the spin waves. It is necessary to go beyond linear spin wave theory by taking into account magnon-magnon interactions within a framework of 1/S expansion. 13/30 Duke University March 9, 2006

  14. Spin wave theory (linear) Classical ground state is an incommensurate spin-spiral along strong-bond (b) direction with the ordering wave vector Q. In order to find ground state energy we introduce a local reference frame: Classical ground state energy: Ordering wave vector: • D = 0 • D = 0.02meV 14/30 Duke University March 9, 2006

  15. 1/S expansion To go beyond linear spin-wave theory we employ Holstein-Primakoff transformation: Where a’s are bosonic spin-wave creation and annihilation operators. The Hamiltonian for the interacting magnons becomes: 15/30 Duke University March 9, 2006

  16. Quadratic part of the Hamiltonian: Magnon-magnon interaction is described by: 16/30 Duke University March 9, 2006

  17. Ground state energy and ordering wave vector In a 1/S expansion quantum corrections of the ground state energy are: Quantum corrections of the ordering wave-vector are: Where etc. 17/30 Duke University March 9, 2006

  18. *Y. Tokiwa, et al., cond-mat/0601272 (2006) ** R. Coldea, et al., PRB 68, 134424 (2003) 18/30 Duke University March 9, 2006

  19. Green’s function To calculate physical observables we need Green’s function for magnons. S’s are the self-energies which we calculate to the order 1/S. 19/30 Duke University March 9, 2006

  20. Sublattice magnetization Staggered magnetization: To the lowest order in 1/S: 20/30 Duke University March 9, 2006

  21. The second order correction has two contributions: Cs2CuCl4 Numerical integration carried using DCUHRE method – Cuba 1.2 library, by T. Hahn) 21/30 Duke University March 9, 2006

  22. Energy spectrum The renormalized magnon energy spectrum is determined by poles of the Green’s function. Which leads to the nonlinear self-consistency equation: 22/30 Duke University March 9, 2006

  23. On renormalization of coupling constants. In order to quantify the “quantum” renormalization of the magnon dispersion relation one fits the 1/S result to a linear spin-wave dispersion with “effective” coupling constants. (R. Coldea, et al., PRB 68, 134424 (2003)) 23/30 Duke University March 9, 2006

  24. Spin structure factor Neutron scattering spectra is expressed in terms of Fourier-transformed real-time dynamical correlation function: Magnon-magnon interaction leads to the mixing of longitudinal (z) and transversal (x) modes (detailed derivation in T. Ohyama&H. Shiba, J. Phys. Soc. Jpn. (1993)) 24/30 Duke University March 9, 2006

  25. G scan Scan along a path at the edge of the Brillouin zone. D = 0.02meV kx = p ky = 2p(1.53-0.32w-0.1w2) linear SW theory wk=0.22meV Energy resolution DE=0.016meV linear SW theory wk+/-Q= 0.28meV Momentum resolution Dk/2p = 0.085 (R. Coldea, et al., PRB 68, 134424 (2003)) two-magnon continuum 25/30 Duke University March 9, 2006

  26. Energy resolution DE=0.016meV Energy resolution DE=0.002meV Momentum resolution Dk/2p = 0.085 Momentum resolution Dk/2p = 0 Near G point the dispersion relation has large modulation along b direction. Significant broadening due to finite momentum resolution. 26/30 Duke University March 9, 2006

  27. What happens if we lower D? Energy resolution DE=0.016meV Momentum resolution Dk/2p = 0.085 G scan theory J = 0.374meV J’ = 0.128meV D = 0.01meV experiment 27/30 Duke University March 9, 2006

  28. Energy resolution DE=0.016meV Momentum resolution Dk/2p = 0.085 D=0.02meV experimental position of the peak w = 0.10(1) meV Smaller value for D fits experiments better! 28/30 Duke University March 9, 2006

  29. Summary and conclusions We have... • derived non-linear spin wave theory for the frustrated triangular magnet Cs2CuCl4. • calculated quantum corrections to the ground state energy and sublattice magnetization • to the 2nd order in 1/S. • calculated spin structure factor and compared it to the recent inelastic neutron scattering • data We find that 1/S theory: • gives good prediction for the ground state energy and ordering wave vector. • significantly underestimates renormalization of the coupling constants. • significant scattering weight is shifted toward higher energies, but not sufficient to • fully explain experiments. 28/30 Duke University March 9, 2006

  30. Other approaches • 1d coupled chains M. Bocquet, et al., PRB (2001) O. Starykh, L. Balents, (unpublished) (2006) • Algebraic vortex liquid J. Alicea, et al., PRL (2005) J. Alicea, et al., PRB (2005) • High-T expansion W.Zheng, et al., PRB (2005) • Proximity of a spin liquid quantum critical point S.V. Isakov, et al., PRB (2005) 29/30 Duke University March 9, 2006

  31. Thank You! 30/30 Duke University March 9, 2006

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