holes in a quantum spin liquid gapped phases in condensed matter gapped spin chains pure systems...
TRANSCRIPT
Holes in a Quantum Spin Liquid
Gapped phases in condensed matterGapped spin chains
Pure systems Doped systems
Conclusions
Collin BroholmJohns Hopkins University and NIST Center for Neutron Research
Viewgraphs posted at http://www.pha.jhu.edu/~broholm/canada/index.htm
Ca2+ Y2-xCaxBaNiO5
Ca2+
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CollaboratorsGuangyong Xu JHU -> University of ChicagoG. Aeppli NECJ. F. DiTusa Louisiana State University I. A. Zaliznyak JHU -> BNLC. D. Frost ISIST. Ito Electro-Technical Lab JapanK. Oka Electro-Technical Lab JapanH. Takagi ISSP and CREST-JST M. E. Bisher NECM. M. J. Treacy NECR. Paul NIST Center for Neutron Research
Publication on Y2-xCaxBaNiO5 to appear in Science (2000)
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Quantum effects in atoms and in solids
Atom (He)Line spectrum
Solid (Mo)continuous spectrumif Q not specified
G. Dieke et al. (1968)
Christensen(1980)
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Quantum effects from a gap
No gap Metal
Gap Insulator
Band gap only possible for even number of electrons per cell
Filled
Empty
Te
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Impurities create states in the gap
0 0.2 0.4 0.6 0.8 1/T (K-1)
Res
isti
vity
(oh
m-c
m)
10-3
10-1
10
10 3
10 5
10 7
10 9
1011
“Dirty” sample
Cleaner sampleTe
1
Insulator
Metal
From Aschroft & Mermin
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Other gapped phases in condensed matter
High energy physics yields low energy gap Band insulator Band semiconductorDimerized spin systems
Phase transition to gapped phase Superconductor Rotons in superfluid 4He Mott Metal-Insulator transitionSpin Peierls transition
Cross over to gapped phase for T<< Quantum Hall effectsUniform integer spin chains
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The beauty of magnetic dielectrics
Well defined low energy Hamiltonian
Chemistry provides qualitatively different HVary H with pressure, magnetic fieldEfficient experimental techniques
lllB
l
zl
llll
ll
g
SD
J
SH
SS
2
''
'H Exchange interaction
Single ion anisotropy
Dipole in magnetic field (Zeeman)
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Spin systems of recent interest
Material Dim. Lattice Spin Anisot. J(meV)
T=0 state
Y2BaNiO5 1 chain 1 no 25 spin liq.NENP 1 chain 1 planar 4.1 spin liq.Cu-Nitrate 1 dimerized 1/2 no 0.5 spin liq.PHCC 2 dimerized 1/2 no 3 spin liq.La2CuO4 2 square 1/2 no 132 NeelK2NiF4 2 square 1 no NeelCr-Jarosite 2 kagome' 3/2 no 1.2 NeelMnF2 3 tetragonal 5/2 easy axis NeelZnCr2O4 3 spinel 3/2 no 4 NeelY2Mo2O7 3 pyrochlore 1 no 1-10 spin glass
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Varying the dimensionality of spin systems
The crystals are actually three dimensionalExchange links spins on low-D network only
Non-magnetic “spacer” molecules in NDMAP
Anisotropic bonding in cubic KCuF3
Chain
dir
ect
ion
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Susceptibility of gapped spin system
l
llB
B SSTk
g
H
M
2
Measure magnetization versus T in infinitesimal field:
Tatsuo et al. (1995)Renard et al (1988)
Te
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Magnetization of gapped spin system
Ajiro et al. (1989)
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Specific heat of gapped spin chain
A way to probe the low energy density of states
1/
TEe
dEE
dT
d
TC
H
NENC Orendac et al. (1995) NENP Tatsuo et al. (1995)
E
TeC
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Magnetic Neutron Scattering
fi kkQ
fi EE
The scattering cross section is proportional to the Fourier transformed dynamic spin correlation function
ik fk
Q
2
''R
)'( )0(S)(S1
2
1),(
RRR
RRQiti teN
edtQ
S
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NIST Center for Neutron Research
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Detection system on SPINS neutron spectrometer
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Neutron scattering from gapped spin chain
gap
Nuclear incoherent scattering Triplet creation peakProton extraction pulse
Copper nitrateT=0.3 K
Xu et al. IRIS, ISIS
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Dispersion relation for triplet waves
Dimerized spin-1/2 system: copper nitrate
JTkB
Xu et al PRL May 2000
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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 chaininfinitesimal bond alternation yields gap
Why a gap in spectrum of dimerized spin system
J0totS
1totS
JJ
J
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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=1 perhaps Uniform spin-1/2 chain 1.1/2 =1/2 no Uniform spin-1 chain 1.1 =1 perhaps
Integer: gap possible
Non-Integer: gap impossible
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Low T excitations in spin-1 AFM chain
Y2BaNiO5 T=10 KMARI chain ki
•Haldane gap =8 meV
•Coherent mode
•S(q,)->0 for Q->2n
pure
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AKLT state for spin-1 chain
• This is exact ground state for spin projection Hamiltonian
• Magnets with 2S=nz have a nearest neighbor singlet covering with full lattice symmetry.
• Excited states are propagating bond triplets separated from the ground state by an energy gap .J
Haldane PRL 1983Affleck, Kennedy, Lieb, and Tasaki PRL 1987
i
iii
iiiii
toti SP 12
131
12 SSSSSSH
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Sum rules and the single mode approximation
'cos1SS1
3
2),(
)(SS1
),(
'
2
)'(
''
ddqJN
qSd
qSeN
qSd
dddd
dd
ddqi
ddd
d
When a coherent mode dominates the spectrum:
)(
'cos1SS1
3
2
)(
),()(
'2
q
ddqJN
q
qSdqS
dddd
dd
Sum-rules link S(q) and (q)
)()(),( qqSqS
The dynamic spin correlation function obeys sum-rules:
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Single mode approximation for spin-1 chain
Dispersion relation
Equal time correlation function
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Haldane mode in Y2BaNiO5 at finite T
•Relaxation rate increases with T due to triplet interactions
•Resonance energy increases with T due to decreasing correlation length
Low T fine structure from spin anisotropy
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Q-scans versus T: energy resolved and energy
integrated
Probing spatialcoherence of Haldane mode
Probing equal time correlation length
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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/(q)
Thermally activated relaxation due to triplet interactions
Thermally activated increase in resonance energy
Coherence length exceeds correlation length for T< /kB
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Impurities in Y2BaNiO5
Ca2+
Y3+
• Mg2+on Ni2+ sites finite length chains• Ca2+ on Y3+ sites mobile bond defectsMg
Ni
Kojima et al. (1995)
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Zeeman resonance of chain-end spinsI(
H=
9 T
)-I(
H=
0 T
) (
cts.
per
min
.)
Hg B
h (
meV
)
H (Tesla)0 2 4 6 8
g=2.16
0 0.5 1 1.5 2
-5
0
10
15
20
Hg B
meV)(
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Form factor of chain-end spins
Q-dependence revealsthat resonating objectis AFM.
The peak resemblesS(Q) for pure system.
Chain end spin carryAFM spin polarizationof length back into chain
Y2BaNi1-xMgxO5 x=4% Hg B
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New excitations in Ca-doped Y2BaNiO5
Pure 9.5% Ca
•Ca-doping creates states below the gap
•sub-gap states have doubly peaked structure factor
Y2-xCaxBaNiO5:
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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
xq
q is single impurity prop.Indep. of
x
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Does q vary with calcium concentration?
q not strongly dependent on x
Double peak is single impurity effect
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(b)
Ca
Ba
Ni
Y
c
b
a
(e)
(f)
(c)
(d)
a
O
Bond Impurities in a spin-1 chain: Y2-
xCaxBaNiO5
(a)
Form-factor for FM-coupled chain-end spins
2/)(Re2)( iqeqMqS
A symmetric AFM droplet
22/*2/ )()()(
ll
iql
iqlll eqMeqMPqS
Ensemble of independent randomly truncated AFM droplets
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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 factorAFM droplets interact through intervening chain
forming disordered random bond 1D magnet
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What sets energy scale for sub gap scattering ?
0
5
10
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.
How to distinguish:
• Neutron scattering in an applied field
• Transport measurements
• Theory
?
q~
h
(meV
)
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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
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 http://www.pha.jhu.edu/~broholm/canada/index.htm