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

2. 2/30 Outline Motivation Motivation Spin waves vs. spinons Spin waves vs. spinons Experiments on Cs 2 CuCl 4 Experiments on Cs 2 CuCl 4 Nonlinear spin wave theory for Cs 2 CuCl 4 Nonlinear spin wave theory for Cs 2 CuCl 4 Summary Summary March 9, 2006Duke University

3. 3/30 Motivation (R. Coldea, et al., PRB 68, 134424 (2003)) Neutron scattering measurements on quantum magnet Cs 2 CuCl 4. extended scattering continuum. Signature of deconfined, fractionalized spin-1/2 (spinon) excitations? Can this broad scattering continuum be explained within a conventional 1/S expansion? (Complementary work: M. Y. Veillette, et al., PRB (2005)) March 9, 2006Duke University

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

5. 5/30 Ground state of an antiferromagnet Antiferromagnetic Heisenberg Hamiltonian: (J>0) State 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. 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. 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 classical energy ~ 1 S Reduction of the staggered magnetization due to quantum fluctuations: March 9, 2006Duke University

6. 6/30 Spin Liquid and Fractionalization in 1d In D=1 quantum fluctuations destroy long range order. In D=1 quantum fluctuations destroy long range order. Spin-spin correlation falls off as a power law. Spin-spin correlation falls off as a power law. Ground state is a singlet with total spin S tot =0 (exactly found using Bethe ansatz). Ground state is a singlet with total spin S tot =0 (exactly found using Bethe ansatz). Excitations are not spin-1 magnons but pairs of fractionalized spin-1/2 spinons. (D.A.Tennant, et al., PRL (1993)) Prototypical system KCuF 3. (half-integer spin) Fractionalization: Excitations have quantum numbers that are fractions of quantum numbers of the local degrees of freedom. Main feature of a fractionalized state  broad scattering continua. March 9, 2006Duke University

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

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

9. Spin crystal Spin Liquid Ground state Semiclassical Néel order Quantum Liquid Order parameter Staggered magnetization No local order parameter Excitations Gapless magnons Gapped deconfined spinons Is there any experimental realization of two dimensional spin liquid? 9/30 March 9, 2006Duke University

10. Cs 2 CuCl 4 - a spin-1/2 frustrated quantum magnet. Crystalline structure: Orthorhombic (Pnma) structure. Orthorhombic (Pnma) structure. Lattice parameters (at T=0.3K) Lattice parameters (at T=0.3K) a = 9.65Å a = 9.65Å b = 7.48Å b = 7.48Å c = 12.26Å. c = 12.26Å. CuCl 4 2- tetrahedra arranged in layers. CuCl 4 2- tetrahedra arranged in layers. (bc plane) separated along a by Cs + ions. (bc plane) separated along a by Cs + ions. 10/30 Cs 2 CuCl 4 is an insulator with each Cu 2+ carrying a spin 1/2. Crystal field quenches the orbital angular momentum resulting in near-isotropic Heisenberg spin on each Cu 2+. Spins interact via antiferromagnetic superexchange Spins interact via antiferromagnetic superexchangecoupling. Superexchange route is mediated by two nonmagnetic Superexchange route is mediated by two nonmagnetic Cl - ions. Superexchange is mainly restricted to the bc planes Superexchange is mainly restricted to the bc planes March 9, 2006Duke University

11. 11/30 Coupling constants Measurements in high magnetic field (12T): J = 0.374(5) meV J’ = 0.128(5) meV J’’= 0.017(2) meV J J’ J’’ High magnetic field experiment also observe small splitting into two magnon branches. 17.7 o D Indication of a weak Dzyaloshinskii-Moriya (DM) interaction. D = 0.020(2) meV DM interaction creates an easy plane anisotropy. Below T N =0.62K the interlayer coupling J’’ stabilizes long range order. The order is an incommensurate spin spiral in the (bc) plane.  0 =0.030(2) March 9, 2006Duke University

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 March 9, 2006Duke University

13. 13/30 A few remarks... A strong scattering continuum does not automatically A strong scattering continuum does not automatically entail a spin liquid phase. entail a spin liquid phase. Magnon-magnon interaction can cause a broad scattering Magnon-magnon interaction can cause a broad scattering continuum in a conventional magnetically ordered phase. In Cs 2 CuCl 4 strong scattering continuum is expected because: low (S=1/2) spin and the frustration lead to a small ordered moment and strong 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 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. March 9, 2006Duke University

14. Spin wave theory 14/30 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: (linear) D = 0 D = 0 D = 0.02meV D = 0.02meV March 9, 2006Duke University

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 March 9, 2006Duke University

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

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

18. 18/30 1/S theory D = 0 meV D = 0.02 meV S 2 E G (0) /J-0.265-0.291 SE G (1) /J-0.157-0.138 E G (2) /J-0.0332-0.0256 E min /J-0.459-0.454 Experiments (Cs 2 CuCl 4 ) E G /J--0.5* *Y. Tokiwa, et al., cond-mat/0601272 (2006) 1/S theory D = 0 meV D = 0.02 meV Q 0 /2 0.55470.5533  Q (1) /2 -0.0324-0.0228  Q (2) /2 -0.011- Q/2 0.51130.5308 Experiments (Cs 2 CuCl 4 ) Q/2  -0.530(2)** ** R. Coldea, et al., PRB 68, 134424 (2003) March 9, 2006Duke University

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

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

21. 21/30 The second order correction has two contributions: Cs 2 CuCl 4 Numerical integration carried using DCUHRE method – Cuba 1.2 library, by T. Hahn) March 9, 2006Duke University

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

23. 23/30 On renormalization of coupling constants. R. Coldea, et al., PRB 68, 134424 (2003)) (R. Coldea, et al., PRB 68, 134424 (2003)) 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. 1/S theory Exp. J ren /J bare1.1311.63(5) J’ ren /J’ bare0.6480.84(9) D ren /D bare0.72- March 9, 2006Duke University

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 (  ) and transversal (  ) modes (detailed derivation in T. Ohyama&H. Shiba, J. Phys. Soc. Jpn. (1993)) 24/30 March 9, 2006Duke University

25. 25/30 R. Coldea, et al., PRB 68, 134424 (2003)) (R. Coldea, et al., PRB 68, 134424 (2003)) G scan Scan along a path at the edge of the Brillouin zone. k x =  k y = 2  (1.53-0.32  -0.1  2 ) Energy resolution  E=0.016meV Momentum resolution  k/2  = 0.085 D = 0.02meV linear SW theory  k  =0.22meV linear SW theory  k+/-Q  = 0.28meV two-magnon continuum March 9, 2006Duke University

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

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

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

29. 28/30 Summary and conclusions derived non-linear spin wave theory for the frustrated triangular magnet Cs 2 CuCl 4. derived non-linear spin wave theory for the frustrated triangular magnet Cs 2 CuCl 4. We have... calculated quantum corrections to the ground state energy and sublattice magnetization calculated quantum corrections to the ground state energy and sublattice magnetization to the 2 nd order in 1/S. calculated spin structure factor and compared it to the recent inelastic neutron scattering calculated spin structure factor and compared it to the recent inelastic neutron scatteringdata We find that 1/S theory: gives good prediction for the ground state energy and ordering wave vector. gives good prediction for the ground state energy and ordering wave vector. significantly underestimates renormalization of the coupling constants. significantly underestimates renormalization of the coupling constants. significant scattering weight is shifted toward higher energies, but not sufficient to significant scattering weight is shifted toward higher energies, but not sufficient to fully explain experiments. March 9, 2006Duke University

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

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