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SPIN STRUCTURE FACTOR OF THE FRUSTRATED QUANTUM MAGNET Cs2CuCl4 Rastko Sknepnek Department of Physics and Astronomy McMaster University In collaboration with: Denis Dalidovich A. John Berlinsky Junhua Zhang Catherine Kallin Iowa State University March 2, 2006 1/30 Outline • Motivation • Spin waves vs. spinons • Experiment on Cs2CuCl4 • Nonlinear spin wave theory for Cs2CuCl4 • Summary Iowa State University March 2, 2006 2/30 Motivation Neutron scattering measurements on quantum magnet Cs2CuCl4. dynamical correlations are dominated by an 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)) Iowa State University March 2, 2006 3/30 Spin Waves Heisenberg Hamiltonian: ˆ ˆ ˆ H J Si S j i, j • 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 (k0): k S | J | (a | k |)2 k JzSa | k | Iowa State University March 2, 2006 (ferromagnet) (antiferromagnet) 4/30 Ground state of an antiferromagnet Antiferromagnetic Heisenberg Hamiltonian: J Hˆ J Sˆiz Sˆ jz Sˆi Sˆ j Sˆi Sˆ j 2 i, j i, j (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 classical energy S Reduction of the staggered magnetization due to quantum fluctuations: Iowa State University March 2, 2006 5/30 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. • Spinons appear in pairs. • Excitation spectrum is a continuum with two soft points (0 and p) Fractionalization: Excitations have quantum numbers that are fractions of quantum numbers of the local degrees of freedom. Iowa State University March 2, 2006 Prototypical system KCuF3. (D.A.Tennant, et al, PRL (1993)) 6/30 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. Iowa State University March 2, 2006 7/30 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. Iowa State University March 2, 2006 8/30 Spin crystal Spin Liquid Ground state Semiclassical Néel order Quantum Liquid Order parameter Staggered magnetization No local order parameter Is there any experimental realization of two dimensional spin liquid? Excitations Gapless magnons Gapped deconfined spinons Iowa State University March 2, 2006 9/30 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 Iowa State University March 2, 2006 10/30 Coupling constants J Measurements in high magnetic field (12T): J’ 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 Below TN=0.62K the interlayer coupling J’’ stabilizes long range order. The order is an incommensurate spin spiral in the (bc) plane. D 17.7o Iowa State University March 2, 2006 1 Q 2p ( 0 )eb 2 0=0.030(2) 11/30 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: H JS R S R J ' S R ( S R S R ) (1) n D S R ( S R S R ) 1 2 1 2 1 2 R Iowa State University March 2, 2006 12/30 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. Iowa State University March 2, 2006 13/30 Spin wave theory (linear) Classical ground state is an incommensurate spin-spiral along strong-bond (b) direction with the ordering wave vector Q. 1 Q 2p ( )eb 2 In order to find ground state energy we introduce a local reference frame: SRx SR cos(Q R) SR sin(Q R) SRz SR sin(Q R) SR cos(Q R) SRy SR Classical ground state energy: J J Q iDQ 3k y kx J k J cos k x 2 J 'cos cos 2 2 3k y kx Dk 2iD sin cos 2 2 S 2 EG(0) (Q ) S 2 J QT Iowa State University T Q Ordering wave vector: 1 J' arcsin 0.0547 •D=0 p 2J • D = 0.02meV March 2, 2006 0.0533 14/30 1/S expansion To go beyond linear spin-wave theory we employ Holstein-Primakoff transformation: 1 † 1 † aR aR aR 2S 1 aR aR aR 2S 4S 1 † 1 † 1 aR aR 2SaR† 1 aR aR 2S 4S SR SR iSR 2S 1 SR SR iSR 2SaR† SR S aR† aR Where a’s are bosonic spin-wave creation and annihilation operators. [aR , aR† ' ] R, R ' [aR , aR ' ] 0 [aR† ' , aR† ' ] 0 The Hamiltonian for the interacting magnons becomes: H S E (Q) (H 2 Iowa State University (0) G (2) March 2, 2006 H (3) H ) (4) 15/30 Quadratic part of the Hamiltonian: H Ak (2) Bk † † † 2S Ak ak ak ak a k ak a k 2 k J 1 T T k JT J J Q Q k 4 Qk 2 Bk Jk 1 J QT k J QT k 2 4 k Ak2 Bk2 Magnon-magnon interaction is described by: H (3) H (4) i S f k1 , k2 ak† ak ak ak† ak† ak k k ,k 1 2 3 1 2 3 1 2 3 2 2 N k1 ,k2 ,k3 f1 k1 , k2 , k3 , k4 a† a† a a k k k k k k , k k 1 1 2 3 4 1 2 3 4 4 N k1 ,k2 ,k3 ,k4 f k , k , k a† a† a† a a † a a a k1 k2 k3 k4 k1 k2 k3 ,k4 2 1 2 3 k1 k2 k3 k4 Iowa State University March 2, 2006 16/30 Ground state energy and ordering wave vector In a 1/S expansion quantum corrections of the ground state energy are: EG (Q) S 2 J QT SEG(1) (Q ) EG(2) (Q2 ) 0 Quantum corrections of the ordering wave-vector are: Q (1) Q (2) Q Q0 2 2S 2S Where EG(1) (Q) J QT Q (1) 2 J QT 2 Q 1 N k k 1 1 N A k k Bk J QT k k Q Q0 etc. Iowa State University March 2, 2006 17/30 1/S theory D = 0 meV D = 0.02 meV S2EG(0)/J -0.265 -0.291 SEG(1)/J -0.157 -0.138 EG(2)/J -0.0332 -0.0256 Emin/J -0.459 -0.454 - -0.5* 1/S theory D = 0 meV D = 0.02 meV Q0/2p 0.5547 0.5533 Q(1)/2p -0.0324 -0.0228 Q(2)/2p -0.011 - Q/2p 0.5113 0.5308 - 0.530(2)** Experiments (Cs2CuCl4) EG/J *Y. Tokiwa, et al., cond-mat/0601272 (2006) Experiments (Cs2CuCl4) Q/2p ** R. Coldea, et al., PRB 68, 134424 (2003) Iowa State University March 2, 2006 18/30 Green’s function To calculate physical observables we need Green’s function for magnons. iGˆ (k , ) dte it ak (t ) † ˆ ak (0) ak (t ) T † a k (t ) ˆ (k , ) Gˆ 1 (k , ) Gˆ (0)1 (k , ) 2SBk 2SAk i (0) 1 ˆ G (k , ) 2 SB 2 SA i k k (4) (3) (4) (3) ( k ) ( k , ) ( k ) ( k , ) 11 11 12 12 (4) (3) ˆ (k , ) ˆ (k ) ˆ (k , ) (4) (3) (4) (3) (k ) (k , ) (k ) ( k , ) 21 22 22 21 ’s are the self-energies which we calculate to the order 1/S. Iowa State University March 2, 2006 19/30 Sublattice magnetization Staggered magnetization: 1 d M S G11 k , ei 0 N k 2p i To the lowest order in 1/S: M (1) 1 2N Iowa State University Ak 1 k k March 2, 2006 20/30 The second order correction has two contributions: M (2) I Q(1) 2S M II(2) 1 N d i 0 ˆ (0) 2p ie G k , ˆ k , Gˆ (0) k , k 11 Cs2CuCl4 Numerical integration carried using DCUHRE method – Cuba 1.2 library, by T. Hahn) Iowa State University March 2, 2006 21/30 Energy spectrum The renormalized magnon energy spectrum is determined by poles of the Green’s function. Re det Gˆ (0)1 k , k ˆ k , k 0 Which leads to the nonlinear self-consistency equation: k f Ak , Bk , k , k Iowa State University March 2, 2006 22/30 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. 1/S theory Exp. Jren/Jbare 1.131 1.63(5) J’ren/J’bare 0.648 0.84(9) Dren/Dbare 0.72 - (R. Coldea, et al., PRB 68, 134424 (2003)) Iowa State University March 2, 2006 23/30 Spin structure factor Neutron scattering spectra is expressed in terms of Fourier-transformed real-time dynamical correlation function: S ik 1 k , 2p i k i (t k R ) dt S (0) S ( t ) e O R R Magnon-magnon interaction leads to the mixing of longitudinal () and transversal () modes (detailed derivation in T. Ohyama&H. Shiba, J. Phys. Soc. Jpn. (1993)) S tot k , px S xx k , p y S yy k , S yy k , S k , Iowa State University S xx k , March 2, 2006 1 S k , S k , S k , 4 24/30 G scan Scan along a path at the edge of the Brillouin zone. kx = p ky = 2p(1.53-0.32-0.12) linear SW theory k =0.22meV Energy resolution E=0.016meV linear SW theory k+/-Q = 0.28meV Momentum resolution k/2p = 0.085 (R. Coldea, et al., PRB 68, 134424 (2003)) Iowa State University D = 0.02meV March 2, 2006 two-magnon continuum 25/30 Energy resolution E=0.016meV Energy resolution E=0.002meV Momentum resolution k/2p = 0.085 Momentum resolution k/2p = 0 Near G point the dispersion relation has large modulation along b direction. Significant broadening due to finite momentum resolution. Iowa State University March 2, 2006 26/30 What happens if we lower D? Energy resolution E=0.016meV D = 0.01meV Momentum resolution k/2p = 0.085 exp. G scan Iowa State University March 2, 2006 27/30 Energy resolution E=0.016meV Momentum resolution k/2p = 0.085 D=0.02meV experimental position of the peak = 0.10(1) meV Smaller value for D fits experiments better! Iowa State University March 2, 2006 28/30 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. Iowa State University March 2, 2006 28/30 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) Iowa State University March 2, 2006 29/30 Thank You! Iowa State University March 2, 2006 30/30