Quantum spin dynamics in 1D antiferromagnets. Igor Zaliznyak Neutron Scattering Group Outline Excitation continuum in 1D S=1 Heisenberg antiferromagnet (Haldane chain). Luttinger liquid behavior in the high-field phase of a Haldane chain Fundamental importance: instability of the coherently propagating excitations in quantum (spin) liquid High-energy spinons in S=1/2 chain copper oxides. Fundamental importance: electronic structure of cuprates, spin-charge separation, log(T) corrections at low T Practical importance: relaxation channel in optoeletronic devices, anisotropic heat transport, …

Collaboration C. Broholm, D. Reich S.-H. Lee, R. Erwin L.-P. Regnault, M. Enderle M. Sieling BENSC Hahn-Meitner Institute P. Vorderwisch, M. Meissner T. Perring, C. Frost S. V. Petrov H. Takagi ISSP University of Tokyo

“Heisenberg model systems”: why bother? H = JS SiSi+1+ JS SiSi+D+ DS(Siz)2 , J/J>> 1 (<<1) What are the “model” assumptions? spins are on localized electrons near(est) neighbor exchange coupling lattice: 1D, 2D, 3D? 1D 2D good approximation for many systems simple and general Hamiltonian great variety of fundamental phenomena Example: CsNiCl3. J = 2.3 meV = 26 K J = 0.03 meV = 0.37 K = 0.014 J D = 0.002 meV = 0.023 K = 0.0009 J 3D magnetic order below TN = 4.84 K

Understanding antiferromagnetic spin chain: history. Spectrum for S=1 is not intermediate case between S=1/2 and S>>1 S >> 1 S = 1/2 S = 1 Single mode, no gap (Anderson, 1952) Continuum, no gap (Bethe, 1931) Gap at q = p (Haldane, 1983) e(q) > p/2 Jsin(q) e(q) < p Jsin(q/2) e(q) = 2J(S(S+1))1/2sin(q) e2 (q) = D2 + (cq)2 e(q)/J/(S(S+1))1/2

Understanding S=1 chain: theory. Mostly - numerical studies on finite chains. Quantum Monte-Carlo: Takahashi (1989), Meshkov (1993), Dietz et. al. (1993), Yamamoto (1995), … Exact diagonalization: Golinelli et. al. (1990), Haas et. al. (1995), … Density matrix renormalization group: White et.al. (1993), ….

Understanding S=1 chain: theory. Quantum Monte-Carlo results for 128-spin chain indeed indicated existence of a continuum. QMC by S. Meshkov (1993). Since S(q)=>0 at q=0, S(q,w)/S(q) is shown.

Interesting detail: excitations are non-interacting fermions! L.P.Regnault, S. Meshkov and I. Zaliznyak J. Phys. Condens. Matter (Letter),1993 How could we know? Receipt is simple: take the spectrum, calculate free energy, and compare with measured thermodynamic quantities.

Understanding S=1 chain: experiment. NENP: Haldane gap confirmed, no continuum observed.

Cross-over from single-mode to continuum in spectrum of 1D S=1 Heisenberg antiferromagnet observed. I. A. Zaliznyak, S.-H. Lee, S. V. Petrov, PRL 017202 (2001) Color contour map of the spectral density of raw magnetic scattering ~ dynamic spin susceptibility

Instability of coherently propagating mode at the top of the excitation band in Haldane spin chain. Haldane mode becomes a continuum at q <~ 0.5 J = 2.275(5) meV, v = 2.49(4)J

Spectral density measured in two configurations. Resolution is better and more round in the “E-resolved 2-axis mode”. Continuum confirmed, starts at q < 0.6.

Conclusions. Instability of the coherent propagating mode at high energies is a universal feature of quantum liquid close to criticality? Dispersion of the in-chain excitation is asymmetric, as expected for disordered S=1 HAFM chain. In the coherent part of the spectrum dispersion parameters agree with those measured in NENP and in Monte Carlo calculations. Single-mode excitation in a Haldane chain becomes unstable around the top of the dispersion band. Continuum excitation spectrum, whose width increases with decreasing q, is observed at q < 0.6 . Acknowledgement This work was carried out under Contract DE-AC02-98CH10886, Division of Materials Sciences, US Department of Energy. The work on SPINS was supported by NSF through DMR-9986442

Haldane chain in magnetic field. L.P. Regnault, I. Zaliznyak, J.P. Renard, C. Vettier, PRB 50, 9174 (1994). 3,5,…-particle continuum 3,5,…-particle continuum Macroscopic quantum phase in the string operator at H>Hc results in the shift in q-space between fermions and magnons. H=0 H~Hc ? particles particles holes I. Zaliznyak, unpublished (2002). H>Hc

Haldane chain in magnetic field: H>Hc. I. Zaliznyak, M. Enderle, C. Broholm, et al, to be published (2002).

Haldane chain in magnetic field H>Hc: Luttinger liquid? I. Zaliznyak, et al (2002).

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

What is interesting about chain copper oxides? Electronic band structure and model Hamiltonian for cuprates Y. Mizuno et al, PRB 58 (1998), W. C. Makrodt, H. J. Gotsis, PRB 62 (2000), H. Rosner et al, PRB 63 (2001)) Spin-charge separation C. Kim et al, PRL 77 (1996), PRB 56 (1997) , H. Fujisawa et al, PRB 59 (1999), H. Suzuura and N. Nagaosa, PRB 56 (1997), F. Essler and A. Tsvelik, PRB (2001) log(T) corrections to low-T magnetic susceptibility N. Motoyama et al, PRL 76 (1996), K. R. Thurber et al, PRL 87 (2001) Heat transport by spinons: giant heat conductance along the chains A. V. Sologubenko et al, PRB 62 (2000), PRB 64 (2001) Ultrafast relaxation of the optical nonlinear absorption T. Ogasawara et al, PRL 85 (2000), H. Kishida et al, PRL 87 (2001)

Electronic band structure of chain copper oxides and spin-charge separation. Effective single-band Hubbard model at half-filling H EH= - 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 Parametri-zation: U, t vc(k-kf) Band gap 1.5 eV Exchange J =? vs(k-kf)  Ef kf k/ ARPES measurement, C. Kim et al, PRL (1996) Essler and Tsvelik cond-mat/0108382

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 (R. Neudert et al, PRL 81 (1998), Rosner et al, PRB 56 (1997)) 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

Two-spinon continuum in SrCuO2: direct measurement MAPS@ISIS, Ei = 98 meV. Color contour map of the scattering intensity. White lines are gaps in the detectorr array. Vertical lines at l = n/2 are spinons. qchain/2

Two-spinon continuum in SrCuO2: direct measurement MAPS@ISIS, Ei = 241 meV. Color contour map of the scattering intensity. qchain/2

Two-spinon continuum in SrCuO2: direct measurement MAPS@ISIS, Ei = 520 meV. Color contour map of the scattering intensity. qchain/2

Two-spinon continuum in SrCuO2: direct measurement Best fit: J = 280(20) meV, higher than usually believed. Agrees with midinfrared absorption result J = 260(10) meV. qchain/2

Conclusions What did we learn? Structure of the non-hydrodynamic part of the excitation spectrum in S=1antiferromagnetic Heisenberg chain cross-over from the coherent propagating Haldane-gap mode to continuum occurs at q<~0.6 experimental studies are the main source of insight instability of the coherent spectrum is a general feature of quantum (spin) liquid close to phase transition (small gap)? Spin excitation spectrum in the Mott-Hubbard insulator SrCuO2 supports evidence for spin-charge separation in -Cu-O- chains two-spinon continuum directly observed electron dynamics is dominated by spinons up to ~ 0.8 eV J = 280(20) meV, not 190 meV => rethink/redo log corrections large energy scale and fractional nature of excitations results in fascinating physical properties

Understanding S=1 chain: theory. Somewhere q=p single mode should crossover to q=0 continuum. 2-particle continuum at q=>0 arise in simple fermion “toy model”, Gomez-Santos (1989) Hierarchy of excited levels after White and Huse (1993) What about continuum? No quantitative theory, difficult to measure: structure factor ->0 at small q. Not observed in NENP down to 0.3

Model S=1 Haldane chain compound CsNiCl3. H = JS SiSi+1+ JS SiSi+D+ DS(Siz)2 , J/J>> 1 (<<1) J = 26 K, J= 0.033 J, D = 0.0009 J, TN = 4.84 K J J “supercritical” J => not important for spin dynamics at high energies the most isotropic of known materials

New measurement: high-luminosity setups with ASD. Area sensitive detector (ASD) gives 4-6 fold increase in throughput without any loss in resolution and with very low background.

SPINS@NG5.NCNR.NIST.GOV Area sensitive detector => 4-6 fold increase in throughput.

Single-mode and continuum spectrum in one-dimensional S=1 Heisenberg antiferromagnet (Haldane chain). NENP (1992): gap+single mode CsNiCl3 (2001): +continuum

Instability of coherently propagating mode at the top of the excitation band in two quantum liquids. Haldane mode becomes a continuum at q <~ 0.5 Top of the band “maxon” excitation in superfluid 4He broadens with pressure Graf, Minkiewicz, Bjerrum Moller, and Passell (1974)

3D corrections to the 1D excitation spectrum: MF-RPA estimates. Except at low energy, spectrum even at T=0 is 1D In mean field random phase approximation (MF-RPA) corrections to the 1D dispersion static structure factor S(q) are within ±10% for 0.2 < q < 0.8 , and within ±5% for 0.3 < q < 0.7 

Weak inter-chain coupling of the S=1/2 -Cu-O-Cu- chains: long-range order and correlated spin glass. TN 5 K k h Q=(h,0.5,0.5) Points: magnetic scattering Line: nuclear scattering SrCuO2: decoupling in zigzag ladder results in short-range anisotropic static order Sr2CuO3: static long-range (Bragg) order

Effect of the inter-chain coupling on spin dynamics in SrCuO2. J  280 meV, TN 5 K, <µ> 0.15µB I. Zaliznyak et al, PRL 83, 5370 (1999). Extremely weak coupling between S=1/2 antiferromagnetic spin chains in Sr2CuO3 and SrCuO2 results in static order but marginal modulation of the inelastic spectrum.

Weak inter-chain coupling of the S=1/2 chains: static order and effect on spin dynamics. A. Zheludev et al, cond-mat/0105223. J  24 meV, TN 9 K, <µ> 0.15µB A C A B B Magnon A C Magnetic Bragg peak

Evidence for spin-charge separation in1D Mott-Hubbard insulator: ARPES in chain copper oxide SrCuO2. C. Kim et al, PRB 56, 15589 (1997).

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

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

Future projects Extend collaboration within BNL femtosecond pump-probe measurements in SrCuO2 /Chemistry gate/photodoping in thin films? Continuum in better-1D Haldane material Y2BaNiO5, check for the effect of inter-chain coupling High-field phase of the Haldane chain (NENP, in works): marginalization of quantum liquid? Two-spinon excitation spectrum in Sr2CuO3 Logarithmic corrections to the susceptibility - by neutrons! Doping SrCuO2 and Sr2CuO3 away from half-filling Doping the gapped (Haldane) -O-Ni-O- spin chains in SrNiO2, isostructural with SrCuO2 – new sub-gap physics?