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SPIN STRUCTURE FACTOR OF THE
FRUSTRATED QUANTUM MAGNET Cs2CuCl4
Rastko Sknepnek
Department of Physics and Astronomy
McMaster University
In collaboration with:
Denis Dalidovich
A. John Berlinsky
Junhua Zhang
Catherine Kallin
Iowa State University
March 2, 2006
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Outline
• Motivation
• Spin waves vs. spinons
• Experiment on Cs2CuCl4
• Nonlinear spin wave theory for Cs2CuCl4
• Summary
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Motivation
Neutron scattering measurements on quantum magnet Cs2CuCl4.
dynamical correlations are
dominated by an extended
scattering continuum.
Signature of deconfined, fractionalized
spin-1/2 (spinon) excitations?
(R. Coldea, et al., PRB 68, 134424 (2003))
Can this broad scattering continuum be explained within a
conventional 1/S expansion?
(Complementary work: M. Y. Veillette, et al., PRB (2005))
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Spin Waves
Heisenberg Hamiltonian:
ˆ ˆ
Hˆ  J  Si  S j
i, j
• J<0 – ferromagnetic ground state
• J>0 – antiferromagnet (Néel ground state)
• Spin waves are excitations of the (anti)-ferromagnetically ordered state.
• Exciting a spin wave means creating a quasi-particle called magnon.
• Magnons are S=1 bosons.
Dispersion relations (k0):
k  S | J | (a | k |)2
k  JzSa | k |
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(ferromagnet)
(antiferromagnet)
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Ground state of an antiferromagnet
Antiferromagnetic Heisenberg Hamiltonian:

J
Hˆ  J  Sˆiz Sˆ jz   Sˆi Sˆ j  Sˆi Sˆ j
2 i, j
i, j

(J>0)
State   can not be the ground state - it is not an eigenstate of the Hamiltonian.
• Antiparallel alignment gains energy only from the z-z part of the Hamiltonian.
• True ground state - the spins fluctuate so the system gains energy from the spin-flip terms.
Ground state of the Heisenberg antiferromagnet shows quantum fluctuations.
How important is the quantum nature of the spin?
quantum correction
~
1
classical energy
S
Reduction of the staggered magnetization due to quantum fluctuations:
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Spin Liquid and Fractionalization in 1d
(half-integer spin)
• In D=1 quantum fluctuations destroy long range order.
• Spin-spin correlation falls off as a power law.
• Ground state is a singlet with total spin Stot=0 (exactly found using Bethe ansatz).
Excitations are not spin-1 magnons but pairs of fractionalized spin-1/2 spinons.
• Spinons appear in pairs.
• Excitation spectrum is a continuum with two soft
points (0 and p)
Fractionalization: Excitations have quantum numbers that
are fractions of quantum numbers of the local degrees of
freedom.
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Prototypical system KCuF3.
(D.A.Tennant, et al, PRL (1993))
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Geometrical Frustration in 2d
• Ising-like ground state is possible only on bipartite lattices.
• Non-bipartite lattices (e.g., triangular) exhibit geometrical
frustration.
• On an isotropic triangular lattice the ground state is a three sub-lattice Néel state.
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Resonating valence bond (RVB)
(P.W. Anderson, Mater. Res. Bull. (1973))
• Ground state – linear superposition of disordered valence bond configurations.
• Each bond is formed by a pair of spins in a singlet state.
RVB state has the following properties:
• spin rotation SU(2) symmetry is not broken.
• spin-spin, dimer-dimer, etc. correlations are exponentially decaying – no LRO.
• excitations are gapped spin-1/2 deconfined spinons.
RVB state is an example of a two dimensional spin liquid.
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Spin crystal
Spin Liquid
Ground state
Semiclassical Néel order
Quantum Liquid
Order parameter
Staggered magnetization
No local order parameter
Is there any experimental realization of
two dimensional
spin liquid?
Excitations
Gapless magnons
Gapped deconfined spinons
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March 2, 2006
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Cs2CuCl4 - a spin-1/2 frustrated quantum magnet.
Crystalline structure:
• Orthorhombic (Pnma) structure.
• Lattice parameters (at T=0.3K)
a = 9.65Å
b = 7.48Å
c = 12.26Å.
• CuCl42- tetrahedra arranged in layers.
(bc plane) separated along a by Cs+ ions.
Cs2CuCl4 is an insulator with each Cu2+ carrying a spin 1/2.
Crystal field quenches the orbital angular momentum resulting in near-isotropic Heisenberg
spin on each Cu2+.
• Spins interact via antiferromagnetic superexchange
coupling.
• Superexchange route is mediated by two nonmagnetic
Cl- ions.
• Superexchange is mainly restricted to the bc planes
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Coupling constants
J
Measurements in high magnetic field (12T):
J’
J’’
J = 0.374(5) meV
J’ = 0.128(5) meV
J’’= 0.017(2) meV
High magnetic field experiment also observe small splitting into two magnon branches.
Indication of a weak Dzyaloshinskii-Moriya (DM) interaction.
DM interaction creates an easy plane anisotropy.
D = 0.020(2) meV
Below TN=0.62K the interlayer coupling
J’’ stabilizes long range order.
The order is an incommensurate spin
spiral in the (bc) plane.
D
17.7o
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March 2, 2006
1
Q  2p (   0 )eb
2
0=0.030(2)
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The Hamiltonian
Relatively large ratio J’/J≈1/3 and considerable dispersion along both b and c directions
indicate two dimensional nature of the system.
Effective Hamiltonian:
H    JS R  S R    J ' S R  (S R   S R  )  (1)n D  S R  (S R   S R  ) 
1
2
1
2
1
2 

R
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A few remarks...
• A strong scattering continuum does not automatically
entail a spin liquid phase.
• Magnon-magnon interaction can cause a broad scattering
continuum in a conventional magnetically ordered phase.
In Cs2CuCl4 strong scattering continuum is expected because:
• low (S=1/2) spin and the frustration lead to a small ordered moment and strong
quantum fluctuations
• the magnon interaction in non-collinear spin structures induces coupling between
transverse and longitudinal spin fluctuations  additional damping of the spin waves.
It is necessary to go beyond linear spin wave theory by taking into account magnon-magnon
interactions within a framework of 1/S expansion.
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Spin wave theory (linear)
Classical ground state is an incommensurate spin-spiral
along strong-bond (b) direction with the ordering wave
vector Q.
1
Q  2p (   )eb
2
In order to find ground state energy we introduce a local reference frame:
SRx  SR cos(Q  R)  S R sin(Q  R)
S Rz  S R sin(Q  R)  S R cos(Q  R)
SRy  SR
Classical ground state energy:
S 2 EG(0) (Q)  S 2 J QT
J  J Q  iDQ
T
Q
3k y
kx
J k  J cos k x  2 J 'cos cos
2
2
3k y
kx
Dk  2iD sin cos
2
2
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Ordering wave vector:
1
J'


arcsin
 0.0547
•D=0
p
2J
• D = 0.02meV
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  0.0533
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1/S expansion
To go beyond linear spin-wave theory we employ Holstein-Primakoff transformation:
1 †
1 † 

S  S R  iS R  2S 1 
aR aR aR  2S 1 
aR aR  aR
2S
 4S

1 †
1 † 



†
† 
S R  S R  iS R  2S aR 1 
aR aR  2S aR 1 
aR aR 
2S
 4S


R


SR  S  aR† aR
Where a’s are bosonic spin-wave creation and annihilation operators.
[aR , aR† ' ]   R , R '
[aR , aR ' ]  0
[aR† ' , aR† ' ]  0
The Hamiltonian for the interacting magnons becomes:
H  S E (Q)  ( H
2
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(0)
G
(2)
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H
(3)
H
(4)
)
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Quadratic part of the Hamiltonian:
H
Ak 
(2)
Bk † †


†
 2S   Ak ak ak 
ak a k  ak a k 
2
k 


J
1 T
T
  k  JT
J

J
Q
Q k 
4  Qk
2
Bk 

Jk
1
  J QT  k  J QT k 
2 4
k  Ak2  Bk2
Magnon-magnon interaction is described by:
H (3) 
H
(4)



i S
f k1 , k2 ak† ak ak  ak† ak† ak  k k ,k

1
2
3
1
2
3
1
2 3
2 2 N k1 ,k2 ,k3



 f1 k1 , k2 , k3 , k4 a † a † a a 


k
k
k
k
k

k
,
k

k
1
1
2
3
4
1
2 3
4






4 N k1 ,k2 ,k3 ,k4  f k , k , k a † a † a † a  a † a a a 
k1 k2 k3 k4
k1  k2  k3 , k4 
 2 1 2 3 k1 k2 k3 k4

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

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Ground state energy and ordering wave vector
In a 1/S expansion quantum corrections of the ground state energy are:
EG (Q)  S 2 J QT  SEG(1) (Q )  EG(2) (Q2 ) 
0
Quantum corrections of the ordering wave-vector are:
Q(1) Q(2)
Q  Q0 


2
2S
 2S 
Where
EG(1) (Q)  J QT 
1

N k k
1
Q (1)
  2 J QT  1
 

2
 Q  N

A
k
k
 Bk  J QT  k
k
Q
Q0
etc.
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1/S theory
D = 0 meV
D = 0.02 meV
S2EG(0)/J
-0.265
-0.291
SEG(1)/J
-0.157
-0.138
EG(2)/J
-0.0332
-0.0256
Emin/J
-0.459
-0.454
-
-0.5*
1/S theory
D = 0 meV
D = 0.02 meV
Q0/2p
0.5547
0.5533
Q(1)/2p
-0.0324
-0.0228
Q(2)/2p
-0.011
-
Q/2p
0.5113
0.5308
-
0.530(2)**
Experiments (Cs2CuCl4)
EG/J
*Y. Tokiwa, et al., cond-mat/0601272 (2006)
Experiments (Cs2CuCl4)
Q/2p
** R. Coldea, et al., PRB 68, 134424 (2003)
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Green’s function
To calculate physical observables we need Green’s function for magnons.
iGˆ (k ,  ) 

 dte

it
 ak (t )  †
ˆ
 ak (0) ak (t ) 
T †

 a k (t ) 
Gˆ 1 (k ,  )  Gˆ (0) 1 (k ,  )  ˆ (k ,  )
2SBk
   2SAk  i

(0) 1
ˆ
G (k ,  )  

2
SB



2
SA

i

k
k


(4)
(3)
(4)
(3)



(
k
)


(
k
,

)

(
k
)


(
k
,

)
11
11
12
12
(4)
(3)
ˆ (k ,  )  ˆ (k )  ˆ (k ,  )   (4)

  (k )  (3) (k ,  ) (4) (k )  (3) (k ,  ) 
21
22
22
 21

’s are the self-energies which we calculate to the order 1/S.
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Sublattice magnetization
Staggered magnetization:
1
M S
N
d
 i 0
G
k
,

e
  2p i 11
k


To the lowest order in 1/S:
M
(1)
1

2N
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 Ak

    1
k  k

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The second order correction has two contributions:
M
(2)
I
Q (1)

2S
M II(2) 
      
1
d i 0 ˆ (0)
ˆ k ,  Gˆ (0) k , 
e
G
k
,



N k  2p i
11
Cs2CuCl4
Numerical integration carried using DCUHRE method – Cuba 1.2 library, by T. Hahn)
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Energy spectrum
The renormalized magnon energy spectrum is determined by poles of the Green’s function.


 

Re det Gˆ (0) 1 k , k  ˆ k , k   0


Which leads to the nonlinear self-consistency equation:


k  f Ak , Bk ,  k , k
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
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On renormalization of coupling constants.
In order to quantify the “quantum” renormalization of the magnon dispersion relation
one fits the 1/S result to a linear spin-wave dispersion with “effective” coupling constants.
1/S theory
Exp.
Jren/Jbare
1.131
1.63(5)
J’ren/J’bare
0.648
0.84(9)
Dren/Dbare
0.72
-
(R. Coldea, et al., PRB 68, 134424 (2003))
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Spin structure factor
Neutron scattering spectra is expressed in terms of Fourier-transformed real-time
dynamical correlation function:
S
ik


1
k , 
2p



dt  SOi (0) S Rk (t ) ei (t k R )
R
Magnon-magnon interaction leads to the mixing of longitudinal () and transversal ()
modes (detailed derivation in T. Ohyama&H. Shiba, J. Phys. Soc. Jpn. (1993))





S tot k ,   px S xx k ,   p y S yy k , 



S yy k ,   S  k , 
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


S xx k ,  
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
  



1 
S k ,   S  k ,   S  k , 
4

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G scan
Scan along a path at the edge of the Brillouin zone.
kx = p
ky = 2p(1.53-0.32-0.12)
linear SW theory k =0.22meV
Energy resolution E=0.016meV
linear SW theory k+/-Q = 0.28meV
Momentum resolution k/2p = 0.085
(R. Coldea, et al., PRB 68, 134424 (2003))
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D = 0.02meV
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two-magnon continuum
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Energy resolution E=0.016meV
Energy resolution E=0.002meV
Momentum resolution k/2p = 0.085
Momentum resolution k/2p = 0
Near G point the dispersion relation has large modulation along b direction.
Significant broadening due to finite momentum resolution.
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What happens if we lower D?
Energy resolution E=0.016meV
D = 0.01meV
Momentum resolution k/2p = 0.085
exp.
G scan
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Energy resolution E=0.016meV
Momentum resolution k/2p = 0.085
D=0.02meV
experimental
position of the peak
 = 0.10(1) meV
Smaller value for D fits experiments better!
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Summary and conclusions
We have...
• derived non-linear spin wave theory for the frustrated triangular magnet Cs2CuCl4.
• calculated quantum corrections to the ground state energy and sublattice magnetization
to the 2nd order in 1/S.
• calculated spin structure factor and compared it to the recent inelastic neutron scattering
data
We find that 1/S theory:
• gives good prediction for the ground state energy and ordering wave vector.
• significantly underestimates renormalization of the coupling constants.
• significant scattering weight is shifted toward higher energies, but not sufficient to
fully explain experiments.
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Other approaches
• 1d coupled chains
M. Bocquet, et al., PRB (2001)
O. Starykh, L. Balents, (unpublished) (2006)
• Algebraic vortex liquid
J. Alicea, et al., PRL (2005)
J. Alicea, et al., PRB (2005)
• High-T expansion
W.Zheng, et al., PRB (2005)
• Proximity of a spin liquid quantum critical point
S.V. Isakov, et al., PRB (2005)
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Thank You!
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