quasiparticle breakdown in quantum spin liquid
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ICNS – 2005, Sydney, AustraliaICNS – 2005, Sydney, Australia
Quasiparticle breakdown in quantum spin liquidQuasiparticle breakdown in quantum spin liquid
CollaboratorsCollaborators
• M. B. Stone
• D. Reich, T. Hong
• C. Broholm
Igor ZaliznyakIgor Zaliznyak
Neutron Scattering Group, Brookhaven National Laboratory
&&
OAK RIDGE NATIONAL LABORATORY
ICNS – 2005, Sydney, AustraliaICNS – 2005, Sydney, Australia
What is quantum liquid?What is quantum liquid?• What is liquid?
− no shear modulus− no elastic scattering = no static correlation of density fluctuations
‹ρ(r1,0)ρ (r2,t)› → 0t → ∞
• What is quantum liquid? − all of the above at T → 0 (i.e. at temperatures much lower than inter-particle interactions in the system)
• Elemental quantum liquids:− H, He and their isotopes− made of light atoms strong quantum fluctuations
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ε(q)
(Kel
vin)
q (Å-1)
phonon
roton
maxonwhatsgoingon?
Excitations in quantum Bose liquid: superfluid Excitations in quantum Bose liquid: superfluid 44HeHe
Woods & Cowley, Rep. Prog. Phys. 36 (1973)
ICNS – 2005, Sydney, AustraliaICNS – 2005, Sydney, Australia
The “cutoff point” of the quasiparticle spectrum in The “cutoff point” of the quasiparticle spectrum in the quantum Bose-liquidthe quantum Bose-liquid
ICNS – 2005, Sydney, AustraliaICNS – 2005, Sydney, Australia
Spectrum termination in Spectrum termination in 44He: experimentHe: experiment
ICNS – 2005, Sydney, AustraliaICNS – 2005, Sydney, Australia
What is quantum spin liquid?What is quantum spin liquid?
• Quantum liquid state for a system of Heisenberg spins
H = J|| SiSi+||+ JSiSi
• Exchange couplings J||, J through orbital overlaps may be different
− J||/J >> 1 (<<1) parameterize quasi-1D (quasi-2D) case
Coupled chains J||/J>> 1
Coupled planes J||/J<<1• no static spin correlations
‹Siα (0)Sj
β (t)› → 0, i.e. ‹Si
α (0)Sjβ (t)› = 0
• hence, no elastic scattering (e.g. no magnetic Bragg peaks)
t → ∞
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Simple example: coupled S=1/2 dimersSimple example: coupled S=1/2 dimers
H = J0 S1S2J0/2 (S1 + S2)2 + const.
Single dimer: antiferromagnetically coupled S=1/2 pair J0 > 0
0 = J0
singlet
triplet
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Simple example: coupled S=1/2 dimersSimple example: coupled S=1/2 dimers(
q)
q/(2)
0 = J0
H = J0 S2iS2i+1J1 (S2i S2i+2)
Chain of weakly coupled dimers
Dispersion (q) ~ J0 + J1cos(q)
J0
J1
triplet
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Dimers in 1D (aka alternating chain)Dimers in 1D (aka alternating chain)
Chains of weakly interacting dimers inCu(NO3)2x2.5D2O
CuCu2+2+ 3d9 S=1/2
E (m
eV)
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Weakly interacting dimers in Cu(NOWeakly interacting dimers in Cu(NO33))22x2.5Dx2.5D22OO
D. A. Tennant, C. Broholm, et. al. PRB 67, 054414 (2003)
Spin excitations never cross into 2-particle continuum and
live happily ever after
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weak interaction
2D quantum spin liquid: a lattice of frustrated 2D quantum spin liquid: a lattice of frustrated dimersdimers
M. B. Stone, I. Zaliznyak, et. al. PRB (2001)(C4H12N2)Cu2Cl6 (PHCC)
− singlet disordered ground state− gapped triplet spin excitation
strong interaction
CuCu2+2+ 3d9 S=1/2
h
l
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PHCC: a two-dimensional quantum spin liquidPHCC: a two-dimensional quantum spin liquid
• gap = 1 meV• bandwidth = 1.8 meV
• Single dispersive mode along h
• Single dispersive mode along l
• Non-dispersive mode along k
ICNS – 2005, Sydney, AustraliaICNS – 2005, Sydney, Australia
Quasiparticle spectrum termination line in PHCCQuasiparticle spectrum termination line in PHCC
max{E2-particle (q)}
min{E2-particle (q)}
E1-particle(q)
Spectrum termination line
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PHCC: dispersion along the diagonalPHCC: dispersion along the diagonal800
600
400
200
0
Q = (0.5,0,-1.5) resolution-corrected fit
400
300
200
100
0
Q = (0.25,0,-1.25)resolution-corrected fit
200
150
100
50
0
7654321
Q = (0.15,0,-1.15) resolution-corrected fit
Inte
nsity
(cou
nts
in 1
m
in)
200
150
100
50
0
Q = (0.15,0,-1.15) resolution-corrected fit
150
100
50
0
Q = (0.1,0,-1.1) resolution-corrected fit
120
80
40
0
7654321
Q = (0,0,1) resolution-corrected fit
E (meV) E (meV)
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2D map of the spectrum along both directions2D map of the spectrum along both directions7
6
5
4
3
2
1
0
E (m
eV)
0.4 0.3 0.2 0.1 0
89
100
2
3
4
5
6
Inte
grat
ed in
t (ar
b.)
0.50.40.30.20.10
Total Triplon Continuum
3.02.52.01.51.0 log(intensity)
(0.5,0,-1-l) (h,0,-1-h)
0.20
0.15
0.10
0.05
0
(meV
)0.5 0.4 0.3 0.2 0.1 0
(h 0 -1-h)
•a
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Compare: spectrum end point in helium-4Compare: spectrum end point in helium-4
4
3
2
1
0
(m
eV)
3210Q (Å-1)
a
2
qc
1.0
0.8
0.6
0.4
0.2
0
S(Q
,
) (1/
meV
)
0.150
S(Q
,
)
6420 (meV)
0.40.2
00.15
0
2.6 Å-1b1.3 K
1.85 K
2.25 K
ICNS – 2005, Sydney, AustraliaICNS – 2005, Sydney, Australia
Summary and conclusionsSummary and conclusions
• Quasiparticle breakdown at E > 2 is a generic property of quantum Bose (spin) fluids– observed in the superfluid 4He
– observed in the Haldane spin chains in CsNiCl3 (I. Zaliznyak, S.-H. Lee and S. V. Petrov, PRL 017202 (2001))
– observed in the 2D frustrated quantum spin liquid in PHCC
• A real physical alternative to the ad-hoc “excitation fractionalization” explanation of scattering continua
• Implications for the high-Tc cuprates: spin gap implies disappearance of coherent spin modes at high E
ICNS – 2005, Sydney, AustraliaICNS – 2005, Sydney, Australia
Temperature dependence in PHCCTemperature dependence in PHCC
40
20
0
6420 (meV)
60
30
0
Inte
nsity
(cou
nts
/ 2 m
in.)
180
120
60
0180
120
60
0
(0.5 0 -1)
a
6420 (meV)
(0.15 0 -1.15)
c T = 1.5 K T = 10 K T = 15 K T = 20 K
6420 (meV)
(0.5 0 -1.5)
b 800
400
0420
400
200
0420
ICNS – 2005, Sydney, AustraliaICNS – 2005, Sydney, Australia
Temperature dependence in copper nitrateTemperature dependence in copper nitrate
ICNS – 2005, Sydney, AustraliaICNS – 2005, Sydney, Australia
Dispersion along the side (Dispersion along the side (ll) in PHCC) in PHCC800
600
400
200
0
Q = (0.5,0,-1.5) resolution-corrected fit
300
200
100
0
Q = (0.5,0,-1.15) resolution-corrected fit
400
300
200
100
0
Q = (0.5,0,-1.1) resolution-corrected fit
400
300
200
100
0
7654321
Q = (0.5 0 -1) resolution-corrected fit
Inte
nsity
(cou
nts
in 1
m
in)
E (meV)
ICNS – 2005, Sydney, AustraliaICNS – 2005, Sydney, Australia
What would be a “spin solid”?What would be a “spin solid”?• Heisenberg antiferromagnet with classical spins, S >> 1S >> 1
− ground state has static Neel order (spin density wave with propagation vector q = )
− elastic magnetic Bragg scattering at q =
n n+1
SSnn = S = S0 0 cos(cos(n)n)
− quasiparticles are gapless Goldstone magnons
(q) ~ sin(q)
(q)
q/(2)
0.0 0.2 0.4 0.6 0.8 1.0
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0.0 0.2 0.4 0.6 0.8 1.0
(q)
− quasiparticles with a gap ≈ 0.4J at q =
2 (q) = 2 + (cq)2
q/(2)
2
1D quantum spin liquid: Haldane spin chain1D quantum spin liquid: Haldane spin chain
− short-range-correlated “spin liquid” Haldane ground state
• Heisenberg antiferromagnetic chain with S = 1S = 1
Quantum Monte-Carlo for 128 spins. Regnault, Zaliznyak & Meshkov, J. Phys. C (1993)
ICNS – 2005, Sydney, AustraliaICNS – 2005, Sydney, Australia
Spin-quasiparticles in Haldane chains in CsNiClSpin-quasiparticles in Haldane chains in CsNiCl33
NiNi2+2+ 3d8
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 Kunimportant for high energies
S=1 S=1 chains
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Spin-quasiparticles in Haldane chains in CsNiClSpin-quasiparticles in Haldane chains in CsNiCl33
ICNS – 2005, Sydney, AustraliaICNS – 2005, Sydney, Australia
Spectrum termination point in CsNiClSpectrum termination point in CsNiCl33
I. A. Zaliznyak, S.-H. Lee, S. V. Petrov, PRL 017202 (2001)
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