Excitation continuum in 1D S=1 Heisenberg antiferromagnet Excitation continuum in 1D S=1 Heisenberg antiferromagnet (Haldane chain).(Haldane chain). Luttinger liquid behavior in the high-field phase of a Haldane Luttinger liquid behavior in the high-field phase of a Haldane chainchain

– Fundamental importance: instability of the coherently propagating excitations in quantum (spin) liquid

High-energy spinons in S=1/2 chain copper oxides.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, …

OutlineOutline

Quantum spin dynamics in 1D antiferromagnets.Quantum spin dynamics in 1D antiferromagnets.Igor Zaliznyak

Neutron Scattering Group

CollaborationCollaboration

• C. Broholm, D. Reich

• T. Perring, C. Frost

• S.-H. Lee, R. Erwin

• L.-P. Regnault, M. Enderle

• H. Takagi ISSPISSPUniversity of TokyoUniversity of Tokyo

• S. V. Petrov

• P. Vorderwisch, M. MeissnerBENSCBENSCHahn-Meitner InstituteHahn-Meitner Institute

• M. Sieling

What are the “model” assumptions?– spins are on localized electrons– near(est) neighbor exchange– coupling lattice: 1D, 2D, 3D?

1D

2D

J = 2.3 meV = 26 K J = 0.03 meV = 0.37 K = 0.014 JD = 0.002 meV = 0.023 K = 0.0009 J3D magnetic order below TN = 4.84 K

Example: CsNiCl CsNiCl33..

• good approximation for many systems• simple and general Hamiltonian • great variety of fundamental phenomena

““Heisenberg model systems”: why bother?Heisenberg model systems”: why bother?

H = J SiSi+1+ JSiSi D(Si

z)2 , J/J>> 1 (<<1)

(q)

/J/(

S(S

+1)

)1/2

(q) = 2J(S(S+1))1/2sin(q) (q) > /2Jsin(q)(q) < Jsin(q/2)

S >> 1S >> 1 S = 1/2S = 1/2 S = 1S = 1

2 (q) = 2 + (cq)2

Single mode, no gap(Anderson, 1952)

Continuum, no gap(Bethe, 1931)

Gap at q = (Haldane, 1983)

Spectrum for S=1 is not intermediate case between S=1/2 and S>>1

Understanding antiferromagnetic spin chain: history.Understanding antiferromagnetic spin chain: history.

Understanding S=1 chain: theory.Understanding S=1 chain: theory.

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), ….

Mostly - numerical studies on finite chains.

Understanding S=1 chain: theory.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,)/S(q) is shown.

L.P.Regnault, S. Meshkov and I. Zaliznyak J. Phys. Condens. Matter (Letter),1993Interesting detail: excitations are non-interacting fermions!Interesting detail: excitations are non-interacting fermions!

How could we know? Receipt is simple: take the spectrum, calculate free energy, and compare with measured thermodynamic quantities.

NENP: Haldane gap confirmed, no continuum observed.

Understanding S=1 chain: experiment.Understanding S=1 chain: experiment.

I. A. Zaliznyak, S.-H. Lee, S. V. Petrov, PRL 017202 (2001)

Cross-over from single-mode to continuum in spectrum Cross-over from single-mode to continuum in spectrum of 1D S=1 Heisenberg antiferromagnet observed.of 1D S=1 Heisenberg antiferromagnet observed.

Color contour map of the spectral density of raw magnetic scattering ~ dynamic spin susceptibility

Instability of coherently propagating mode at the top of Instability of coherently propagating mode at the top of the excitation band in Haldane spin chain. 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.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.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 .

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

Acknowledgement

Haldane chain in magnetic field.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

H=0 H~Hc

H>Hc

?particles

holesparticles

I. Zaliznyak, unpublished (2002).

Macroscopic quantum phase in the string operator at H>Hc results in the shift in q-space between fermions and magnons.

Haldane chain in magnetic field: H>HHaldane chain in magnetic field: H>Hcc..

I. Zaliznyak, M. Enderle, C. Broholm, et al, to be published (2002).

Haldane chain in magnetic field H>Hc: Luttinger liquid?Haldane chain in magnetic field H>Hc: Luttinger liquid?

I. Zaliznyak, et al (2002).

H>~Hc

H>Hc

H~Hc

H<Hc

H=0

Chain copper oxides: 1D Mott-Hubbard insulators.Chain copper oxides: 1D Mott-Hubbard insulators.

SrSr22CuOCuO33 SrCuOSrCuO22

Cu-O bond length 1.95 Å, exchange coupling J ~ 0.2-0.3 eV (!)

What is interesting about chain copper oxides?What is interesting about chain copper oxides?

• Electronic band structure and model Hamiltonian for cupratesElectronic 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 separationSpin-charge separationC. 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 susceptibilitylog(T) corrections to low-T magnetic susceptibilityN. Motoyama et al, PRL 76 (1996), K. R. Thurber et al, PRL 87 (2001)

• Heat transport by spinons: giant heat conductance along the Heat transport by spinons: giant heat conductance along the chainschains

A. V. Sologubenko et al, PRB 62 (2000), PRB 64 (2001)

• Ultrafast relaxation of the optical nonlinear absorptionUltrafast relaxation of the optical nonlinear absorptionT. Ogasawara et al, PRL 85 (2000), H. Kishida et al, PRL 87 (2001)

Electronic band structure of chain copper oxides and Electronic band structure of chain copper oxides and spin-charge separation.spin-charge separation.

Essler and Tsvelik cond-mat/0108382

H EH= - tm(cj,+

cj+m, + H.c.) + (U/2) (n jn j- + H.c.) + V(n jn j+1 + H.c.) - |K|SjSj

Effective single-band Hubbard model at half-filling

t1

t2

Ef

k/

kf

vc(k-kf)

vs(k-kf)

Electron spectral function A(k,): holon-spinon continuum

ARPES measurement, C. Kim et al, PRL (1996)

• Band gap 1.5 eV • Exchange J =?

Parametri-zation: U, t

How do we know exchange coupling J?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

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 eVt 0.55 eV

J = 4t2/(U-V) - |K| ~ 0.25-0.36 eV (!?)

J~ 0.5 - 1 meV

Record-high Record-high J, J, record-low record-low JJ/J/J

Infrared absorption below the optical band gap (H. Suzuura et al, PRL (1996))

Two-spinon continuum in SrCuOTwo-spinon continuum in SrCuO22: direct measurement: direct measurementMAPS@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 SrCuOTwo-spinon continuum in SrCuO22: direct measurement: direct measurement

MAPS@ISIS, Ei = 241 meV.Color contour map of the scattering intensity.

qchain/2

Two-spinon continuum in SrCuOTwo-spinon continuum in SrCuO22: direct measurement: direct measurement

MAPS@ISIS, Ei = 520 meV.Color contour map of the scattering intensity.

qchain/2

Two-spinon continuum in SrCuOTwo-spinon continuum in SrCuO22: direct measurement: direct measurement

Best fit: J = 280(20) meV, higher than usually believed.

Agrees with midinfrared absorption result J = 260(10) meV.

qchain/2

ConclusionsConclusions

Structure of the non-hydrodynamic part of the excitation Structure of the non-hydrodynamic part of the excitation spectrum in S=1antiferromagnetic Heisenberg chainspectrum 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 SrCuOSpin excitation spectrum in the Mott-Hubbard insulator SrCuO22

supports evidence for spin-charge separation in -Cu-O- chainssupports 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

What did we learn?

Understanding S=1 chain: theory.Understanding S=1 chain: theory.

What about continuum? No quantitative theory, difficult to measure: structure factor ->0 at small q. Not observed in NENP down to 0.3

2-particle continuum at q=>0 arise in simple fermion “toy model”, Gomez-Santos (1989)

Hierarchy of excited levels after White and Huse (1993)

Somewhere q= single mode should crossover to q=0 continuum.

H = J SiSi+1+ JSiSi D(Si

z)2 , J/J>> 1 (<<1)

J = 26 K, J= 0.033 J, D = 0.0009 J, TN = 4.84 K

•“supercritical” J => not important for spin dynamics at high energies

• the most isotropic of known materials

Model S=1 Haldane chain compound Model S=1 Haldane chain compound CsNiClCsNiCl33..

J

J

New measurement: high-luminosity setups with ASD. 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 SPINS@NG5.NCNR.NIST.GOV Area sensitive detector => 4-6 fold increase in throughput.

Single-mode and continuum spectrum in one-dimensional Single-mode and continuum spectrum in one-dimensional S=1 Heisenberg antiferromagnet (Haldane chain).S=1 Heisenberg antiferromagnet (Haldane chain).

NENP (1992): gap+single mode CsNiCl3 (2001): +continuum

Instability of coherently propagating mode at the top of Instability of coherently propagating mode at the top of the excitation band in two quantum liquids. the excitation band in two quantum liquids. Haldane mode becomes a continuum at q <~ 0.5

Graf, Minkiewicz, Bjerrum Moller, and Passell (1974)

Top of the band “maxon” excitation in superfluid 4He broadens with pressure

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

3D corrections to the 1D excitation spectrum: MF-RPA 3D corrections to the 1D excitation spectrum: MF-RPA estimates.estimates.

Except at low energy, spectrum even at T=0 is 1D

Weak inter-chain coupling of the S=1/2 -Cu-O-Cu- Weak inter-chain coupling of the S=1/2 -Cu-O-Cu- chains: long-range order and correlated spin glass.chains: long-range order and correlated spin glass.

Sr2CuO3: static long-range (Bragg) order

Q=(h,0.5,0.5)Points: magnetic scatteringLine: nuclear scattering

TTNN 5 K 5 K

k h

SrCuO2: decoupling in zigzag ladder results in short-range anisotropic static order

Effect of the inter-chain coupling on spin dynamics in Effect of the inter-chain coupling on spin dynamics in SrCuOSrCuO22. .

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 Weak inter-chain coupling of the S=1/2 chains: static order and effect on spin dynamics.order and effect on spin dynamics.

A. Zheludev et al, cond-mat/0105223. J 24 meV, TN 9 K, <µ> 0.15µB

Magnetic Bragg peak

MagnonA

A

AB

C

C

B

C. Kim et al, PRB 56, 15589 (1997).

Evidence for spin-charge separation in1D Mott-Hubbard Evidence for spin-charge separation in1D Mott-Hubbard insulator: ARPES in chain copper oxide SrCuOinsulator: ARPES in chain copper oxide SrCuO22..

Spinons in chain copper oxides: picosecond relaxation of Spinons in chain copper oxides: picosecond relaxation of optical nonlinearity.optical nonlinearity.

T. Ogasawara et al, PRL 85, 2204 (2000).SrSr22CuOCuO33

Spinons in chain copper oxides: giant heat conductance.Spinons in chain copper oxides: giant heat conductance.

A. V. Sologubenko et al, PRB 64, 054412 (2001).

Future projectsFuture projects

• 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?

Extend collaboration within BNLExtend collaboration within BNL• femtosecond pump-probe measurements infemtosecond pump-probe measurements in SrCuO SrCuO22 / /ChemistryChemistry• gate/photodoping in thin films?gate/photodoping in thin films?