Holes in a quantum spin liquid
*
+
+
Guangyong Xu , G. Aeppli , M.E. Bisher ,
C. Broholm*,§, J. F. DiTusa, C. D. Frost#,
T.Ito**, K. Oka**, H. Takagi++& M.M.J. Treacy+
*
Department of Physics and Astronomy, Johns Hopkins University, Baltimore,
Maryland 21218, USA
+
NEC Research Institute, 4 Independence Way, Princeton, New Jersey 08540, USA
§
NIST Center for Neutron Research, Gaithersburg, Maryland 20899, USA
||
Department of Physics and Astronomy, Louisiana State University, Baton Rouge,
Louisiana 70803, USA
#
ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX,
UK
**
Electrotechnical Laboratory, Tsukuba 305, Japan
++
The Institute of Solid State Physics, University of Tokyo, Roppongi, Tokyo 106
Japan
Because of possible connections to high temperature
superconductivity, spin density modulations in transition
metal oxides are receiving huge attention. The modulations
appear upon introduction of charge carriers, via chemical
substitution, into an insulating and antiferromagnetic parent
compound. They tend to be static when the carriers are frozen,
dynamic when they are mobile. Until now, the evidence for
such modulations has been largely confined to lamellar
compounds1,2,3,4, i.e. materials whose magnetism and charge
transport are essentially two-dimensional. It has also been
restricted to materials whose parent insulators are ordered
antiferromagnets. In this Letter, we provide the first evidence
for analogous phenomena in a doped transition metal oxide
chain compound5, Y2-xCaxBaNiO5, for which the parent is a
spin liquid by virtue of quantum fluctuations6,7,8,9and which
has motivated considerable theoretical activity10,11,12,13,14,15
The key features of this orthorhombic material are the chains
of NiO6 octahedra shown in Fig. 1(a). The octahedra are cornerrather than edge-sharing, which results in the dominance of the
magnetism by the very simple ..O-Ni-O-Ni-O.. back-bone. The
electrical conduction is also primarily along the chains, as
indicated by the more than three orders of magnitude smaller
resistivity recorded along the crystallographic a axis, which is
parallel to the chains, than along the perpendicular b and c axes5.
The orbital moments on the Ni2+ ions are quenched, so that the
magnetic degree of freedom at each Ni site is the spin S=1
associated with the d8 configuration of Ni2+. Each S=1 ion is
coupled to its neighbors via antiferromagnetic super-exchange
through intermediate O2- ions. Replacing the off-chain Y3+ ions
by Ca2+ ions introduces holes primarily onto apical oxygen atoms
in the chains, and increases the electrical conductivity. Thus, Y2xCaxBaNiO5 is a one-dimensional analog of the cuprates, where
off-(CuO)-plane chemical impurities donate holes to the CuO
planes.
Magnetic one-dimensionality causes the x=0 parent
compound of the Y2-xCaxBaNiO5 series to be a spin liquid
prevented from ordering antiferromagnetically by quantum
fluctuations. The material is not an ordinary paramagnet with
heavily damped spin fluctuations, but rather it is the magnetic
analog of super-fluid 4He in the sense that it has a macroscopically
coherent quantum ground state even while lacking classically
discernible order. Affleck, Kennedy, Lieb and Tasaki (AKLT)
provided a prescription for this state which we have illustrated in
Fig. 1(b) 16,17. They considered each S=1 degree of freedom as the
triplet ground state formed between two ferromagnetically coupled
S=1/2 degrees of freedom, in accord with atomic physics which
posits a large and positive Hund’s rule coupling between the two
S=1/2 holes in the d-shell of Ni2+. The S=1 antiferromagnetic
chain can thus be redrawn as an S=1/2 chain with twice the
number of sites and bonds of alternating sign (see Fig 1(a)). If we
could tune the ferromagnetic Hund coupling to a value well below
the antiferromagnetic interaction between different S=1 sites, the
ground state would be a chain of singlets formed between S=1/2
spins on neighboring sites. A gap equal to the antiferromagnetic
coupling J would separate the ground state bond singlets from the
excited state triplets. A large body of work performed over the last
15 years shows that the AKLT ground and excited states evolve
adiabatically into the Haldane ground state as the ferromagnetic, or
Hund’s rule, coupling is taken to much larger values than J. The
major change is that for energies below the intra-atomic exchange
energy, S=1/2 degrees of freedom are never observed singly (i.e.
they are confined much as the fractional charges of quarks are
confined), and that the triplet band exhibits dispersion, with a
minimum energy, called the Haldane gap18 =0.41 J.
Fig 2(a) shows that the triplet band can be observed
experimentally for Y2BaNiO5. The figure, an image of the
magnetic fluctuations as a function of wave-number (along the
chains) Q and energy  , was obtained using the multi-detector
time-of-flight neutron scattering spectrometer MARI at the pulsed
spallation source ISIS in the UK. Magnetic fluctuations are found
only when  and Q lie on a well-defined dispersion relation,
which is direct evidence for a coherent propagating triplet mode.
What distinguishes this mode from a spin wave is that it
propagates in a medium with no static magnetic order. Key
features are (i) a 9 meV Haldane gap, which corresponds to the
threshold energy for triplet creation, (ii) the magnetic excitation
bandwidth of 62 meV that translates into an antiferromagnetic
exchange constant J=21 meV, and (iii) the vanishing intensity for
Q=n2 that is evidence for an isotropic singlet ground state.
What happens when we add carriers by Ca doping the
quantum spin liquid of Fig 2 (a)? Fig 2 (b), collected for our
x  0.095  0.005 sample, reveals excitations following the same
underlying dispersion relation as for pure Y2BaNiO5. However
near the former minimum at Q= and  =9 meV, their intensity
is substantially reduced and new magnetic scattering appears
below the Haldane gap. Fig. 3 (b) shows more detailed data
collected near and below the gap. Near the gap energy, the
scattering is clearly peaked at the commensurate Q= position but
for 3-7 meV it is most intense along two vertical lines, displaced
symmetrically to the left and right of Q=.
In addition to the energy-resolved data of Fig.3, we have
obtained higher statistics energy-integrated Q-scans to better
quantify the incommensurate peaks. Fig. 4 shows the results for
two doped single crystals. Increased doping results in a peak
splitting which is only slightly larger for the x=0.14 sample.
Specifically, if we fit both data sets by the sum of two Lorentzians
centered at  with equal amplitudes and half-widths-at-halfmaximum , we find that all parameters after division by  are of
the same order and also comparable to the impurity density x.
More specifically, =0.085(3) and 0.095(4), and =0.071(4)
and 0.072(5) for x=0.095(5) and 0.14(1) respectively.
There are various models which will produce the
incommensurate fluctuations we have discovered in Y2xCaxBaNiO5. The first proceeds from a charge-ordering hypothesis
where the holes order periodically to minimize their mutual
electrostatic repulsion and the spin system acquires a period
commensurate to that of the hole lattice. The magnetic fluctuations
are then spin waves broadened as a function of wave-number
because the hole lattice is somewhat disordered. Such a scenario is
commonly supposed for the two-dimensional analog of Y2xCaxBaNiO5, namely doped La2NiO4. Key experimental support
was from electron19 and neutron diffraction20,21 measurements,
which revealed the lattice distortions that might follow from hole
ordering. Thus inspired, we have used electron diffraction to
search for hole ordering in Y2-xCaxBaNiO5 down to 15K. The
failure of the search makes the charge ordering explanation of our
experiments unlikely. Furthermore, in the simplest picture, the hole
density has the same period as the spin density squared, so that the
hole density 1/x=/, i.e. one half the wavelength for the spin
density wave. The experimental values for  imply spin density
wavelengths which are 10% and 40% too long for x=0.095 and
0.014 respectively.
A second possibility is to identify the incommensurate
magnetic fluctuations with electron-hole pair excitations in the
one-dimensional hole liquid associated with mobile S=1/2 holes
dressed by their propagation through the Haldane paramagnet10.
The wave-vectors  would be vectors spanning the Fermi
surface while the vertical nature of the incommensurate streaks in
Fig. 3 would imply a Fermi velocity in excess of 0.5 eVÅngstrom.
The problem with this picture is not so much that the sample
resistitivities are high at low T – the resistivity data5 are consistent
with metallic behavior on the scale of the Haldane length - as that
the Fermi sea is expected to grow in proportion to x, a result again
contradicted by the almost negligible increase in  on raising x
from 0.095 to 0.14.
A last possibility starts from consideration12 of the magnetic
structure factor for a single static hole donated to the NiO chain by
the Ca ions. Because it carries a spin, a localized hole on a superexchange mediating oxygen atom favors parallel alignment of
neighboring Ni spins and hence induces an effective ferromagnetic
nearest neighbor interaction. Thus, the infinite chain is divided into
two semi-infinite segments, and the ground and excited state
wavefunctions can be constructed from a basis set which is the
direct product of wavefunctions for the left and right segments and
for the hole on the intervening oxygen. The ground state
wavefunction for a semi-infinite segment is a doublet rather than
the singlet wavefunction for the unterminated infinite chain.
Fig. 1(b)-(c) give a pictorial justification: the chain ends have
unpaired spins because only one of the two AKLT spins which
together represent the S=1 sites can be paired with an AKLT spin
on a neighboring spin S=1 site. As we move from the alternating
valence bond representation to the real S=1 chain, the doublet
wavefunction is not confined to the chain end, but extends into the
chain over a distance of order the finite magnetic correlation length
associated with the Haldane gap. A ferromagnetic bond inserted
between two semi-infinite chains combines the doublets found on
either side of the bond into a triplet. To the left or right of the
ferromagnetic bond, the spin polarization is antiferromagnetic with
a characteristic decay length =-1. The net result is that the
correlations <S0Sj> between spins at site 0 and site j, where the
ferromagnetic bond occurs between sites -1 and 0, are (-1)je-|j|/
for j>0, 1 for j=0 and -1, and (-1)j+1e-|j+1|/ for j<-1.
Neutron scattering measures the Fourier transform of the spin
polarization, which for the triplet ground state associated with a
single ferromagnetic bond (neglecting the spin on the oxygen), is
S (Q )  F (Q )
(1  e ) cos Q / 2
cosh   cos Q
2
(1)
Eq. (1) describes a function with nodes at odd multiples of 
and pairs of peaks, with half widths and incommensurability of
order . Thus, the description leading to Eq. (1) does have a
chance of accounting for a key feature of our experiments. Even
so, there are two clear difficulties, the first being that the nodes
predicted by (1) are not seen experimentally. The second problem
is that because Eq. (1) describes the ground state for a single
ferromagnetic bond, neutron scattering should not involve any
energy loss, implying that the scattering should be elastic rather
than inelastic as seen experimentally.
The first difficulty has a simple resolution. At finite hole (and
therefore ferromagnetic bond) densities, the polarization clouds
associated with the holes overlap and the expression (1), derived
for isolated impurities, becomes invalid. A crude model, which
considers the overlaps, simply truncates the polarization clouds at
neighboring impurity sites. Because the bonds are randomly
distributed, the inversion symmetry characterizing isolated
impurities is broken and intensity becomes allowed at Q=. The
only parameters in such a description are the FM bond density and,
as for Eq (1), the extent of the polarization cloud 1/. The first is
equated with the measured Ca concentration. We adjusted the
second to optimize the fit of the corresponding structure factor to
our data. We also allowed for a spin on the oxygen atom whose
main effect is to change the relative intensity of the two peaks.
Best agreement with the data was obtained for an oxygen dipole
moment O=0.01(1)B antiferromagnetically correlated with the
neighboring Ni spin. As shown by the red lines in Fig. 4 the model
provides a good account of the data for = -1 =7.5(4) and 7.4(4)
for the two concentrations. Both values for  are close to the equal
time spin correlation length, 6 lattice spacings22, for the unbroken
spin-1 chain.
How do we account for the inelasticity of the incommensurate
signal? One approach is to view the chain as consisting not of the
original S=1 degrees of freedom from the Ni2+ ions but of the
effective spin degrees of freedom associated with the randomly
placed holes. The latter interact via the overlapping tails of their
antiferromagnetic polarization clouds, which will imply effective
couplings of random sign because the distances between two holes
are as probable to be even as odd multiples of the Ni-Ni separation.
Neglecting quantum fluctuations, the ground state is likely to be a
spin glass, as deduced from other experiments23on Y2-xCaxBaNiO5.
The incommensurate nature of the excitations simply follows from
the structure factor of the composite spin induced by a hole and
should dissolve only when these effective degrees of freedom are
themselves dissolved as either the Haldane gap energy or the
ferromagnetic bond strength is exceeded.
In summary, we have performed the first measurements of the
magnetic fluctuations in single crystals of a doped spin-1 chain
with antiferromagnetic couplings. At high energies, well above the
Haldane gap the triplet excitations associated with the spin liquid
ground state of the parent compound persist with doping.
However, below the Haldane gap, new excitations appear with a
broad spectrum and characteristic wave vectors, which are,
displaced from the zone boundary by an amount of order the
inverse correlation-length in the bulk spin chains. Our results are
significant in realms beyond that of nickel chain compounds. This
follows from the strong similarities of our data to those for the
superconducting cuprates, where we also see incommensurate
fluctuations at low energies and remnant spin-wave-like signals at
high energies. The one-dimensionality makes Y2BaNiO5 much
easier to model and our analysis reveals that “incommensurate”
peaks arise naturally even without hole order because of the
characteristic spin density modulation that develops around a
defect in the singlet ground state of a quantum magnet. This
phenomenon accounts for the saturation of the incommensurability
with increased doping in the one-dimensional nickelate, and could
account for the same phenomenon in the two-dimensional
cuprates. Indeed, the one-dimensional nickelate gives us our first
quantitative impression of the magnetic polarization cloud
associated with the holes in a doped transition metal oxide. Our
results imply that the spin part of the hole wave-function is actually
the edge state nucleated by the hole in a quantum spin fluid.
1
Cheong, S.-W. et al. Incommensurate magnetic fluctuations in La2-xSrxCuO4. Phys.
Rev. Lett. 67, 1791-1794 (1991).
2
Hayden, S. M. et al. Incommensurate magnetic correlations in La1.8Sr0.2NiO4. Phys.
Rev. Lett. 68, 1061-1064 (1992).
3
Mook, H. A. et al. Spin fluctuations in YBa2Cu3O6.6. Nature 395, 580-582 (1998).
4
Tranquada, J. M., Sternlieb, B. J., Axe, J.D., Nakamura, Y. & Uchida, S. Evidence
for stripe correlations of spins and holes in copper oxide superconductors. Nature
375, 561-563 (1995).
5
DiTusa, J. F. et al. Magnetic and charge dynamics in a doped one-dimensional
transition metal oxide. Phys. Rev. Lett. 73, 1857-1860 (1994).
6
Darriet, J. & Regnault, L.P. The compound Y2BaNiO5: a new example of a
Haldane gap in a S=1 magnetic chain. Solid State Commun. 86, 409-412 (1993).
7
DiTusa, J. F. et al. One-dimensional spin fluctuations in a transition metal oxide.
Physica B 194-196, 181-183 (1994).
8
Xu, G. et al. Y2BaNiO5: a nearly ideal realization of the S=1 Heisenberg chain with
antiferromagnetic interactions. Phys. Rev. B 54, R6827-6830 (1996).
9
Sakaguchi, T., Kakurai, K., Yokoo, T. & Akimitsu, J. Neutron scattering study of
magnetic excitations in the spin S=1 one-dimensional Heisenberg antiferromagnet
Y2BaNiO5. J. Phys. Soc. Japan 65, 3025-3031 (1996).
10
Dagotto, E., Riera, J., Sandvik, A., & Moreo, A. Spin dynamics of hole doped Y2-x
CaxBaNiO5 Phys. Rev. Lett. 76, 1731-1734 (1996).
11
Lu, Z.-Y., Su, Z.-B., & Yu, L. Doping-induced quantum bound states within the
Haldane gap. Phys. Rev. Lett. 74, 4297-4300 (1995).
12
Penc,K. & Shiba,H. Propagating S=1/2 particles in S=1 Haldane-gap systems.
Phys. Rev. B 52, R715-R718 (1995).
13
Fujimoto, S. & Kawakami, N. Weak-coupling approach to hole-doped S=1
Haldane systems. Phys. Rev. B 52, 6189-6192 (1995).
14
Sorensen, E. S. & Affleck, I. Impurities in s=1 Heisenberg antiferromagnets.
Phys.Rev. B 51, 16115-16127 (1995).
15
Hyman, R. A., Yang, K., Bhatt, R.N. & Girvin, S.M. Random Bonds and
Topological Stability in Gapped Quantum Spin Chains. Phys. Rev. Lett. 76, 839-842
(1996).
16
Affleck, I., Kennedy, T., Lieb, E. H. & Tasaki, H. Rigorous results on valencebond ground states in antiferromagnets. Phys. Rev. Lett. 59, 799-802 (1987).
17
Kennedy, T. Exact diagonalisations of open spin-1 chains. J. Phys. Cond. Matter
2, 5737-5745 (1990).
18
Haldane, F. D. M. Continuum dynamics of the 1-D Heisenberg antiferromagnet:
identification with the O(3) nonlinear sigma model. Phys. Lett. 93A, 464-468
(1983).
19
Chen, C. H., Cheong, S.-W., & Cooper, A. S. Charge modulations in La2xSrxNiO4+y: ordering of polarons. Phys. Rev. Lett. 71, 2461-2464 (1993).
20
Tranquada, J.M., Buttrey, D.J., Sachan, V. & Lorenzo, J.E. Simultaneous ordering
of holes and spins in La2NiO4.125. Phys. Rev. Lett. 73, 1003-1006 (1994).
21
Lee, S.-H. & Cheong, S.-W. Melting of quasi-two-dimensional charge stripes in
La5/3Sr1/3NiO4. Phys. Rev. Lett. 79, 2514-2517 (1997)
22
White, S. R. & Huse, D. A. Numerical renormalization-group study of low-lying
eigenstates of the antiferromagnetic S=1 Heisenberg chain. Phys. Rev. B 48, 38443852 (1993).
23
Kojima, K. et al. Muon spin relaxation and magnetic susceptibility measurements
in the Haldane system Y2-xCaxBaNi1-yMgyO5. Phys. Rev. Lett. 74, 3471-3474 (1995).
Acknowledgements. Work at JHU and NIST was supported by the US National
Science Foundation (NSF). Work at LSU was supported by the NSF and the State of
Louisiana Board of Regents. We thank T. M. Rice, A. J. Millis, and Q. Huang for
useful discussions, A. Krishnan for help with TEM measurements, and Rick Paul for
performing neutron activation analysis.
Correspondence and requests for materials should be addressed to C. B. (e-mail:
broholm@jhu.edu)
Figure 1. (a) Chain unit of Y2BaNiO5 extending along the a-direction and featuring
nickel atoms with octahedral oxygen coordination and Ca2+ impurities on Y3+ sites.
(b) Schematic of spin-1/2 degrees of freedom at the edges of the AKLT state. (c)
Ferromagnetically coupled chain end spin-1/2 degrees of freedom.
Figure 2. Coarse-resolution overview of the magnetic fluctuations in (a) pure and
(b) 9.5% doped Y2BaNiO5 measured at T=10K. The initial neutron beam energy
was 90 meV and the sample was oriented with its chain axis perpendicular to the
incident beam direction. The boxes indicate the region of the energy-wave-vector
phase space examined in the high-resolution survey of Fig.3
Figure 3. Low energy detail of magnetic excitations in (a) pure and (b) 9.5%
doped Y2BaNiO5. (a) shows MARI data at T=10K while (b) shows data
measured at T=1.5K, and the spectrometers used were SPINS and BT2 at NIST
with final neutron energies fixed at 5 and 14.7 meV respectively.
Figure 4. Sweeps as a function of wave-vector through the incommensurate peaks
at fixed energy transfer of 4.6 meV for two different hole concentrations. The
energy resolution of the spectrometer was 2 meV FWHM. The yellow line in the
top frame shows the single impurity model Eq. (1). The red lines take into account
that neighboring impurities truncate the spin polarization cloud around an impurity
bond. The green lines show the dual-Lorentzian model.