Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Conservation laws and magnon Conservation laws and magnon decay in quantum spin liquidsdecay in quantum spin liquids
Igor ZaliznyakIgor Zaliznyak
Neutron Scattering Group, Brookhaven National Laboratory
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
CollaboratorsCollaborators
• M. B. Stone
• C. Broholm, D. Reich, T. Hong
• S.-H. Lee
• S. V. Petrov
/ U. Virginia/ U. Virginia
OAK RIDGE NATIONAL LABORATORY
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Particles in the UniverseParticles in the Universe
MeVMeV GeVGeV
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Quasiparticles in condensed matterQuasiparticles in condensed matter
neutron out
neutron outkkff
meV, meV, μμeVeV
Quasiparticle:Quasiparticle:
phonon, magnonphonon, magnon
q = kq = kii - k - kff
neutron in
neutron in
kk ii
1 meV = 11.6 K1 meV = 11.6 K
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Neutron scattering: how neutrons measure Neutron scattering: how neutrons measure quasiparticles.quasiparticles.
fi
fiffiii
zif
zf
i
f
m
k
m
kηEηEηSηS
k
k
dEdΩ
Ed
,
,b,,
22
2222 q
q
fi kkq m
k
m
kηEηEE fi
iiff 22
22
, ,2
22
20
,
dt
tMMeeq
qqr
k
k
dEdΩ
Ed
jjjj
iEti
mi
fmag jj RRqq
magnetic scattering length, rm = -5.39*10-13 cm
jj
tiiEti
jji
fnuc dteeebb
k
k
dEdΩ
Edjj
,
*,
20
2RqRqq
nuclear scattering length, b ~ 10-13 cm qqq
ESdEdΩ
Ed~
,2
Long-lived quasiparticle (magnon)
delta-function singularity in cross-section
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
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
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
ε(q
) (K
elvi
n)
q (Å-1)
phonon
roton
maxonwhatsgoingon?
Excitations in quantum Bose liquid: Excitations in quantum Bose liquid: superfluid superfluid 44HeHe
Woods & Cowley, Rep. Prog. Phys. 36 (1973)
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
The “cutoff point” of the quasiparticle The “cutoff point” of the quasiparticle spectrum in the quantum Bose-liquidspectrum in the quantum Bose-liquid
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Breakdown of the excitations in Breakdown of the excitations in 44He: He: experimentexperiment
H = qε (q) aq+
aq + q,q′ Vq,q′(aqa+q′a+
q-q′ + H.c.) + …
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Roton decays and conservation lawsRoton decays and conservation laws
• Breakdown of roton quasiparticle spectrum at E > 2 due to pair decays satisfies:
– Particle non-conservation: cubic terms in the boson Hamiltonian
=> Vq,q′(aqa+q′a+
q-q′ + H.c.)
– Energy-momentum conservation
qq’
q”
q = q’ + q”
(q) = (q’) + (q”)
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Quantum spin liquid: what is it?Quantum spin liquid: what is it?
• 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>> 1Coupled 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 → ∞
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
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
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
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
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
1D array of dimers (aka alternating chain)1D array of dimers (aka alternating chain)
Chains of weakly interacting dimers in
Cu(NO3)2x2.5D2O
CuCu2+2+ 3d9
S=1/2
E (
me
V)
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Weakly interacting dimers in Weakly interacting dimers in Cu(NOCu(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
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
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)
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Spin-quasiparticles in Haldane chains in Spin-quasiparticles in Haldane chains in CsNiClCsNiCl33
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
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Spin-quasiparticles in Haldane chains in Spin-quasiparticles in Haldane chains in CsNiClCsNiCl33
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Magnon quasiparticle breakdown in CsNiClMagnon quasiparticle breakdown in CsNiCl33
I. A. Zaliznyak, S.-H. Lee, S. V. Petrov, PRL 017202 (2001)
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Spectrum termination in the dimer-chain Spectrum termination in the dimer-chain material IPA-CuClmaterial IPA-CuCl33
T. Masuda, A. Zheludev, et. al., PRL 96 047210 (2006)
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
weak interaction
2D quantum spin liquid: a lattice of 2D quantum spin liquid: a lattice of frustrated dimersfrustrated dimers
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
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Magnon spectrum termination line in PHCCMagnon spectrum termination line in PHCC
max{E2-particle (q)}
min{E2-particle (q)}
E1-particle(q)
Spectrum termination line
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
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
(co
unts
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)
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
2D map of the spectrum along both 2D map of the spectrum along both directionsdirections
7
6
5
4
3
2
1
0
E (
meV
)
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
M. B. Stone, I. Zaliznyak,
T. Hong, C. L. Broholm, D. H. Reich, Nature 440 (2006)
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Magnon breakdown: theoryMagnon breakdown: theory
Kolezhuk and Sachdev, PRL 96 087203 (2006)
Zhitomirsky, PRB 73 100404R (2006)
Coherent magnon disappearsWidth appears at the crossing point
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Spectrum end point in helium-4 and Spectrum end point in helium-4 and quantum spin liquid in PHCCquantum spin liquid in PHCC
4
3
2
1
0
(meV
)3210
Q (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)
0.4
0.2
00.15
0
2.6 Å-1b1.3 K
1.85 K
2.25 K
M. B. Stone, I. Zaliznyak, T. Hong, C. L. Broholm, D. H. Reich, Nature 440 (2006)
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Spectrum breakdown in quantum spin liquid Spectrum breakdown in quantum spin liquid in PHCC in magnetic fieldin PHCC in magnetic field
I. Zaliznyak, T. Hong, M. B. Stone, C. L. Broholm, D. H. Reich, unpublished
gB
gB Sz=+1
Sz=-1Sz=0
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Spectrum end point in PHCC in magnetic Spectrum end point in PHCC in magnetic field: spin conservationfield: spin conservation
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
SummarySummary
• Quasiparticle spectrum breakdown at E > 2 is a generic property of quantum Bose (spin) fluids
• Governed by conservation laws
• Roton breakdown in He-4
– particle non-conservation
– energy-momentum conservation
• Magnon breakdown in quantum magnets
– particle non-conservation
– energy-momentum conservation
– spin angular momentum conservation => apparent in magnetic field
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
How do neutrons measure excitations.How do neutrons measure excitations.
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Breakdown of the roton excitation in Breakdown of the roton excitation in 44He: He: early experimentsearly experiments
Graf, Minkiewicz, Bjerum Moller & Passell, Phys. Rev. A (1974)Fak & Bossy, J. Low Temp. Phys. (1998)
Montfrooij & Svensson, J. Low Temp. Phys. (2000)
H = qε (q) aq+
aq + q,q′ Vq,q′(aqa+q′a+
q-q′ + H.c.) + …
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
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
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Temperature dependence in copper nitrateTemperature dependence in copper nitrate
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Temperature dependence in PHCCTemperature dependence in PHCC
40
20
0
6420 (meV)
60
30
0
Inte
nsi
ty (
cou
nts
/ 2
min
.)
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
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
PHCC: a two-dimensional quantum spin PHCC: a two-dimensional quantum spin liquidliquid
• gap = 1 meV• bandwidth = 1.8 meV
• Single dispersive mode along h
• Single dispersive mode along l
• Non-dispersive mode along k
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
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
(co
unts
in 1
m
in)
E (meV)
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
PHCC: a two-dimensional quantum spin PHCC: a two-dimensional quantum spin liquidliquid
• = 1 meV, bandwidth = 1.8 meV
• Single dispersive mode along L
• Non-dispersive mode along K40
20
0
40
20
04.03.02.01.0
(meV)
40
20
0
T = 1.8 KT = 50 K
(0, k, 0.5)
k = 0.5
k = 0.75
k = 1.0
Inte
nsity
(co
unts
/min
)
T=1.4K30
20
10
0
30
20
10
03.02.01.00
(meV)
30
20
10
0
Inte
nsity
(C
ount
s / m
in)
(h, 0, 1.5)
h = 0.6
h = 0.7
h = 0.8
• Single dispersive mode along H
80
40
03.02.01.00
(meV)
80
40
0
80
40
0
(0.5, 0, l)
l = 1.5
l = 1.6
l = 1.8
Inte
nsity
(co
unts
/min
)
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Neutron scattering cross-sectionNeutron scattering cross-section
fi
fiffiii
zif
zf
i
f
m
k
m
kηEηEηSηS
k
k
dEdΩ
Ed
,
,b,,
22
2222 q
q
fi kkq m
k
m
kηEηEE fi
iiff 22
22
, ,
,
22
22
dttMMee
q
qqr
k
k
dEdΩ
Ed
jjjj
iEti
mi
fmag jj RRqq
magnetic scattering length, rm = -5.39*10-13 cm
jj
tiiEti
jji
fnuc dteeebb
k
k
dEdΩ
Edjj
,
*,
20
2RqRqq
nuclear scattering length, b ~ 10-13 cm
m
k
m
kEEηSηS fi
fiiziif
zfffi 22
2222
,,T,, kk
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
Quasiparticle cross-sectionQuasiparticle cross-section
qqq
ESdEdΩ
Ed~
,2
Quasiparticle (undamped)
singularity in cross-section (delta-function)
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
How do neutrons measure quasiparticles.How do neutrons measure quasiparticles.
I. A. Zaliznyak and S.-H. Lee,
in Modern Techniques for Characterizing Magnetic Materials, Ed. Y. Zhu, Springer (2005)
M o n o ch ro m a to r (2 ) s
F o cu s in g an a ly ze r
S am p le
D e tec to r
B. Brokhouse (1961)
Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg
How neutrons measure excitations now.How neutrons measure excitations now.
B. Brokhouse (1961) Gain up to factor 10
M o n o ch ro m a to r (2 ) s
F o cu s in g an a ly ze r
S am p le
D e tec to r
Gain up to factor 5
I. A. Zaliznyak and S.-H. Lee,
in Modern Techniques for Characterizing Magnetic Materials, Ed. Y. Zhu, Springer (2005)
M o n o ch ro m a to r
A n a ly ze r
S am p le
D e tec to r