1. Holes in a Quantum Spin Liquid
Collin Broholm
Johns Hopkins University and NIST Center for Neutron Research
Gapped phases in condensed matter
Gapped spin chains
Pure systems
Doped systems
Conclusions
Y3+
I am grateful for opportunity to speak today
An unusual type of magnetism: magnetic ions but large non-magnetic molecule
The new results : impurity induced magnetism
Diverse audience so develop an analogy to doped semiconductors
I encourage you to stop me with questions
viewgraphs are posted on the web
Ca2+
Y2-xCaxBaNiO5
Viewgraphs posted at

2. Collaborators Y2BaNiO5 Copper Nitrate [Cu(NO3)2.2.5D2O]
Guangyong Xu JHU -> University of Chicago
G. Aeppli NEC
J. F. DiTusa Louisiana State University
I. A. Zaliznyak JHU -> BNL
C. D. Frost ISIS
T. Ito Electro-Technical Lab Japan
K. Oka Electro-Technical Lab Japan
H. Takagi ISSP and CREST-JST
M. E. Bisher NEC
M. M. J. Treacy NEC
R. Paul NIST Center for Neutron Research
“Holes in a Quantum Spin Liquid” to appear in Science (2000)
Copper Nitrate [Cu(NO3)2.2.5D2O]
Daniel Reich JHU
M. A. Adams ISIS facility
“Triplet Waves in a Quantum Spin Liquid” to appear in PRL May 1 (2000)
Acknowledge competent collaborators
former student Guangyong Xu who did most of the analysis that you shall see Ito, Oka, and Takagi that grew the crystals that made the experiments possible
A paper describing the main results presented will appear in Science magazine shortly

3. Quantum effects in atoms and in solids
Atom (He)
Line spectrum
G. Dieke et al. (1968)
Solid (Mo)
continuous spectrum
if Q not specified
Long before my time a proud tradition for atomic spectroscopy in the department of physics and astronomy at Johns Hopkins University
Henry Rowland who founded our department invented the diffraction grating used to resolve these spectra.
Later experimentalist such as Gerhard Dieke with the help I am told of 30 graduate students at a time collected spectra from the far infrared to the far ultraviolet for atoms in the gas phase or disolved in various solid matrixes.
In these spectra we see direct evidence for the quantum nature of matter: discrete transitions between atomic levels that produce light of specific or wavelength.
Shown in the figure is the spectrum produced by a helium lamp recorded on a photographic plate using a quartz prizm as monochromator.
Forming a solid the energy levels for outer electrons acquires dispersion as a function of wave vector transfer. Essentially this comes about because of Bragg diffraction of electrons in the periodic lattice structure.
Q must be specified to appreciate these quantum levels and this means that bulk measurements probe crude averages of this intricate structure. While measurements exist that are sensitive to these details they are not versatile so all I can show is a numerical calculation.
But there are circumstances where the nature of the bandstructure has profound effects on the properties of solids.
Christensen
(1980)
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4. Quantum effects from a gap
Empty
Band gap only
possible for even
number of electrons
per cell
Filled
No gap Metal
Gap Insulator
If you ever wondered what the gap means here is the explanation.
It happens in many cases that an energy gap separates filled from empty levels. Such a gap has direct and profound effect on all materials properties.
As an example I show electrical transport properties for a material with and without a gap.
When there is no gap in the spectrum an external electric field can shift the population of these states ever slightly in an amount that depends on the life time of these states. The shifted occupation carries electrical current and we end up with a metal. We show the resistivity of copper versus temperature and see that the resistance decreases to values less than 1uW cm for low T and low level of defect.
In contrast if there is a gap in the spectrum the state is robust and unaffected by a small external electric field. So the zero field state persists. That state is invarient under time reversal symmetry and so carries no electrical current, that is we have an insulator.
The figure shows the conductivity versus inverse temperature in so-called Arhenius plot. The ability of the material to conduct depends entirely upon thermal excitation of electrons across the gap and so is exponentially activated. The gap has profound consequences for all materials property that one might choose to consider.
Because each band can accommodate two electrons per unit cell can only have a band gap when there are an even number of electrons per cell
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5. Impurities create states in the gap
Cleaner sample
10 5
10 7
10 9
1011
Insulator
Resistivity (ohm-cm)
10-3
10-1 10
10 3
Impurities may not be something you normally would like to think about but in gapped systems they lead to interesting and very useful properties.
Impure semiconductors are the basis for the information technological revolution that we are in the midst of.
I am showing electrical transport data for doped silicon as resistivity versus inverse temperature. In clean samples we see the exponentially activated behavior. Perversely the low T resistivity is less for the dirtier samples. Suitable impurities eventually drive the material from insulator to metal.
In the pure system for every electron in the filled band there is another electron moving in the exact opposite direction. Impurities create states in the gap that can donate or accept electrons from the band such that this delicate balance is destroyed and the material can conduct.
All gapped systems have dramatic impurity effects because their unusual properties are a result of perfect cancellation of certain electronic properties in the pure and symmetric state.
Metal
“Dirty” sample
1/T (K-1)
From Aschroft & Mermin
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6. Other gapped phases in condensed matter
High energy physics yields low energy gap
Band insulator
Band semiconductor
Dimerized spin systems
Phase transition to gapped phase
Superconductor
Rotons in superfluid 4He
Mott Metal-Insulator transition
Spin Peierls transition
Cross over to gapped phase for T<<D
Quantum Hall effects
Uniform integer spin chains
In this talk I am trying to draw out the analogies between a vast range of different systems where an energy gap dominates bulk properties.
Looking closer these systems are of course really very different. Specifically there are very different ways that a gap can come about that I have tried to classify in this transparency.
In blue I have pointed out the magnetic analogues of each of these types of gapped phases. In the remainder of the talk we shall focus on the magnetic systems. In doing so we will get tied up in their specific details but it is important to bear in mind the connections to these other physical systems.
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7. The beauty of magnetic dielectrics
Well defined low energy Hamiltonian
Chemistry provides qualitatively different H
Vary H with pressure, magnetic field
Efficient experimental techniques
Exchange interaction
Single ion anisotropy
Dipole in magnetic field (Zeeman)
Insulating magnets are unique model systems for a very wide range of collective phenomena in physics.
Here are reasons for this….
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8. Spin systems of recent interest
To give you a feel for the range of physics that can be explored through experiments in magnetic materials I provide this list that summarizes key properties.
Lattice dimensionality, lattice symmetry, spin quantum number, and spin space anisotropy are the factors that lead to a variety of different ground states and collective behavior in these systems.
You will notice that the dominant exchange constant varies over two or three orders of magnitude in these systems. While the magnitude of J is unimportant for the physics of these materials it has a profound effect on how we must carry out the experiments. By selecting materials with specific values we can facilitate experiments that probe specific aspects of the system.
I would like to dwell for a moment on what we mean when we say that a spin system has a dimensionality less than the three spatial dimensions that we live in.
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9. Varying the dimensionality of spin systems
The crystals are actually three dimensional
Exchange links spins on low-D network only
The materials are real three dimensional solids but the magnetic interactions link lower dimensional networks of spins.
As a result in what we would call a one dimensional magnet we really have an ensemble of chains of magnetic spins that to a good approximation act independently. However we need them all to build enough signal for our experiments to be successful.
Spin chains can be separated by non-magnetic ions in low symmetry structures such as orthorhombic NDMAP shown to the left.
But the isolation of spin chains can occur in materials that are almost cubic such as KCuF3 where the anisotropic electronic orbital of the hole on the copper atom twists and turns leaving only one contiguous path for magnetic interactions.
Now we specialize within this class of materials on those magnets that have a gap in their excitation spectra and we would like see what we can learn about such systems by studying magnetic dielectrics.
So that you can understand the new results that we have obtained concerning impurities in these systems, I will start off by summarizing key experimental properties of spin systems with a gap in their excitation spectra.
Chain direction
Non-magnetic “spacer” molecules in NDMAP
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Anisotropic bonding in cubic KCuF3

10. Susceptibility of gapped spin system
Measure magnetization versus T in infinitesimal field:
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Renard et al (1988)
Tatsuo et al. (1995)

11. Magnetization of gapped spin system
Ajiro et al. (1989)
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12. Specific heat of gapped spin chain
A way to probe the low energy density of states
NENC Orendac et al. (1995)
NENP Tatsuo et al. (1995)
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13. Magnetic Neutron Scattering
The scattering cross section is proportional to the
Fourier transformed dynamic spin correlation function
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14. NIST Center for Neutron Research
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15. JHU 4/26-13/00

16. Detection system on SPINS neutron spectrometer
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17. Neutron scattering from gapped spin chain
Copper nitrate
T=0.3 K
gap
Xu et al.
IRIS, ISIS
Nuclear incoherent scattering
Proton extraction pulse
Triplet creation peak
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18. Dispersion relation for triplet waves
Dimerized spin-1/2 system: copper nitrate
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Xu et al PRL May 2000

19. Why a gap in spectrum of dimerized spin system
A spin-1/2 pair has a singlet - triplet gap:
Weak inter-dimer coupling cannot close gap
Bond alternation is relevant operator for quantum critical uniform spin chain
infinitesimal bond alternation yields gap J
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20. Gapped phases in isotropic spin systems?
n = number of spins per primitive unit cell
S = the spin quantum number
m = the magnetization per spin
n(S-m)=
Oshikawa, Yamanaka, and Affleck (1997) and Oshikawa (2000)
gaps in non-magnetized spin chains?
Alternating spin-1/2 chain 2.1/2= perhaps
Uniform spin-1/2 chain /2 =1/2 no
Uniform spin-1 chain =1 perhaps
Integer: gap possible
Non-Integer: gap impossible
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21. Low T excitations in spin-1 AFM chain
pure
Y2BaNiO5 T=10 K
MARI chain ki
Haldane gap D=8 meV
Coherent mode
S(q,w)->0 for Q->2np
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22. AKLT state for spin-1 chain
Magnets with 2S=nz have a nearest neighbor singlet covering
with full lattice symmetry.
This is exact ground state for spin projection Hamiltonian
Excited states are propagating bond triplets separated from the
ground state by an energy gap
Haldane PRL 1983
Affleck, Kennedy, Lieb, and Tasaki PRL 1987
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23. Sum rules and the single mode approximation
The dynamic spin correlation function obeys sum-rules:
When a coherent mode dominates the spectrum:
Sum-rules link S(q) and e(q)
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24. Single mode approximation for spin-1 chain
Dispersion relation
Equal time
correlation function
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25. Haldane mode in Y2BaNiO5 at finite T
Relaxation rate increases
with T due to
triplet interactions
Resonance energy increases
decreasing correlation length
Low T fine structure
from spin anisotropy
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26. T-dependence of relaxation rate and “resonance” energy
Parameter free comparison:
Semi-classical theory of triplet
scattering by Damle and Sachdev 3 k T æ D ö ( ) G T = B
exp ç - ÷ ç ÷ p è k T ø B
Quantum non linear s model æ D ö ( ) D T = D + 2 p D k T
exp ç - ÷ ç ÷ B è k T ø B
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27. Q-scans versus T: energy resolved and energy integratedh w D
Probing equal time
correlation length h w = D
Probing spatial
coherence of
Haldane mode
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28. Coherence and correlation lengths versus T
Coherence length
exceeds correlation
length for kBT<D
becoming very long
as T
Equal-time
correlation length
saturates at x=8.
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(Solid line from Quantum non linear s model)

29. Pure quantum spin chains - at zero and finite T
Gap is possible when n(S-m) is integer
gapped systems: alternating spin-1/2 chain, integer chain,…
gapless systems: uniform spin-1/2 chain
gapped spin systems have coherent collective mode
For appreciable gap SMA applies: S(q) ~ 1/e(q)
Thermally activated relaxation due to triplet interactions
Thermally activated increase in resonance energy
Coherence length exceeds correlation length for T< D/kB
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30. Impurities in Y2BaNiO5 Mg2+on Ni2+ sites finite length chains
Ca2+ on Y3+ sites mobile bond defects Mg
Ca2+ Ni
Y3+
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Kojima et al. (1995)

31. Zeeman resonance of chain-end spins20
g=2.16
hw (meV) 15
H (Tesla) 10
I(H=9 T)-I(H=0 T) (cts. per min.) -5
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32. Form factor of chain-end spins
Y2BaNi1-xMgxO5 x=4%
Q-dependence reveals
that resonating object
is AFM.
The peak resembles
S(Q) for pure system.
Chain end spin carry
AFM spin polarization
of length x back into
chain
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33. New excitations in Ca-doped Y2BaNiO5
Pure
9.5% Ca
Y2-xCaxBaNiO5:
Ca-doping
creates states
below the gap
sub-gap states
have doubly
peaked structure
factor
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34. Why a double ridge below the gap in Y2-xCaxBaNiO5 ?
Charge ordering yields incommensurate spin order
Quasi-particle Quasi-hole pair excitations in Luttinger liquid
Anomalous form factor for independent spin degrees of freedom associated with each donated hole d q x q d
is single
impurity prop.
Indep. of x
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35. Does dq vary with calcium concentration?
dq not strongly dependent on x
Double peak is
single impurity effect
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36. Bond Impurities in a spin-1 chain: Y2-xCaxBaNiO5
(f)
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37. Form-factor for FM-coupled chain-end spins
A symmetric AFM droplet
Ensemble of independent
randomly truncated AFM droplets

38. Calcium doping Y2BaNiO5 Experimental facts:
Ca doping creates sub-gap excitations with doubly peaked structure factor and bandwidth
The structure factor is insensitive to concentration and temperature for 0.04<x<0.14 (and T<100 K)
Analysis:
Ca2+ creates FM impurity bonds which nucleate
AFM droplets with doubly peaked structure factor
AFM droplets interact through intervening chain forming disordered random bond 1D magnet
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39. Incommensurate modulations in high TC superconductors
YBa2Cu3O6.6 T=13 K E=25 meV
h (rlu)
k (rlu)
Hayden et al. 1998
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40. What sets energy scale for sub gap scattering ?
Possibilities:
Residual spin interactions through
Haldane state. A Random bond AFM.
Hole motion induces additional
interaction between static AFM droplets
AFM droplets move with holes:
scattering from a Luttinger liquid of
holes. 10
hw (meV) 5 ?
How to distinguish:
Neutron scattering in an applied field
Transport measurements
Theory
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41. Conclusions:
Dilute impurities in the Haldane spin chain create sub-gap composite spin degrees of freedom.
Edge states have an AFM wave function that extends into the bulk over distances of order the Haldane length.
Holes in Y2-xCaxBaNiO5 are surrounded by AFM spin polaron with central phase shift of p
Neutron scattering can detect the structure of composite impurity spins in gapped quantum magnets.
The technique may be applicable to probe impurities in other gapped systems eg. high TC superconductors.
Microscopic details of gapped spin systems may help understand related systems where there is no direct info.
Viewgraphs posted at
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