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Ground states and excitations of spatially anisotropic quantum antiferromagnets Oleg Starykh, University of Utah Leon Balents, KITP Masanori Kohno, NIMS, Tsukuba Jason Alicea, Caltech/UC Irvine Andrey Chubukov, U Wisconsin Ground states and excitations of spatially anisotropic triangular lattice antiferromagnets Oleg Starykh, University of Utah Leon Balents, KITP Masanori Kohno, NIMS, Tsukuba Jason Alicea, Caltech/UC Irvine Andrey Chubukov, U Wisconsin Main idea: exploit spatial anisotropy J’/J << 1 • Approach from the limit of decoupled chains (frustration helps -- it greatly enhances spatial anisotropy) • Allow for ALL symmetry-allowed inter-chain interactions to develop • Most relevant perturbations of decoupled chains drive spin order • High energy excitations: not very sensitive to details of the ground state J’/J Decoupled chains J’=0 critical (infinitely degenerate) state Coupled chains J’/J = 0.34 quantum collinear order (OAS and Balents, PRL 98 077205 (2007)) How small should J’/J be ? • Frustration greatly enhances region of “small” interchain J’ • Example: spatially anisotropic triangular AFM • Numerics: no interchain correlations for J’/J < 0.6 - 0.7 Weng et al 2006, Hayashi Ogata 2007, Heidarian Sorella Becca 2009 interchain spin correlations, J’=0.6 J; exponential decay Weng et al 2006 Collinear AFM state, generated by quantum fluctuations, coupling between NN chains (J’/J)4 (J’)4/J3 Pardini Singh 2008 - no; Bishop et al 2008 - yes Starykh, Balents 2007 Cs2CuCl4: structure and parameters - + moderate frustration f = 6 J = 0.374 meV J’/J = 0.34 D/J = 0.05 J”/J= 0.05 transverse to chain Along the chain J’ J k Very unusual response: broad and strong continuum! Numerous 2D theories • Arguments for 2D: J’/J = 0.3 not very small Significant transverse dispersion ★ Exotic theories: ★ Series expansion: • W. Zheng, J. Fjaerestad, R. R. P. Singh, R. H. McKenzie, R. Coldea 2006. • S. Yunoki and S. Sorella, 2006. • M. Q. Weng, D. N. Sheng, Z. Y. Weng, R. J. Bursill, 2006 ★ ★ d=1: Bocquet, Essler, Tsvelik, Gogolin, PRB 64, 094425 (2001) Spin waves: Two-spinon continuum Spinon energy S=1 excitation Energy ε Upper boundary Variables: kx1 and kx2 or ε and Qx Q Lower boundary x Low-energy sector QAFM=π Dynamic structure factor of copper pyrazine dinitirate (CuPzN) Stone et al, PRL 91, 037205 (2003) No single particle peaks! Effective Schrödinger equation in two-spinon basis • Study two spinon subspace (two spinons on chain y with Sz tot =+1) – Momentum conservation: 1d Schrödinger equation in ε space (k = (kx, ky)) • Crucial matrix elements known exactly Bougourzi et al, 1996 Structure Factor • Spectral Representation J.S. Caux et al, 2006 Weight in 1d: 73% in 2 spinon states 99% in 2+4 spinons – Can obtain closed-form “RPA-like” expression for 2d S(k,ω) in 2-spinon approximation 1 e ! Θ[ωkU − ω]Θ[ωkL − ω] 2 2 2π ωU (k) − ω −I(ρ(k,ω)) S1d (k, ω) = I(ρ) = ! 0 t ∞ dt e cosh(2t) cos(4ρt) − 1 t cosh t sinh(2t) ρ(k, ω) = 4 acosh π ! 2 (k) − ω 2 (k) ωU L 2 (k) ω 2 − ωL Types of behavior • Behavior depends upon spinon interaction J’(kx,ky) Bound “triplon” Identical to 1D J’(k) = 4J1’cos2[kx/2] cos[ky] J’2 = J’3 = J’1/2 Upward shift of spectral weight. Broad resonance in continuum or antibound state (small k) Details of triplon dispersion • Energy separation from the continuum δE ~ [J’(k)]2 • Spectral weight in the triplon pole Z ~ |J’(k)| bound k = (π/4,π) • Anti-bound triplon when J’(k) > J’critical(k) and • Expect at small k~0 where continuum is narrow. • Always of finite width due to 4-spinon contributions. Cs2CuCl4 -- Broad lineshape: “free spinons” • “Power law” fits well to free spinon result – Fit determines normalization J’(k)=0 here ! scan G triangular geometry: kx’ = kx , ky’ = kx + 2ky Triplon: S=1 bound state of two spinons • Effect of finite experimental resolution: ‣ Dashed curve: theoretical line shape scan E scan F Curves: 2-spinon RPA with experimental resolution Curves: 4-spinon RPA with experimental resolution Transverse dispersion J’(k)=0 Bound state and resonance Solid symbols: experiment Note that peak (blue diamonds) coincides with bottom edge only for J’(k)<0 Kohno, Starykh, Balents, Nature Physics 2007 Spectral asymmetry • Comparison: Vertical green lines: J’(k)=4J’ cos[k’x/2] cos[k’y/2] = 0. “Spinons unchained” Summary (I) ❖ Mystery of Cs2CuCl4 is (almost) solved: • not a 2d spin liquid • Dynamic response: high-energy elementary excitations are descendants of 1d spinons - (kinematically) bind in S=1 pairs (triplons) - significant variation of spectral weight with twodimensional momentum (kx,ky) ★ Geometric frustration preserves 1d features, promotes multi-particle excitations Low-energy properties • Ordered ground state – understood as an instability of weakly coupled chains – need to follow renormalization of all (bare and dynamically generated) symmetry-permitted interactions • Examples: – dimerized phases in quasi-1d J1-J2 and checkerboard models – non-coplanar in quasi-1d kagome – collinear AF (CoAF) in triangular antiferromagnet • Here: ground states of Cs2CuCl4 and Cs2CuBr4 in magnetic field Heisenberg spin chain via free Dirac fermions • Spin-1/2 AFM chain = half-filled (1 electron per site, kF=π/2a ) fermion chain • Spin-charge separation q=0 fluctuations: right- and left- spin currents 2kF (= π/a) fluctuations: charge density wave ε Staggered Magnetization N Staggered Dimerization ε = (-1) S S x x , spin density wave N Spin flip ΔS=1 -kF kF Susceptibility 1/q 1/q ΔS=0 -kF x+a • Must be careful: often spin-charge separation must be enforced by hand kF 1/q S=1/2 AFM Chain in a Field • Field-split Fermi momenta: Uniform magnetization Half-filled condition • Affleck and Oshikawa, 1999 1 Sz component (ΔS=0) peaked at scaling dimension increases 1/2 • Sx,y components (ΔS=1) remain at scaling dimension decreases π • Derived for free electrons but correct always - Luttinger Theorem 0 1/2 M h/hsat 1 hsat=2J 0 • XY AF correlations grow with h and remain commensurate • Ising “SDW” correlations decrease with h and shift from π h/hsat 1 Weakly coupled Heisenberg chains in magnetic field z z y y x x ! ! ! + N N S · S → N ! • non - frustrated inter-chain coupling r r r Nr ! + Nr Nr ! r r most relevant less relevant 2πR2 < 1/(2πR2) J’ spins order in the plane perpendicular to the direction of magnetic field (z): umbrella / cone / spin-flop states • frustrated inter-chain coupling y+1 y y z z x !x,y ·(S !x,y+1 +S !x+1,y+1 ) → Nyx ∂x Ny+1 (y+1) (y)Sπ+2δ +sin(δ)Sπ−2δ +Nyy ∂x Ny+1 S less relevant most relevant (small to intermediate fields) 1/(2πR2) 1+2πR2 > ★ frustration promotes collinear SDW order Real material: relevant inter-chain DM interaction, B||D • Even though D=0.05 : DM beats CoAF and dimerization zero-field instabilities [y = chain index] relevant: dim = 2πR2 • DM allows relevant coupling of Nx and Ny on neighboring chains – immediately stabilizes spiral state • orthogonal spins on neighboring chains c b – - + -+ - + -+ small J’ perturbatively makes spiral weakly incommensurate Dzyaloshinskii-Moriya interaction (DM) controls phase diagram of Cs2CuCl4 for B || D (B along crystal a axis) Finite D, but J’=0 Finite D and J’ Transverse Field: B || D B || a • DM term becomes more relevant • b-c spin components (XY) remain commensurate: spin simply tilt in the direction of the field • Spiral (cone) state just persists for all fields. Experiment vs Theory Tc Order increases with h here due to increasing relevance of DM term h Order decreases with h here due to vanishing amplitude as hsat is approached (density of magnons -> 0 ) physics is controlled by weak (~1/20 of J) Dzyaloshinskii-Moriya interaction BEC Coldea et al 2002; Radu et al 2005; Veillette et al 2006; Kovrizhin et al 2006 Longitudinal Field: B D removes DM DM term involves Sz (at π − 2δ) and Sx (at π): • Leads to momentum mis-match for h>0: DM “irrelevant” for h > D • With DM killed, sub-dominant instabilities take hold • Two important couplings for h>0: kF ↓ − kF ↑ = 2δ = 2πM Magnetic field relieves frustration! dim 1/2πR2: 1 -> 2 dim 1+2πR2: 2 -> 3/2 “collinear” SDW spiral “cone” state • “Critical point”: 1+2πR2 = 1/2πR2 at M = 0.3 1 Tc sdw gives cone M 1/2 h/hsat (somewhat naïve) Phase Diagram T Theory “collinear” SDW (DM) “cycloid” “cone” ? 0 polarized 0.9 1 D/J ~ 0.1 h/hsat Experiment “commensurate” AF order ? DM cycloid sdw cone ? ? ? “Commensurate Collinear” order of some sort has been observed recently: - seems to have orthogonal spins on neighboring chains (Coldea // Veillette Chalker 2006) RG length: shortest wins sdw / cone cone ComAF sdw h Our theory does predict fluctuations-induced term of right sign, but too small an amplitude + Beyond the naïve: commensurate locking • “Collinear” SDW state locks to the lattice at low-T -“irrelevant” (1d) umklapp terms become relevant once SDW order is present (when commensurate): multiparticle umklapp scattering -strongest locking is at M=1/3 Msat Observed in Cs2CuBr4 (Ono 2004, Tsuji 07, Fortune 09) • down-spins at the centers of hexagons T (DM) “cycloid” 0 ! Ψ†R ΨL "n “collinear” SDW Cs2CuBr4 Fortune et al PRL 2009 “cone” polarized ? 0.9 1 uud ~ 0.1 h/hsat → (π − 2δ)n = 2πm → 2M = 1 − 2m/n n 3 4 5 5 6 m 1 1 1 2 1 2M 1/3 1/2 3/5 1/5 2/3 1/3 ~1/2 5/9 2/3 M=1/3 magnetization plateau ★ Observed in Cs2CuBr4 (Ono 2004, Tsuji 2007) J’/J = 0.75 but not Cs2CuCl4 [J’/J = 0.34] S=1/2 J’ J • UUD (up-up-down) structure -- down-spins at the centers of hexagons; commensurate structure -- one down spin per every triangle ★ “up-up-down” state predicted by large-S expansion for spatially isotropic triangular lattice antiferromagnet (Chubukov, Golosov 1991): special to J=J’ situation Classical isotropic Δ AFM in magnetic field • Spiral state in the absence of the field • Apply field H: two degenerate states ✦ planar - magnetic field is in the plane of spin spiral ✦ cone (umbrella) - field perpendicular to the spiral plane, spins tilt out of the plane towards the field direction • Energies (and susceptibilities) of the two states are equal for J=J’: Eclass = - c J S2 (accidental degeneracy) • 1/3 plateau is possible at collinear point (φ=0): develops into finite interval by thermal and quantum fluctuations Planar plateau is possible at “collinear” point Umbrella (cone) No plateau possible Order-by-Disorder mechanism plateau RbFe(MoO4)2: S=5/2 Fe3+ Svistov et al PRB (2003) Smirnov et al PRB (2007) Tsuji et al (2007) NMR spectra: Fujii et al (2004) Cs2CuBr4: S=1/2 Ono et al 2003 δh The problem: spatial anisotropy stabilizes umbrella ! Spatially anisotropic model: classical analysis fails H = ∑ J ijSi ⋅ S j − h ∑ S 〈 ij 〉 ! J != J • z i J’ i hsat 0 0 J 1/3-plateau hsat h h Umbrella state: favored classically Planar states: favored by quantum fluctuations J’/J = 0.75 not particularly small: OK to use semiclassical spin wave expansion • Technical formulation: spatial anisotropy J-J’ causes softening of interacting (including 1/S correction) spin waves Our semiclassical approach: treat spatial anisotropy (J-J’) as a perturbation to interacting spin waves • single dimensionless parameter δ=(40/3)S(1 - J’/J)2: classical anisotropy energy ~ (J - J’)/J fully polarized state spin-wave gap ~ J/S h distorted umbrella (2) planar hc2 UUD plateau BEC k = 0 hc1 commensurate BEC k != 0 roton condensation 2 low-energy gapped modes planar distorted umbrella (1) shaded: local minimum only; pulsed field measurements? incommensurate zero-field spiral 1 Cs2CuBr4 2 Alicea, Chubukov, Starykh PRL 102, 137201 (2009) 3 Cs2CuCl4 4 ! ~ S(1 - J’/J) 2 Summary (II) ✦ Rich, interesting physics: much can be understood by viewing the problem/material from 1d perspective ✦ Good for weakly ordered states: abundance of multi-particle excitations ✦ Delicate details of 3d ordering are determined by minute, and often anisotropic, sub-leading interactions • T-H phase diagram of Cs2CuCl4 and Cs2CuBr4 remain to be understood – commensurate AF state in Cs2CuCl4 – 2/3 and other fractional states in Cs2CuBr4 • Evolution of high-energy continuum into low-energy spin waves? NMR data: persistence of planar structures RG for quantum spins: • – collinear tendency persists among orthogonal to the field components (Sx,y) top view: umbrella – generates collinear coupling between nextnearest chains (J’/J)4 [ S+x,y S-x,y+2 + h.c.] • Observed in magnetic field in Cs2CuCl4 (M. Takigawa, presentation at HFM-08): 2-sublattice collinear order between chains in the plane perpendicular to the applied magnetic field Tokiwa et al, 2007 2-sublattice collinear Magnetization measurements M(h) is smooth: not sensitive to low-energy (long-distance) fluctuations. Determined by uncorrelated (but magnetized) chains. Tokiwa et al, 2006 “Molecular” field J’= 0 J’=0.34 J Error in saturation field is (J’)2/2J ≈ 2% dM/dh delineates phase boundaries: divergent derivative = phase transition Cs2CuBr4 • Isostructural to Cs2CuCl4 but believed to be less quasi-1d: J’/J = 0.75 T. Ono et al, 2004 • Magnetization plateau at M=1/3 Msat observed for longitudinal but not transverse fields (additional feature at 2/3 Msat) - ? Alicea, Chubukov, Starykh 2008 Order-by-Disorder: Quantum fluctuations (1/S) • fluctuation spectra of different spin structures are different: E = Eclass + ΔEsw • quantum fluctuations prefer planar arrangement sw = ∆Eplanar S! S! sw = ωplanar (k) < ∆Eumbrella ωumbrella (k) 2 2 k k • prefer collinear configuration even more, when possible: state with maximum number of soft modes wins. • plateau is a quantum effect, width δh = 1.8 J/S (hsaturation = 9 J) • plateau is the effect of interactions (hence, width ~ 1/S) between spin waves Tsuji et al (2007) Chubukov, Golosov (1991) NMR spectra: Fujii et al (2004) δh The problem: spatial anisotropy stabilizes umbrella Continuum of excitations is a high-energy feature! Coldea et al, PRB 2003 J J’ Low-energy degrees of freedom • Quantum triad: uniform magnetization M = JR + JL , staggered magnetization N and staggered dimerization ε = (-1)x Sx Sx+1 Components of Wess-Zumino-Witten-Novikov SU(2) matrix • Hamiltonian H ~ JRJR + JLJL + γbs JRJL marginal perturbation • Operator product expansion • Scaling dimension 1/2 (relevant) • Scaling dimension 1 (marginal) (similar to commutation relations) EXPERIMENT: Longitudinal Field ,Tc vs B; SDW, …, cone • Very different behavior for B along b, c axes (both orthogonal to DM direction a) • Additional anisotropies in the problem? Nature of AFM and AFM’ phases - the biggest puzzle? AFM FM Cone states AFM’ R. Coldea et al, 2001; T. Radu et al, 2005; Y. Tokiwa et al, 2006 FM