leon balents- spin liquid states in frustrated antiferromagnets
TRANSCRIPT
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Spin Liquid States in
FrustratedAntiferroma nets
Leon BalentsKITP
Non-equilibrium Dynamics and Correlations in StronglyInteracting Atomic, Optical and Solid State Systems, 2009
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Spin Liquid States in
FrustratedAntiferroma nets
Leon BalentsKITP
Lookingfor
and findingsurprises
Non-equilibrium Dynamics and Correlations in StronglyInteracting Atomic, Optical and Solid State Systems, 2009
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Collaborators
Oleg Starykh, U.Utah
Masanori Kohno,NIMS, Japan
Gang Chen, UCSB
Andreas Schnyder,KITP
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Quantum Spin Liquids
QSL: a state of a magnet in which quantumfluctuations prevent order even at T=0.
Many theoretical suggestions since Anderson(73)
Resonating Valence Bond QSL states
+ +
=
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Magnons
Basic excitation: spin flip
Carries Sz= 1
Periodic Bloch states: spinwaves
Quasi-classical picture:small precession
Image: B. Keimer
k
= (
k)
MnF2
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One dimensionHeisenberg model is a spin liquid
No magnetic order
Power law correlations of spins and dimers
Excitations are s=1/2 spinons
General for 1d chains
S(x) S(x) (1)xx
|x x|+
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Spinons by neutronsBethe ansatz:
Spinon energy
Spin-1 states
Theory vs experimentfor KCuF3withanisotropy 30
B. Lake, HMI
s(k) =
J
2| sin k|
= s(k1) + s(k2)
k = k1 + k2 2-particle
continuum
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Many spin liquidsDimension?
d=2
Spin gap?
yes
Z2 state
no
Cv?
U(1) FS
T2/3
Z2 dirtyDirac
T
RW = 1?T2
d=3
Z2 DiracU(1) Dirac
ASL
yes no
in theory!
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A diagnostic flowchartT>0 transition
d=2
Spin gap?yes
U(1)
noCv?
z2 FSTln(1/T) Z2 line
node
T T2
d=3
U(1) FS U(1) ??
Z2. Spin gap?
yes
yes
Z2
no
Cv?
?
disordered possibilities neglected
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QSL candidatesCsCu2Cl4 - spin-1/2 anisotropic triangularlattice
NiGa2S4 - spin-1 triangular lattice
-(BEDT-TTF)2Cu2(CN)3 , EtMe3Sb[Pd(dmit)2]2 -triangular lattice organics
FeSc2S4 - orbitally degenerate spinel
Na4Ir3O8 - hyperkagome
ZnCu3(OH)6Cl2 - kagome
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Preview
CsCu2Cl4:
frustration enhances 1d correlationsdispersing bound states are a generalfeature in weakly coupled chains
FeSc2S4:
Spin-orbit coupling + orbital degeneracystrongly enhances quantum fluctuations
Competition with exchange leads to a QCP
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Cs2CuCl4
Spatially anisotropictriangular lattice
Cu2+ spin-1/2 spins
couplings:
H=1
2
ij
Jij
Si Sj
Dij Si
Sj
J=0.37meV
J=0.3J D=0.05J
D = Da
R. Coldea et al
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Neutron scatteringColdea et al, 2001/03: a 2d spin liquid?
Very broad spectrumsimilar to 1d (in somedirections of k space).
Roughly fits power law.
Fit of peak dispersion tospin wave theory requiresadjustment of J,J by 40%
- in opposite directions!
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2d theoriesArguments for 2d:
J/J = 0.3 not very small
Transverse dispersionExotic theories:
Spin waves
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Dimensional reduction
Frustration of interchain coupling makes itless relevant
First order energy correction vanishes
Leading effects on correlations are in factO[(J)4/J3]!
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Dimensional reduction
Frustration of interchain coupling makes itless relevant
First order energy correction vanishes.Numerics: J/J < 0.7 is weak
Weng et al,2006
Very different from
spin wave theory
Very weak inter-chain
correlations
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Build 2d excitations from 1d spinons
Exchange:
Expect spinon binding to lower inter-chainkinetic energy
Use 2-spinon Schroedinger equation
Excitations
J
2
S+
i S
j + S
i S+
j
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Broad lineshape: free spinonsPower law fits well to free spinon result
Fit determines normalization
J(k)=0 here
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Bound stateCompare spectra at J(k)0:
Curves: 2-spinon theory w/ experimental resolution Curves: 4-spinon RPA w/ experimental resolution
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Transverse dispersion
Bound state and
resonance
Solid symbols: experiment
Note peak (blue diamonds) coincideswith bottom edge only for J(k)
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Spectral asymmetry
Comparison:
Vertical lines: J(k)=0.
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Conclusions on Cs2CuCl4Simple theory works well for frustratedquasi-1d antiferromagnets
Frustration leads to a strong enhancementof one-dimensionality
The mystery of Cs2CuCl4 should beconsidered solved
Many (nearly all) other details of diverseexperiments on this material may beunderstood in the same framework
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AB2X4 spinels
One of the most
common mineralstructures
Common valence:
A2+,B3+,X2-
X=O,S,Se
A
X
B
cubic Fd3m
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Deconstructing thespinel
A atoms: diamond lattice
Bipartite: notgeometricallyfrustrated
A
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Frustration Signature
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A-site spinelsSpectrum of materials
900
FeSc2S4
V. Fritsch et al. PRL 92, 116401 (2004); N. Tristan et al. PRB 72, 174404 (2005); T. Suzuki etal. (2006)
Orbitaldegeneracy
1 10 205
CoAl2O4
MnSc2S4
MnAl2O4
CoRh2O4 Co3O4s = 5/2
s = 3/2
s = 2
=
|CW|
TN
Gang Chen
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Orbital degeneracy inFeSc2S4
Chemistry:
Fe2+: 3d6
1 hole in eg level
Spin S=2
Orbital pseudospin 1/2
Static Jahn-Tellerdoes not appear
S
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Atomic Spin Orbit
Separate orbital and spin degeneracy can be split!
Energy spectrum: singlet GS with gap =
Microscopically,
Naive estimate 25K
should be reduced by dynamic JT
HSO =
1
3
x
(Sx)2
(Sy)2+
z
(Sz)2
S(S+1)3
=62
0
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Spin orbital singlet
Ground state of>0 term:
Due to gap, there is a stable SOS phase for >> J.
Sz=0 12 Sz=2( Sz=-2+ )
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Exchange
Inelastic neutrons showsignificant dispersion
indicating exchange
Bandwidth 20K similarorder as CW andestimated
Gap (?) 1-2K
Small gap is classicindicator of incipientorder
( ( ( ( (
( ( (
:(;%,
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Exchange
Most general symmetry-allowed form ofexchange coupling (neglecting SOI)
Hex =1
2
ij
JijSi Sj +Kiji j + Kij
y
i y
j
+
Liji j + Lij
y
i y
j
Si Sj
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Exchange
Neglecting SOI, a simplified superexchangecalculation gives
Largest coupling is AF spin interaction
More exchange processes
Hex =1
2
ij
{JijSi Sj + Kij (4 + Si Sj) (1 + 4i j)}
Si Sj
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Ordered Phase (J >> )
Ground state of Hex is almost certainly
ordered
SiSj coupling is strongest
Complex multi-spiral ground states possible
Inclusion of weak SOI favors simplercommensurate cubic spin arrangements
spin order leads to induced orbital order
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Quantum Critical Point
Full Hamiltonian H = HSO + Hex
/J
T
AF
QCP
SO singlet
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Minimal Model
Neutron scatteringsuggests peak closeto 2(100)
Indicates J2 >> J1
( "4 et al:=(>?@):(A"B:(C"$$:(94=(0DE1F0=(0FFG
Hmin = J2
ij
Si Sj +HSO
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Quantum Critical Point
Mean field phase diagram
/J2
T
2(100) AF
Ferro OO
16
SO singlet
FeSc2S4
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Consequences of QCP
Power-law spin correlations
Scaling form for (T1T)-1
f(/T)
Specific heat Cv T3 f(/T)
Possibility of pressure-induced ordering
Impurity effects?
Behavior in field? Can triplet be made to
condense?
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c.f. dimer antiferromagnet
Behavior in Field
/J
H
AF
QCP
dimer singlet
saturation
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Behavior in field
This model
/J2
H
2(100) AF
Ferro OO
16
SO singlet(disordered)
FeSc2S4
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Conclusions on FeSc2S4
Orbital degeneracy and spin orbit provides anexciting route to quantum paramagnetism andquantum criticality
entangled spin-orbital singlet ground statein an S=2 magnet!
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More for the future!CsCu2Cl4 - spin-1/2 anisotropic triangularlattice
NiGa2S4 - spin-1 triangular lattice-(BEDT-TTF)2Cu2(CN)3 , EtMe3Sb[Pd(dmit)2]2 -triangular lattice organics
FeSc2S4 - orbitally degenerate spinel
Na4Ir3O8 - hyperkagome
ZnCu3(OH)6Cl2 - kagome
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and in cold atoms?