Excitation continuum in 1D S1 Heisenberg antiferromagnet Haldane chain'

1
Quantum spin dynamics in 1D antiferromagnets.
Igor Zaliznyak
Neutron Scattering Group
Outline

• Excitation continuum in 1D S1 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 S1/2 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,

2
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
3
Heisenberg model systems why bother?
H JS SiSi1 J?S SiSiD? DS(Siz)2 , J/J?gtgt 1
(ltlt1)
1D

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

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
4
Understanding antiferromagnetic spin chain
history.
Spectrum for S1 is not intermediate case between
S1/2 and Sgtgt1
S gtgt 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) gt p/2 Jsin(q) e(q) lt p Jsin(q/2)
e(q) 2J(S(S1))1/2sin(q)
e2 (q) D2 (cq)2
e(q)/J/(S(S1))1/2
5
Understanding S1 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), .

6
Understanding S1 chain theory.
Quantum Monte-Carlo results for 128-spin chain
indeed indicated existence of a continuum.
QMC by S. Meshkov (1993).
Since S(q)gt0 at q0, S(q,w)/S(q) is shown.
7
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.
8
Understanding S1 chain experiment.
NENP Haldane gap confirmed, no continuum
observed.
9
Cross-over from single-mode to continuum in
spectrum of 1D S1 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
10
Instability of coherently propagating mode at the
top of the excitation band in Haldane spin chain.
Haldane mode becomes a continuum at q lt 0.5?
J 2.275(5) meV, v 2.49(4)J
11
Spectral density measured in two configurations.
Resolution is better and more round in the
E-resolved 2-axis mode. Continuum confirmed,
starts at q lt 0.6?.
12
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 S1 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 lt
  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
13
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 HgtHc results in the shift in q-space between
fermions and magnons.
H0
HHc
?
particles
particles
holes
I. Zaliznyak, unpublished (2002).
HgtHc
14
Haldane chain in magnetic field HgtHc.
I. Zaliznyak, M. Enderle, C. Broholm, et al, to
be published (2002).
15
Haldane chain in magnetic field HgtHc Luttinger
liquid?
H0
HgtHc
HltHc
HgtHc
HHc
I. Zaliznyak, et al (2002).
16
Chain copper oxides 1D Mott-Hubbard insulators.
Cu-O bond length ? 1.95 Å, exchange coupling J
0.2-0.3 eV (!)
Sr2CuO3
SrCuO2
17
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)

18
Electronic band structure of chain copper oxides
and spin-charge separation.
Effective single-band Hubbard model at
half-filling
H EH - S tm(cj,? cjm,? H.c.) (U/2) S(n j?n
j-? H.c.) VS(n jn j1 H.c.) - KS SjSj1
Electron spectral function A(k,?) holon-spinon
continuum
Parametri-zation U, t
vc(k-kf)
vs(k-kf)

• Band gap ??1.5 eV
• Exchange J ?

?
Ef
kf
k/?
ARPES measurement, C. Kim et al, PRL (1996)
Essler and Tsvelik cond-mat/0108382
19
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
20
Two-spinon continuum in SrCuO2 direct measurement
MAPS_at_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?
21
Two-spinon continuum in SrCuO2 direct measurement
MAPS_at_ISIS, Ei 241 meV. Color contour map of the
scattering intensity.
qchain/2?
22
Two-spinon continuum in SrCuO2 direct measurement
MAPS_at_ISIS, Ei 520 meV. Color contour map of the
scattering intensity.
qchain/2?
23
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?
24
Conclusions
What did we learn?

• Structure of the non-hydrodynamic part of the
  excitation spectrum in S1antiferromagnetic
  Heisenberg chain
• cross-over from the coherent propagating
  Haldane-gap mode to continuum occurs at qlt0.6?
• 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 gt rethink/redo log
  corrections
• large energy scale and fractional nature of
  excitations results in fascinating physical
  properties

25
Understanding S1 chain theory.
Somewhere qp single mode should crossover to q0
continuum.
2-particle continuum at qgt0 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 -gt0 at
small q. Not observed in NENP down to 0.3?
26
Model S1 Haldane chain compound CsNiCl3.
H JS SiSi1 J?S SiSiD? DS(Siz)2 , J/J?gtgt 1
(ltlt1)
J 26 K, J? 0.033 J, D 0.0009 J, TN 4.84 K
J
J?

• supercritical J? gt not important for spin
  dynamics at high energies
• the most isotropic of known materials

27
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.
28
SPINS_at_NG5.NCNR.NIST.GOV
Area sensitive detector gt 4-6 fold increase in
throughput.
29
Single-mode and continuum spectrum in
one-dimensional S1 Heisenberg antiferromagnet
(Haldane chain).
NENP (1992) gapsingle mode
CsNiCl3 (2001) continuum
30
Instability of coherently propagating mode at the
top of the excitation band in two quantum
liquids.
Haldane mode becomes a continuum at q lt 0.5?
Top of the band maxon excitation in superfluid
4He broadens with pressure
Graf, Minkiewicz, Bjerrum Moller, and Passell
(1974)
31
3D corrections to the 1D excitation spectrum
MF-RPA estimates.
Except at low energy, spectrum even at T0 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? lt q lt 0.8 ?, and within
  5 for 0.3? lt q lt 0.7 ?

32
Weak inter-chain coupling of the S1/2 -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
Sr2CuO3 static long-range (Bragg) order
SrCuO2 decoupling in zigzag ladder results in
short-range anisotropic static order
33
Effect of the inter-chain coupling on spin
dynamics in SrCuO2.
J ? 280 meV, TN? 5 K, ltµgt? 0.15µB I. Zaliznyak
et al, PRL 83, 5370 (1999).

• Extremely weak coupling between S1/2
  antiferromagnetic spin chains in Sr2CuO3 and
  SrCuO2 results in static order but marginal
  modulation of the inelastic spectrum.

34
Weak inter-chain coupling of the S1/2 chains
static order and effect on spin dynamics.
A. Zheludev et al, cond-mat/0105223. J ? 24 meV,
TN? 9 K, ltµgt? 0.15µB
A
C
A
B
B
Magnon
A
C
Magnetic Bragg peak
35
Evidence for spin-charge separation in1D
Mott-Hubbard insulator ARPES in chain copper
oxide SrCuO2.
C. Kim et al, PRB 56, 15589 (1997).
36
Spinons in chain copper oxides picosecond
relaxation of optical nonlinearity.
T. Ogasawara et al, PRL 85, 2204 (2000).
Sr2CuO3
37
Spinons in chain copper oxides giant heat
conductance.
A. V. Sologubenko et al, PRB 64, 054412 (2001).
38
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?