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Spinons in strongly correlated chain cuprates

Spinons in strongly correlated chain cuprates. Igor Zaliznyak Neutron Scattering Group Condensed Matter Physics and Material Science Department. Outline Introduction: who, what, why, and how? Structure and electronic properties of chain cuprates

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Spinons in strongly correlated chain cuprates

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  1. Spinons in strongly correlated chain cuprates Igor Zaliznyak Neutron Scattering Group Condensed Matter Physics and Material Science Department Outline • Introduction: who, what, why, and how? • Structure and electronic properties of chain cuprates • high-energy spinons, itinerancy, and spin-charge separation • Dimensional cross-over and low-energy excitations • unbound spinons in 2D?

  2. Contributors and Collaborators Chain cuprates SrCuO2, Sr2CuO3, Sr2CuO3+y Material syntesis and crystal growth G. Gu (BNL) H. Takagi (U. Tokyo) Neutron Scattering A. Walters, T. Perring, C. Frost (ISIS) H. Woo (BNL/ISIS) S. Park (NIST NCNR) C. Broholm (JHU) Theory & Advice J.-S. Caux (U. Amsterdam) F. Essler (Oxford, BNL) J. Bhaseen (Oxford, BNL) A. Tsvelik (BNL)

  3. Low-dimensional electron/spin systems: fundamental importance • Quantum effects and fluctuations are important • disorder, quantum criticality, spin-liquid states • Excitation fractionalization by dimensional confinement • topological excitations with fractional quantum numbers (spinons) • spin-charge separation, Luttinger liquid phase • Dimensional cross-over and confinement of fractional excitations • Mott-insulating state in 1D

  4. Low-dimensional electron/spin systems: practical significance • Chains/planes are the building blocks of • high-Tc superconductors, chain cuprates, MgB2, NaCoO2, … • Low-D physics governs nano-scale functional materials • nanotubes (Luttinger liquid, 1D Mott insulator) • thin films, multilayers, graphene (2D) • Low-D systems are generated by electronic phase segregation, e.g. in doped oxides, etc. • stripes in cuprates => spin ladders (J. Tranquada, et. al.)

  5. Low-dimensional electron systems: practical significance Low-dimensional orbital networks in copper oxides SrCuO2, YBa2Cu4O8,Sr14Co24O41, other ladder systems. Sr2CuO3, YBa2Cu3O7, Sr14Cu24O41, … - O – Cu – O – Cu – O – Cu – O – Cu – O– Cu – O - La2CuO4, etc. Fundamental minimal model: 1-band Hubbard Hamiltonian HH= -S tm(cj,+cj+m, + H.c.) + (U/2) S(n jn j- + H.c.)

  6. Spin excitations in S=1/2 Cu2+ chains: spinons in KCuF3 U >> t charge gap (~ U, few eV) >> spin bandwidth ~ 0.1 eV (J  34 meV) Dimensional cross-over: 3D order at TN 39 K, <µ> 0.5µB S=1/2 Heisenberg spin chain H = J S Si Si+1 Mean-field (MF) picture of spinon attraction in the ordered phase. [A. Tennant et al (2001)].

  7. Follows from half-filled Hubbard model R. Coldea, et al, PRL (2001) Spin waves in La2CuO4: itinerancy effects in 2D Mott insulator J  0.13 eV, optic gap ~ 1.2 eV => spin bandwidth ~ 0.3 eV is NOT MUCH SMALLER than charge gap. S=1/2 square lattice Heisenberg AFM with ring exchange H = J S Si Sj+JcS [(SiSj)(SkSl)+ (SiSl)(SkSj)- (SiSk)(SjSl)] U  2.5 eV t  0.3 eV Jc  0.27J

  8. Itinerancy effects in 1D Mott insulator? What happens in 1D when spin bandwidth IS NOT MUCH SMALLER than charge gap, i.e. for 2t ~ U? • Spin sector separates: same spinon dispersion • Spectral weight transferred to upper boundary M. J. Bhaseen, et al, PRB (2005)

  9. Excitation: phonon, magnon q = ki - kf meV, μeV neutron out kf neutron in ki Neutron scattering: how neutrons measure excitations. 1 meV = 11.6 K

  10. Long-lived quasiparticle (magnon)  delta-function singularity in cross-section nuclear scattering length, b ~ 10-13cm magnetic scattering length, rm = -5.39*10-13 cm Neutron scattering: how neutrons measure excitations.

  11. Chain copper oxides: 1D Mott-Hubbard insulators. Sr2CuO3 SrCuO2 Cu-O bond length  1.95 Å, exchange coupling J ~ 0.2-0.3 eV (!)

  12. Weakly coupled S=1/2 -Cu-O-Cu- chains: frustration and spin freezing vs long-range order I. Zaliznyak et al, PRL (1999) <µ> 0.035µB K. Kojima et al, PRL (1997) <µ> 0.06µB TN 5 K k h Q=(h,0.5,0.5) Points: magnetic scattering Line: nuclear scattering SrCuO2: decoupling in zigzag ladders leads to short-range anisotropic static order Sr2CuO3: static, long-range (Bragg) order

  13. How do we know exchange coupling J? “Inelastic neutron scattering experiments are much desired”, Maekawa & Tohyama, Rep. Prog. Phys. (2001), T. Rice, Physica B (1992). • Temperature dependence of the magnetic susceptibility (N. Motoyama et al, PRL (1996)) J = 0.19(2) eV ? • Infrared absorption below the optical band gap (H. Suzuura et al, PRL (1996)) J = 0.26(1) eV • Electron + Xray spectroscopy + band structure calculations (Neudert et al, PRL 81 (1998), Rosner et al, PRB 56 (1997), Kim, et al (2006), Koitzsch, et al (2006)) • U  4.2 eV • V  0.8 eV • t  0.55 eV J = 4t2/(U-V) - |K| ~ 0.25-0.36 eV (!?) J~ 0.5 - 1 meV Record-high J, record-low J/J

  14. t1 t2 ARPES measurement, C. Kim et al, PRL (1996) Koitzsch, et al, PRB (2006) Electronic band structure of copper oxide chains and spin-charge separation. Effective single-band Hubbard model at half-filling HEH= -S tm(cj,+cj+m, + H.c.) + (U/2) S(n jn j- + H.c.) + VS(n jn j+1 + H.c.) - |K|S SjSj+1 Electron spectral function A(k,): holon-spinon continuum Parameteri-zation: U, t vc(k-kf) • Band gap   1.5 eV • Exchange J =? vs(k-kf)  Ef kf k/ Essler,Tsvelik PRB (2002)

  15. Measure the spin part of one-dimensional electrons directly Triplet spectrum of two-spinon states, combined set of 4 measurements on MAPS at ISIS SrCuO2 Ei = 850 meV Ei = 550 meV spinons Single crystal sample (H. Takagi) m = 3.9 g Mosaic ~ 0.5 T= 12 K Ei = 250 meV phonons Ei = 100 meV I. Zaliznyak, S.-H. Lee, in “Modern Techniques for Characterizing Magnetic Materials”, Ed. Y. Zhu, Springer (2005)

  16. Measure the spin part of one-dimensional electrons directly 2-parameter fit (J, A) to “Muller ansatz”:

  17. Measure the spin part of one-dimensional electrons directly Intensity variation: anisotropic Cu2+ magnetic formfactor!

  18. I. Zaliznyak, S.-H. Lee, in “Modern Techniques for Characterizing Magnetic Materials”, Ed. Y. Zhu, Springer (2005) Cu2+ anisotropic magnetic formfactor

  19. Measure the spin part of one-dimensional electrons directly Account for anisotropic Cu2+ magnetic formfactor => A is ~ E-independent

  20. -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 600 500 400 300 (a) (e) 0.1 0.0 S(Q,E) (mbarn/meV/sr/Cu) Energy transfer, E, (meV) 400 300 200 (b) (f) 0.1 0.0 200 150 100 (c) (g) 0.6 0.3 0.0 70 50 30 (d) (h) 2.0 1.0 0.0 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Momentum Ql (units of 2/c) Spin part of one-dimensional electrons in SrCuO2 • Holon (charge) gap: m  0.75 eV • Optic gap:  = 2m  1.5 eV • Two-spinon band: πJ  0.7 eV Effective single-band 1D Hubbard model at half-filling, only 2 parameters: U  4.2 eV, t  0.55 eV Calculated (left) and measured (right) magnetic scattering in 1D Mott insulator SrCuO2. (I. Zaliznyak, et. al., PRL, 2004) Maps of the absolute net scattering intensity measured in SrCuO2 for incident neutron energies (e) EI = 1003 meV, (f) EI = 517 meV, (g) EI = 240 meV, (h) EI = 98 meV. Corresponding resolution-broadened intensity maps calculated from S(Q,E) for free spinons (Müller ansatz) are shown in (a)-(d).

  21. Exact 2-spinon Muller atsatz Benthien & Jeckelmann, PRB (2007) Karbach, et al, PRB (1997) Summary I • dCP continuum dominates electron spin dynamics in -Cu-O- chains up to 0.7 eV • no evidence for itinerancy effects in spinon spectral weight • consistent with theoretical calculation by Benthien & Jeckelmann, PRB (2007) • “Muller alsatz” (MA) fits data quite well • disappointing, as there are notable differences at higher energies between MA and exact results by Bougourzi, et al (1997), Caux et al (2005, 2006).

  22. Spin excitations in Sr2CuO3 Triplet spectrum of two-spinon states, combined set of 3 measurements on MAPS at ISIS is shown Sr2CuO3 Ei = 794 meV Ei = 516 meV spinons Ei = 240 meV Three co-aligned crystals mtotal = 18.45 g Mosaic < 0.3 T= 6 K

  23. Normalized INS data from Sr2CuO3 crystals Ei=1088 meV • Four neutron incident energies used • Ei = [240, 516, 794,1088] meV • Energy resolution  Ei • Detailed fitting done on multiple “cuts” at constant energy transfer for each Ei • Figure: comparison of background-subtracted normalized data (left column) and best fit to Muller ansatz expression corrected for the instrumenal resolution (right column) Ei=794 meV Energy Transfer (meV) S(Q,) (mbarn/sr/meV/Cu2+) Ei=516 meV Ei=240 meV Qchain (2/b)

  24. Fits of the constant-E cuts in Sr2CuO3

  25. 240 meV 516 meV 794 meV 1088 meV Fits to the Muller ansatz Incident Energies: <J> = 240(7) meV J (meV) • Results are qualitatively similar to SrCuO2 • However, now observe clear deviations outside error bars • Energy-dependent exchange coupling J decreasing at low E • Energy-dependent intensity prefactor A decreasing at high E • Sr2CuO3 data clearly deviate from Muller ansatz! Deviation #1 <A> = 0.44(8) Deviation #2 A Energy Transfer (meV)

  26. Fits to exact expressions for 2- and 4-spinon continua 2-spinon <A> ≈ 0.43 <J> = 243(6) meV 2-spinon + 4-spinon <A> ≈ 0.32 J.-S. Caux et al. J. Stat. Mech. (2006)

  27. Incident Energies: 240 meV 516 meV 794 meV 1088 meV Fits to exact expressions for 2- and 4-spinon continua J.-S. Caux et al. J. Stat. Mech. (2006) <J> = 243(6) meV J (meV) • Both J and intensity prefactor A are now E-independent • agrees with physical expectation • A is dramatically reduced compared to expected value A ≈ 1 • two-spinon and four-spinon excitations must account for ≈ 98% of total spin spectral function • Mystery of missing factor ~3 ≈ π ? <A> = 0.32(5) A Energy Transfer (meV)

  28. Summary II • Spin response in Sr2CuO3 is similar to SrCuO2 but measured with much better precision • multispinon dCP continuum dominates dynamical electronic properties in -Cu-O- chains up to 0.7 eV • no evidence for itinerancy effects in spin spectral weight, consistent with theoretical calculation by Benthien & Jeckelmann, PRB (2007) • “Muller alsatz” (MA) DOES NOT fit the data ! • notable differences at higher energies between MA and exact results by Bougourzi, et al (1997), Caux et al (2005, 2006) are experimentally observed • Mystery of anomalously small intensity (prefactor A) does not (yet) allow to claim observation of 4-spinon contribution

  29. Dimensional cross-over: effect of inter-chain coupling What happens in the ordered phase? T << TN≈ 5 K << J ≈ 2,800 K

  30. Effect of inter-chain coupling on spin dynamics: SrCuO2. J  226 meV, TN 5 K, <µ> 0.035µB I. Zaliznyak et al, PRL (1999,2004). Extremely weak coupling between S=1/2 antiferromagnetic spin chains in SrCuO2 and Sr2CuO3 results in static order but marginal modulation of the inelastic spectrum.

  31. Weak inter-chain coupling of the S=1/2 chains: effect of the static order on spin dynamics. A. Zheludev et al, PRB 65 014402 (2001). J  24 meV, TN 9 K, <µ> 0.15µB A C A B Magnon B A C Magnetic Bragg peak

  32. Inter-chain dispersion of spin excitations in Sr2CuO3 Where are magnons? A C B D No evidence for any coherent quasiparticle excitation at all!?

  33. Inter-chain dispersion of spin excitations in Sr2CuO3 Intensity and continuum gap fixed at values found for l=0.575 Intensity and continuum gap refined in a fit

  34. Inter-chain dispersion of spin excitations in Sr2CuO3 Try higher energy resolution

  35. Inter-chain dispersion of spin excitations in Sr2CuO3 Intensity and continuum gap fixed at values for l=0.5 Intensity and continuum gap refined I a fit

  36. “Confederate flag”: a fine-tuned model with compensated mean field • A. Nersesyan and A. M. Tsvelik, PRB 67 024422 (2003). • J’/J <<1, J’ = 2J” Frustration relieves the mean-field spinon attraction and a need for their confinement into magnons: 2D propagating (weakly bound?) S=1/2 spinons J’/2 J’ J’/2 J J

  37. Realization of the frustrated “confederate flag” model in Sr2CuO3 2J’ b≈3.5 Ǻ • Use realistic ionic radii • Interchain hopping proceeds through Cu-O-O-Cu path! • J’/J ~ (tCu-OtO-O)/DO2 c≈3.9 Ǻ J J’ Seems to be a realization of Tsvelik-Nersesyan’s “Confederate flag” model!

  38. Summary • Spin excitations dominate electron dynamics in chain cuprates up to 0.7 eV • no evidence for itinerancy corrections to spectral weight  spin-charge separation in 1D • spin-only Hamiltonian is sufficient • Muller ansatz fails  exact result fits well except for mystery of small A • Weakly bound/unbound 2D spinons in the Neel-ordered phase? • 2D-dispersive continuum at low E in the ordered phase • no evidence for the coherent quasiparticle excitations • no energy-scale separation for a continuum • fine-tuned Tsvelik-Nersesyan’s “frustrated flag” model is realized by the inter-chain Cu-O-O-Cu hopping ?

  39. Thank you!

  40. Spinons in chain copper oxides: giant heat conductance. A. V. Sologubenko et al, PRB 64, 054412 (2001).

  41. Spinons in chain copper oxides: picosecond relaxation of optical nonlinearity. T. Ogasawara et al, PRL 85, 2204 (2000). Sr2CuO3

  42. M. J. Bhaseen, F. S. H. Essler, and A. Grage, cond-mat/0312055 T. Valla, et. al.,cond-mat/0403486 Planned projects and directions for future studies • Itinerancy effects in the two-spinon spectrum of chain cuprates • refine analysis of SrCuO2 data, measure Sr2CuO3 • Oxygen-doping chain cuprate Sr2CuO3+y • two-spinon response (incommensurability?) • effect on the inter-chain hopping and dimensional cross-over • Model Heisenberg spin systems: ABX3x2D2O, PHCC, … • instability of Bose-quasiparticles in spin liquids • dimensional and quantum-classical cross-over in spin chains • Layered perovskites La2-xSrx(Co,Mn)O4 • effect of charge-order superstructure, disorder, etc. on spin dynamics

  43. Crystal structure of chain cuprates 1D Hubbard model: only 2 parameters, U  4.2 eV, t  0.55 eV Spin bandwidth, πJ ≈ 4πt2/U ~ 0.75 eV Sr2CuO3 SrCuO2 Optic gap   1.5 eV

  44. t1 t2 Spin part of one-dimensional electrons Triplet spectrum of two-spinon states, combined set of 4 measurements on MAPS at ISIS • Effective single-band Hubbard model at half-filling SrCuO2 • Spin-charge separation Ei = 850 meV • Holon gap • m  0.75 eV • Optic gap •  = 2m  1.5 eV • Two-spinon band • πJ  0.7 eV spinons Ei = 550 meV Ei = 250 meV Ei = 100 meV phonons

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