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Spin 1/2 Triangular Antiferromagnet Cs2CuCl4
Martin Veillette,
Department Of Theoretical Physics, Oxford University
May/2005
Collaborators:
John Chalker, Radu Coldea
Fabian Essler, Andrew James
Ref: cond-mat/0501347
1
Outline
• Geometric Frustration
• Fundamentals of Cs2CuCl4
? Structure and Properties
? Experimental Results: Phase Diagram, Excitations
• Mean Field in Zero Field
• Role of Quantum Fluctuations
? Large S expansion: Spinwaves
? Quantum fluctuation renormalization
? Dynamical Correlation Functions
? Comparison to Experiments
• Conclusion
2
Frustrated Magnets: Unhappy Magnets
Quantum Fluctuations are generic of Antiferromagnets
i dNdt
= [N, H] 6= 0
Order parameter is not a constant of motion.
Phase space for fluctuations is enhanced
by geometric frustration
• Competition between mean-field ordering and quantum fluctuations
• New collective behavior may emerge
? Order by Disorder
? Quantum Spin Liquid: Cs2CuCl4 ??
3
Crystal Structure and Magnetism of Cs2CuCl4
Magnetic Unit
−
2+
Cl
Cu2+
Cl −
Magnetic Pathways
Cu
Layers ofS = 1/2 Cu2+ ions coupled in triangular geometry
4
Cs2CuCl4: Quasi 2-D Spin 1/2 Triangular Antiferromagnet
c
bc
ba δ
δ + δ
δ
1
2
1 2
J’J
J’
H0 =12
∑R,δ
JδSR · SR+δ
• J ′ = 0→ Non-Interacting Chain
• J = J ′ → Fully Frustrated Triangular Lattice
Experiment:J ′/J = 0.34(3), J = 0.37(4)meV ≈ 4K
5
Additional Interactions:
• CouplingJ ′′ = 0.017meV between Stacked Layers
Small but ultimately responsible for LRO
• Dzyaloshinskii-Moriya (DM) Interaction :
Due to absence of inversion symmetry
Small (D = 0.020meV) butBreaks SU(2) symmetry
Transverse Field6= Longitudinal Field
HDM= −D∑R
(−1)n (SR × [SR+δ1 + SR+δ2 ])a
No Inversion Symmetrycb
a
DChirality SymmetryBroken Explicitely
6
Mean Field ResultIn Zero field: Cycloid Phase→ Incommensurate LRO
SbR = S cos (φR)
ScR = S sin (φR)
φR = Q ·R + α
Q = (0, π + 2πε, 0)
ε ' 1/π sin−1
(J ′
2J
)Non-Collinearity promoted by frus-
tration,ε = 0.053
φ
7
In Transverse field, i.e.Ba 6= 0:
SaR = S sin θ
SbR = S cos θ cos (φR)
ScR = S cos θ sin (φR)
sin θ = ha/hacr
hacr = 2(JT0 − JTQ)
Bacr = 8.36T
Exp.:Bacr = 8.44T
φR = Q ·R independent ofBa
Perpendicular field
stabilizes Cone State
Spins cant along
the magnetic field
8
Experimental Phase DiagramR.Coldea et al. PRL, 2001, Ibid, 2002 and Ibid, PRB 2003
• Very Sensitive to Anisotropy
• Incommensurate Ordering in
Transverse and Zero field
• Quantum Spin Liquid Phase in
Longitudinal Field ?
9
Phase and ExcitationsSpin Solid Spin Liquid
Long Range Order belowTc Short Range Order
Bragg Peaks 〈SrS0〉 ∼ e−r/ξ
Mean-Field Quantum fluctuations
Conventional Melting of Crystal Order
Excitations:
Magnons Spinons
Goldstone Mode (Gapless) Gap/Gapless
Spin 1 Spin 1/2
Bosons Fermions or Bosons
10
Spin Spectral FunctionSpin Solid Spin Liquid
S=1∆
Neutron Scattering
E
I
from Magnons
S=1
Ek
SharpPeaks
S=1∆from Spinons
Neutron Scattering
S=1/2
S=1/2
E
maxEE min
I Extended Continuum
11
Spin Spectral FunctionContinuum Scattering in zero-field: DeconfinedS = 1/2 spinons?
12
13
Experimental Papers
• R. Coldea, D. A. Tennant, R. A. Cowley, D. F. McMorrow, B. Dorner, and Z.
Tylczynski, Phys. Rev. Lett.79,151 (1997).
? Quasi-1D S=1/2 antiferromagnet Cs2CuCl4 in a Magnetic Field.
• R. Coldea, D. A. Tennant, A. M. Tsvelik, and Z. Tylczynski, Phys. Rev. Lett.86, 1335
(2001).
? Experimental realization of a 2D fractional quantum spin liquid.
• R. Coldea, D. A. Tennant, K. Habicht, P. Smeibidl, C. Wolters, and Z. Tylczynski,
Phys. Rev. Lett.88, 137203 (2002).
? Direct Measurement of the Spin Hamiltonian and Observation of Condensation of Magnons in
the 2D Frustrated Quantum Magnet.
• R. Coldea, D. A. Tennant, and Z. Tylczynski, Phys. Rev. B68, 134424 (2003).
? Extended scattering continua characteristic of spin fractionalization in the two-dimensional
frustrated quantum magnet Cs2CuCl4 observed by neutron scattering.
14
Theoretical Papers
• M. Bocquet, F. H. L. Essler, A. M. Tsvelik, and A. O. Gogolin, Phys. Rev. B64,094425 (2001).
? Finite-temperature dynamical magnetic susceptibility of quasi-one-dimensional frustrated
spin-1/2 Heisenberg antiferromagnets.
• C. H. Chung, J. B. Marston, and R. H. McKenzie, J. Phys. : Condens. Matter13, 5159(2001).
? Large-N solutions of the Heisenberg and Hubbard-Heisenberg models on the anisotropic
triangular lattice: application to Cs2CuCl4 and to the layered organic
superconductorsκ-(BEDT-TTF)2X.
• S. -Q. Shen and F. C. Zhang, Phys. Rev. B66, 172407 (2002).
? Antiferromagnetic Heisenberg model on an anisotropic triangular lattice in the presence of a
magnetic field.
• C. -H. Chung, K. Voelker, and Y. B. Kim, Phys. Rev. B68, 094412 (2003).
? Statistics of spinons in the spin liquid of Cs2CuCl4.
• J. Y. Gan, F. C. Zhang, and Z. B. Su, Phys. Rev. B67, 144427 (2003).
? Spin wave theory for antiferromagnetic XXZ spin model on a triangle lattice in the presence of
an external magnetic field.
15
• S. Takei, C.-H. Chung and Y.B. Kim, Phys. Rev. B.70, 104402 (2004),
? Evolution of the single-hole spectral function across a quantum phase transition in the
anisotropic-triangular-lattice antiferromagnet.
• Y. Zhou, and X. -G. Wen, in cond-mat/0210662.
? Quantum Orders and Spin Liquids in Cs2CuCl4.
• W. Zheng, R. R. P. Singh, R. H. McKenzie, and R. Coldea in cond-mat/0410381.
? Temperature Dependence of the Magnetic Susceptibility for Triangular-Lattice
Antiferromagnets with spatially anisotropic exchange constants.
• T. Radu, H. Wilhelm, V. Yushankhai, D. Kovrizhin, R. Coldea, Z. Tylczynski, T.
Luehmann, and F. Steglich, in cond-mat/0505058.
? Bose-Einstein Condensation of Magnons in Cs2CuCl4.
• S. V. Isakov, T. Senthil, and Y. B. Kim, in cond-mat/0503241.
? Ordering in Cs2CuCl4 : Is there a proximate spin liquid.
• J. Alicea, O. I. Motrunich, M. Hermele, and M. P. A. Fisher in cond-mat/0503399.
? Criticality in quantum triangular antiferromagnets via fermionized vortices.
16
What Makes Cs2CuCl4 So Special?
• Very few evidences of spin liquid behavior for D> 1
? NMR on Organic Mott Insulatorκ−(ET)2Cu2(CN)3
• Why conventional spin wave theory fails?
? Ockham’s Razor Principle
∗ Long Range Order at T=0 in most of phase diagram
? Strategy: How far can we push spin-wave theory?
? Going beyond linear spin-wave theory
∗ Putting back Quantum Mechanics:[Si, Sj
]= i~εijkSk
∗ Quantitative 1/S expansion
∗ Zero point fluctuations:EQM = 12
∑k ωk
∗ Expansion parameter for unfrustrated magnets:1−( h
hc)2
2zS
17
Beyond Mean-FieldUse Holstein-Primakoff Bosons to describe fluctuations around classical state
Sz′
R = S − φ†RφR,
Sx′
R =
√2S
2
(φ†R + φR
), Sy
′
R = i
√2S
2
(φ†R − φR
).
Asymptotic expansion in 1/S.Bosonic Hamiltonian:H = H0 +H2 +H3 +H4 + · · ·
Hn ' S2−n/2φ†1 . . . φn
H0 = NS2
JTQ − (ha)2
4[JT0 − JTQ
] ,
H2 = NSJTQ +
S
2
∑k
(Ak + Ck)(φ†kφk + φkφ
†k
)− Bk
(φ†−kφ
†k + φ−kφk
),
Bogoliubov transformation onH2
H2 = NSJTQ + S
∑k
ωk
(γ†kγk +
1
2
),
18
19
Quantum Fluctuations To leading order in 1/S
E = NS(S + 1)JTQ −NS2(ha)2
4[JT0 − JTQ
] +S
2
∑k
ωk.
All ground state parameters get renormalized by the quantumfluctuations:Q, θ, ...
SublatticeMagnetization
〈S〉 = S− 1
N
∑k
〈φ†kφk〉
Strong reduction
of ordered moment 0 0.2 0.4 0.6 0.8 1
hahacr0.1
0.2
0.3
0.4
0.5
XS\
20
Magnetization
At T = 0,
ma = − dE
dBa
2 4 6 8 10
Ba HTL0.2
0.4
0.6
0.8
1
Magnetization
HΜ BL
Ba
Energy
Ecr
Bacr
Mean−Field
Exp. data (T=30mK)Linear Spinwave (1/S)
21
Ordering Wavevector
E = NS(S + 1)JTQ −NS2(ha)2
4[JT0 − JTQ
] +S
2
∑k
ωk.
2 4 6 8
Ba HTL0.01
0.02
0.03
0.04
0.05
ΕHr.l.
u.
L Εcl
HBaLcr
Exp. data (T=20mK)Linear Spinwave (1/S)Dilute Bose Gas
22
Perpendicular Ordered Moment
ST = 〈√
(SbR)2 + (ScR)2〉
2 4 6 8
Ba HTL0.1
0.2
0.3
0.4
0.5
XST\
Mean−FieldLinear Spinwave (1/S)Exp. data (T<0.1K)
Quenching fluctations increases order
23
Interlayer Coupling in Transverse Field
The interlayer couplingJ ′′ frustrates ordering
SR =
Sa
(−)nSb cos Q ·R
Sc sin Q ·R
Sb 6= Sc
Eccentricity
I = SbSc
D=0
Even Layers Odd Layers
J’’=0
J’’=0D=0
Quantum Superposition
24
Energy FunctionalIn terms of two variables:
mixing angleχ and field amplituder
E = 4Sr2 [(JQ − J0 + ha/2) +DQ sin 2χ− J ′′ cos 2χ]
+ r4 [const+ cstf(χ)] +O(r6).
I =SbSc
= tanχ
At the transition field,r = 0+,
tanχ =DQ
J ′′ −√
(J ′′)2 + (DQ)2
Independent ofJ, J ′ !!0 2 4 6 8
Ba HTL1
1.1
1.2
1.3
1.4
1.5
1.6
I
25
Longitudinal Field More complicated
States not known even at classical level!!Transverse Longitudinal
hcr
cc
haa
crh
hFM Cone State FM
DistortedCycloid
c
b
ac
c
bb
0 0.351 1
Tilted Cone State
26
27
Low Field: Distorted Cycloid
SR = S(0, cosφR, sinφR)
Spins are pulled along the field:generate anharmonicities
φR = Q ·R + β sin Q ·R,
β =hc
JT2Q + JT0 − 2JTQ
b
c
Strong Field: Tilted Cone
SR = S
(cos θ cosφR cos η + sin θ sin η
cos θ sinφR
sin θ cos η − cos θ cosφR sin η
)The tilting angle
tan η ≈ Dhccr
(1−
(hccrhc
)2)
Energy functional:
invariant underφR → −φR andη → −ηIsing Symmetry:
Two Chiral Cone Statesc
b
28
Renormalization of the ordering wavevector
Low Field:
0.5 1 1.5 2 2.5 3
Bc HTL0.02
0.04
0.06
0.08
ΕHr.l.
u.
LHigh Field
7.2 7.4 7.6 7.8 8
Bc HTL0.01
0.02
0.03
0.04
0.05
0.06
ΕHr.l.
u.
L Ε*cl
HBcLcr
Exp. data (T=20mK)Linear Spinwave (1/S)Dilute Bose Gas
The incommensuration at the transition field does not agree with classical
value !!
→ weak interactions not included in the Hamiltonian must be present
29
Inelastic Neutron Scattering
• How to explain large scattering continua within spin wave theory ?
Sµν(k, ω) =1
2π~
∫ ∞−∞
dt∑R
〈Sµ0SνR(t)〉e+iωt−k·R
The (unpolarized) inelastic neutron scattering cross section is
d2σ
dωdΩ= |fk|2
∑µν
(δµν − kµkν
)Sµνk,ω,
Sum rule for the total scattering per spin= S(S+1)
Three contributions:
? ω = 0→ elastic processes: Bragg Peaks' (S −∆S)2
? ω 6= 0, inelastic processes
∗ one-magnon scattering' (S −∆S)(1 + 2∆S)∗ two-magnon scattering' ∆S(1 + ∆S)
30
Spin Wave Interaction
• HI = H(3) +H(4) + · · ·
H (3)( )2 H(4)
= + +
1/S1/S
• Strong Interaction
? Low SpinS = 1/2
? Frustrated interactions
? Non-collinear order→ H(3) 6= 0
∗ Couples longitudinal to transverse spin fluctuations
∗ Frequency dependent diagrams→ linear spin waves can decay
∗ Finite linewidth for spin waves
31
Dynamical Spin Correlation
(000)
(011)
(010)
32
Spin Waves of a HelimagnetThree Polarizations:
• ∆Sa = 0, ω0k = ωk −→ Out-of-Plane Fluctuations
• ∆Sa = 1, ω+k = ωk+Q −→ In-Plane Fluctuations
• ∆Sa = −1, ω−k = ωk−Q −→ In-Plane Fluctuations
000 0.5 010 0.5 011 0.5 000Reduced Wavevector
0.1
0.2
0.3
0.4
Energy
meV
0.0
0.1
0.2
0.3
0.4
00 01 0
k
k
0k
33
Spin Waves Renormalizationωk = ωk + Σk,ωk
000 0.5 010 0.5 011 0.5 000Reduced Wavevector
0.1
0.2
0.3
0.4
Energy
meV
0.0
0.1
0.2
0.3
0.4
00 01 0
kk
The1/S expansion givesJJ = 1.131, J′
J′ = 0.648 and DD = 0.72.
Experimentally, we haveJJ = 1.63(5), J′
J′ = 0.83(3).
34
Significant Two-Magnon Scattering for In-plane Fluctuations
011000 0.25 0.5 0.75 010 0.25 0.5 0.75 0.75 0.5 0.25 000Reduced Wavevector
0.0
0.2
0.4
0.6
0.8
0.0
0.2
0.4
0.6
0.8
1.0
Ra
tio00 01 0
0.0
0.2
0.4
0.6
0.8
0.0
0.2
0.4
0.6
0.8
1.0
Inte
nsi
ty
−
0Modes
(A)
(B)
35
Excitation Lineshapes
36
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.Energy meV
2.5
5
7.5
10
12.5
15
Intensity
meV
1
With 1 S CorrectionsLinear Spin WaveScan C
CSpectral shift to higher energy
37
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.Energy meV
5
10
15
20
25Intensity
meV
1
With 1 S CorrectionsLinear Spin WaveScan E
E
Strong Spin Wave Renormalization
38
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.Energy meV
5
10
15
20Intensity
meV
1
With 1 S CorrectionsLinear Spin WaveScan H
H
39
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.Energy meV
5
10
15
20Intensity
meV
1
With 1 S CorrectionsLinear Spin WaveScan H
0.2 0.4 0.6 0.8 1
Energy meV
0.5
1
1.5
2
2.5
3
3.5
4
Intensity
1/meV
40
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.Energy meV
5
10
15
20
25
Intensity
meV
1
With 1 S CorrectionsLinear Spin WaveScan E
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Energy meV
1
2
3
4
5
6
7
8
Intensity
1/meV
“Sharp peaks were observed at high energies near special wavevectorswhere the 2D dispersionωk is at a “saddle” point.”
41
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.Energy meV
10
20
30
40Intensity
meV
1
With 1 S CorrectionsLinear Spin WaveScan G
0.0 0.2 0.4 0.6 0.8 1.0
Energy meV
0
2
4
6
8
10
12
14
Intensity
1/meV
42
0.4 0.6 0.8 1.Energy meV
0.5
1.
1.5
2.
Int.
meV
1
Anomalous peak at high energyMulti-Bound state ωk = ωk + Σk,ωk
Vanishing of damping ImΣk,ωk∼ 0
Artifact of 1/S expansion
(A)
43
44
0.2 0.4 0.6 0.8 1Energy meV
0.5
1
1.5
2
2.5
3
3.5
4Energy
meV
0.4 0.6 0.8 1.Energy meV
0.5
1.
1.5
2.
Int.
meV
1
(C)(K)
45
Conclusion
• Phase Diagram: Spin structure dependent on magnetic field
orientation
• Dzyaloshinskii-Moriya interaction plays an important role
• Quantum fluctuations in leading order gives good agreement with
? Magnetization,
? Incommensurate Ordering Wavevector
? Transverse Spin Order
• Dynamical Spin correlation can account for scattering continuum
? Low Spin, Non-collinear Order, Frustration enhance two-magnon
processes
? Finite Field effect for future
46
d2σ
dωdΩ= |fk|2
(1− k2
a
)Saak,ω +
(1 + k2
a
)Sbbk,ω
. (17)
Saak,ω = − 1π=Θ0
k,ω, (18)
Sbbk,ω = Scck,ω = − 1π=[Θ+
k+Q,ω + Θ−k−Q,ω
], (19)
Sbck,ω = −Scbk,ω = − iπ=[Θ+
k+Q,ω −Θ−k−Q,ω
], (20)
47
Magnons in Saturated Field Image the Hamiltonian
• In general ground statea and excitations are non-trivial.
• BeyondBcr → Fully-Polarized state:Eigenstateof Hamiltonian, No
AF correlations
|FM〉 = | ↑↑↑↑↑↑〉
• Excitation in the saturated phase:Coherentexcitations∆Sz = 1 of
spin-flip states
|φk〉 =1√N
∑R
e−ik·RS−R
• MagnetizationSz is a good quantum number.
• No Decay, No dressing−→ No Quantum Fluctuations,Σ(k, ω) = 0
• Magnons dispersionωk images the Hamiltonian
48
49
The exchange interac-tions can be read offfrom the spinwaves
~ω(k) = JTk −JT0 +gµBB
JTk =1
2
∑δ
JTδ exp(ik · δ)
JTk = Jk ±Dk
R. Coldeaet al., PRL 88, 137203, (2002)
Jk = J cos(2πk) + 2J′cos(πk) cos(πl) + J
′′cos(2πh)
Dk = 2D sin(πk) cos(πl)
J = 0.37(4)meV , J′
= 0.12(8)meV
J′′
= 0.01(7)meV , D = 0.02(0)meV
50
Origin of Dzyaloshinskii-Moriya InteractionOne-electron Hamiltonian
H0 =∑R
∑σ
ε(R)a†σ(R)aσ(R) +∑
R6=R′
∑σ
b(R−R′)a†σ(R)aσ(R′)
+∑
R6=R′
∑σσ′
a†σ(R) [C(R−R′) · σ]σσ′ aσ′(R′)
Transfer Integrals
b(R′ −R) =∑σ
∫drψ?σ(r−R′)H1ψσ(r−R)
C(R′ −R) =∑σσ′
∫drψ?σ(r−R′)σσσ′H1ψσ′(r−R)
and
H1 =p2
2m+ V (R) +
~
2m2c2S · [∇V (R)× p]
51
Interaction term (Hubbard U)
HI = U∑R
∑σσ′
a†σ(R)a†σ′(R)aσ′(R)aσ(R)
Interaction between spins in second-order perturbation theory
E(2)R,R′ = JR,R′S(R)·S(R′)+DM
R,R′ ·S(R)×S(R′)+S(R)↔ΓR,R′ S(R′)
JR,R′ = 4/U |b(R−R′)|2
DMR,R′ = 4i/U [b(R−R′)C(R′ −R)− b(R′ −R)C(R−R′)]↔ΓR,R′ = 4/U [C(R−R′)C(R′ −R) + C(R′ −R)C(R−R′)
−↔I C(R−R′) ·C(R′ −R)
]In general
b(R−R′) = [b(R′ −R)]?
C(R−R′) = [C(R′ −R)]?
52
Symmetry of HamiltonianSpin symmetries
• In Zero-field:U(1)× Z2
? Spin rotation aroundD→ U(1)
? Z2 results from invariance underR→ −R and
Sa → −Sa, Sb → Sb, Sc → −Sc
? U(1) symmetry→ Conservation of quantum numberSa
? Chiral scalar associated withZ2 symmetry
1
2
3
K =∑4 S1 · (S2 × S3)
• In transverse field (Ba 6= 0): Z2 broken explicitly, onlyU(1) left
• In longitudinal field: U(1) broken, onlyZ2 left.
53
Field-induced incommensuration due to Spinons
kF = 1/2 + ε
Incommensuration in-
duced by the applied field
ε =gµBB
ν
ε can exceedεcl-0.4 -0.2 0 0.2 0.4
k
-1
-0.5
0
0.5
1
Esplitting g Bµ
Band Filling of spinons
54
Bosonic Hamiltonian
H0 = NS2
(JTQ −
(ha)2
4[JT0 − JTQ
]) ,H2 = NSJTQ +
S
2
∑k
(Ak + Ck)(φ†kφk + φkφ
†k
)−Bk
(φ†−kφ
†k + φ−kφk
),
Ak =1
2
2Jk + JTQ+k + J
TQ−k − 4JTQ +
[JTQ+k + J
TQ−k − 2Jk
] (ha)2
(hacr)2
Bk =1
2
[2Jk − J
TQ+k − J
TQ−k
] 1 −
ha
hacr
2 ,Ck =
[JTQ+k − J
TQ−k
] hahacr
.
H2 = NSJTQ + S∑k
ωk
(γ†kγk +
12
),
55
whereωk =√A2
k −B2k + Ck .
H = H0 +H2 +H3 +H4 + · · · , (31)
whereHn is proportional toS2−n/2 and consists of normal ordered
products ofn boson operators. TheH1 term is absent, because the
ordering wave vector is determined by minimizing the mean-field energy.
Linear spin wave theory takes into account only the termsH0 andH2 of
the expansion. The higher order terms represent interactions between
magnons. The leading terms in the expansion are
56
H0 = NS2JTQ, (32)
H2 = NSJTQ + S∑k
Ak
(φ†kφk + φ−kφ
†−k
)−Bk
(φ†−kφ
†k + φ−kφk
), (33)
H3 =i
2
√S
2N
∑1,2,3
δ1+2+3 (C1 + C2)(φ†−3φ2φ1 − φ
†1φ†2φ−3
), (34)
H4 =1
4N
∑1,2,3,4
[(A1+3 +A1+4 +A2+3 +A2+4)− (B1+3 +B1+4 +B2+3 +B2+4)− (A1 +A2 +A3 +A4)]φ†1φ
†2φ−3φ−4
+23
(B2 +B3 +B4)(φ†1φ−2φ−3φ−4 + φ†1φ
†2φ†3φ−4
)δ1+2+3+4. (35)
57
Theoretical Approaches to Cs2CuCl4
• Bosonic Sp(N) Large-N Mean Field Theory
C.H. Chung, J.B. Marston and R.H. McKenzie, J. Phys. Cond. Matt.13, 5159 (2001)
? Based on SU(2)≈ Sp(1), Expansion in 1/N
? Support Deconfined Spin 1/2 Bosonic Spinons
? Spinons are gapped
• SU(2) Slave-Boson Mean Field TheoryY. Zhou and X.-G Wen, cond-mat/0210662
? Support Deconfined Spin 1/2 Fermionic Spinons
? Spinons are gapless
Spin Spectral function in terms of
spinons:Experimental signature
C.H. Chung, K. Voelkler and Y.B. Kim, PRB68, 094412 (2003)
S=1∆from Spinons
Neutron Scattering
S=1/2
S=1/2
58
Transverse vs Longitudinal Field:Why such a large difference ?Study the high field region by mapping to a Low Density Bose Gas:controlled approximationTransverse Field:
SaR = 1/2− φ†RφR, S+R = φR, S
− = φ†R
Hardcore Constraint:nR = φ†RφR = 0, 1εk − µ = Jk − J0 +Dk +Ba andVk = Jk + U
Jk = 1/2∑δ Jδ cos(k · δ)
H = (εk − µ)φ†kφk + Vqφ†k+qφ
†k′−qφkφk′
Degeneracy Lifted by the DM term
RG Language: DM term is relevant.
"Spin Split"ε
−Q Q k
59
Low Effective Action⇒ Scalar Bose Gas
H =(k2
2m− µ
)a†↑ka↑k + Vqa
†↑k+qa
†↑k′−qa↑ka↑k′
U(1) symmetry→ O(2) Spin Rotation
Only Two Types of Order at T=0M.P.A. Fisher et al. PRB, 1989Condensation ofBosons↔ BEC of magnons
BEC 〈a†↑k=0〉 =√n↑0 for Bacr −Ba > 0
Mean Field:n↑0 =(Bacr−B
a)
2Veffk=0
SaR = 1/2− n↑0SbR =
√n↑0 cos(Q ·R+ α)
ScR =√n↑0 sin(Q ·R+ α)
cB B
n
60
Longitudinal Direction: Accessing from High Fields:
H = (εk − µ)φ†kφk +1
2√N
(Dk,k′φ
†k+k′φkφk′ + h.c
)+
Vq
2Nφ†k+qφ
†k′−qφkφk′
whereεk − µ = Jk − J0 + h
Low Energy Description: Spin-1/2 Bose Gas
Degenerate Minima atk = −Q,QZ2 symmetry is present to all order.
φk → −φ−k
φ†k → −φ†−k
Isospin
kQ−Q
ε
61
Effective Hamiltonian
Slowly fluctuating isospin variables (a↑(x), a↓(x))
φR =(a↑ke
i(k+Q)·R + a↓kei(k−Q)·R
)Θ(Λ− |k|)
H =(
k2
2m− µ
)[a†↓ka↓k + a†↑ka↑k
]+ V 0(n↓ + n↑)2 + V z(n↓n↑) + g(n↓ + n↑)
(a†↑a†↓ + h.c
)V z > 0 unlike bosons less repulsive than like Bosons.
62
Spin 1/2: Internal degree of Freedom
Internal degrees of Freedom yield Richer Structures:
Single particle condensate|ΨSPC〉 = exp(−φαa†α0
)|0〉
〈a↑〉 6= 0 and〈a↓〉 = 0 −→ Cone state with positive cyclicity
〈a↑〉 = 0 and〈a↓〉 6= 0 −→ Cone state with negative cyclicity
〈a↑〉 = −〈a↓〉 6= 0 −→ Spin fan phase
All these states have long range order
Proposal: Pair CondensateP. Nozieres et al., J.Phys. 43, 1133 (1982)
|ΨPC〉 = exp
(∑k
λαβ(k)a†α ka†β −k
)|0〉
Only the composite particle has long range order. No long range order in
terms of spin-spin correlation function.
63
Pair Condensate
• Favoured when bosons with unlike spinsattract
i.e. V 0 − 2V z < 0
? Bare Values: Unlike bosons are repulsive
? Favours cone state (state with LRO)
Large Fluctuations in 2-D, We need Renormalized Values.
Renormalization Group Flow to first loop
At the critical point, namely, T=0, atµ = 0
dV 0
dl= −Γ
[(V 0)2 − (V z)2 + 2g2
]dV z
dl= −Γ
[(V 0)2 + (V z)2 − 2g2
]dg
dl= −Γg
(V 0 + V z
)
LadderDiagrams
64
In the largel limit:
V 0(0) ∼ 1
2l
V z(l)
V 0(l)=
√1−
(4g(0)
V 0(0)
)2
− 1√1−
(4g(0)
V 0(0)
)2
+ 2
< 0
g(l)
V 0(l) + V z(l)=
g(0)
V 0(0) + V z(0)
• The renormalized interactions are weaker but do not generate an
attraction betweena↓ anda↑.
• Pair Condensate is not favoured.
• Conventional Ordering is favoured in high field (cone state)
65
The boson Green’s function at zero temperature is expressed as
Gk,ω = −i∫ ∞−∞
dteiωt
⟨T
[φk(t)
φ†−k(t)
] [φ†k(0)φ−k(0)
]⟩, (38)
whereT stands for the time ordering operator and〈...〉 denotes a ground state average. The inverse ofthe unperturbed Green’s function is given by a2× 2 matrix that can be represented in terms of theidentity matrixσ0 and the Pauli matricesσ
G(0)−1k,ω = (−2SAk + iη)σ
0+ 2SBkσ
x+ ωσ
z, (39)
whereη = 0+. The self-energy is defined by the Dyson equation,
G−1k,ω = G
(0)−1k,ω − Σk,ω, (40)
and can be parameterized as
Σk,ω = Ok,ωσ0
+Xk,ωσx
+ Zk,ωσz. (41)
The leading order (in1/S) contributions to the self-energy can be divided into two parts
Σk,ω = Σ(4)k + Σ
(3)k,ω. (42)
HereΣ(4)k denotes the vacuum polarization contribution that arises in first order perturbation theory in
66
H4. It is frequency independent and purely real. On the other hand,Σ(3)k,ω denotes the contribution in
second order perturbation theory in the three-magnon interactionH3. It incorporates the effects ofmagnon decay. Using Eq. (39), theΣ(4) contribution to the self-energy is found to be of the form
O(4)k = Ak +
2S
N
∑k′
1
ωk′
[(1
2Bk + Bk′
)Bk′
+(Ak−k′ − Bk−k′ − Ak′ − Ak
)Ak′
],
X(4)k = −Bk +
2S
N
∑k′
1
ωk′[(Bk + Bk′ )Ak′
+
(Ak−k′ − Bk−k′ − Ak′ −
1
2Ak
)Bk′
],
Z(4)k = 0. (43)
67
The contributionΣ(3) is most easily evaluated in the Boguliobov basis (γ) and is equal to
O(3)k,ω =
−S16N
∑k′
[Φ
(1)(k′,k− k
′)]2
+[Φ
(2)(k′,k− k
′)]2[ 1
ωk′ + ωk−k′ − ω − iη+
1
ωk′ + ωk−k′ + ω − iη
],
X(3)k,ω =
−S16N
∑k′
[Φ
(1)(k′,k− k
′)]2−[Φ
(2)(k′,k− k
′)]2[ 1
ωk′ + ωk−k′ − ω − iη+
1
ωk′ + ωk−k′ + ω − iη
],
Z(3)k,ω =
−S16N
∑k′
2Φ
(1)(k′,k− k
′)Φ
(2)(k′,k− k
′)[ 1
ωk′ + ωk−k′ − ω − iη−
1
ωk′ + ωk−k′ + ω − iη
], (44)
where
Φ(1)
(k′,k− k
′) =
(Ck′ + Ck−k′
)(uk′ + vk′ )
(uk−k′ + vk−k′
)− 2Ck
(uk′vk−k′ + vk′uk−k′
),
Φ(2)
(k′,k− k
′) = Ck′ (uk′ + vk′ )
(uk−k′ − vk−k′
)+ Ck−k′
(uk−k′ + vk−k′
)(uk′ − vk′ ) . (45)
68
1 Dynamical Correlation Function
Inelastic neutron scattering experiments probe the dynamical structure factorSµνk,ω . The latter is definedas the Fourier transform of the dynamical spin-spin correlation function
Sµνk,ω =
∫ ∞−∞
dt
2π~e−iωt〈Sµ−k(0)S
νk(t)〉. (46)
Hereµ, ν = (a, b, c) and the Fourier-transformed spin operators are defined bySµk = 1√
N
∑R SµRe
−ik·R.
It is convenient to introduce the time-ordered correlation function in the rotated coordinate system
Fαβk,ω = −i
∫ ∞−∞
dte−iωt〈TSα−k(0)S
βk (t)〉, (47)
whereα, β = (x, y, z). The dynamical structure factor is related to the imaginary part of the timeordered correlation function in the following way
Saak,ω = −
1
π=Fxxk,ω, (48)
Sbbk,ω = S
cck,ω = −
1
π=[Θ
+k+Q,ω + Θ
−k−Q,ω
], (49)
Sbck,ω = −Scbk,ω = −
i
π=[Θ
+k+Q,ω −Θ
−k−Q,ω
], (50)
69
where
Θ±k,ω =
1
4
Fzzk,ω + F
yyk,ω ± i
(Fzyk,ω − F
yzk,ω
). (51)
To proceed further, we expand the dynamical correlation functions up to the first subleading order in1/S. The two diagonal parts of the transverse fluctuations are
Fxxk,ω =
S
2c2xTr
[(σ
0 − σx)Gk,ω
],
Fyyk,ω =
S
2c2yTr
[(σ
0+ σ
x)Gk,ω
], (52)
where the Green’s function is given by Eq. (40) and where
cx = 1−1
4SN
∑k
(2v
2k − ukvk
),
cy = 1−1
4SN
∑k
(2v
2k + ukvk
). (53)
We note that these results are valid only up to the first subleading order in1/S. The mixing of transverseand longitudinal fluctuations is expressed as
i(Fyzk,ω − F
zyk,ω
)= cy
P
(1)k,ωTr
[(1 + σ
x)Gk,ω
]+ P
(2)k,ωTr
[σzGk,ω
], (54)
70
where
P(1)k,ω = S
4N
∑k′
Φ(1) (
k′,k− k
′) (uk′vk−k′ + vk′uk−k′
) [ 1
ωk′ + ωk−k′ − ω − iη+
1
ωk′ + ωk−k′ + ω − iη
],(55)
P(2)k,ω = S
4N
∑k′
Φ(2) (
k′,k− k
′) (uk′vk−k′ + vk′uk−k′
) [ 1
ωk′ + ωk−k′ − ω − iη−
1
ωk′ + ωk−k′ + ω − iη
].(56)
Finally the longitudinal fluctuations are
Fzzk,ω = F
(0)zzk,ω + F
(1)zzk,ω . (57)
HereF (0)zz andF (1)zz denote the leading and subleading contributions respectively and are given by
F(0)zzk,ω = −
1
2N
∑k′
(uk′vk−k′ + vk′uk−k′
)2 [ 1
ωk′ + ωk−k′ − ω − iη+
1
ωk′ + ωk−k′ + ω − iη
], (58)
F(1)zzk,ω =
1
2S
(P
(1)k,ω
)2Tr[(σ
0+ σ
x)Gk,ω
]+(P
(2)k,ω
)2Tr[(σ
0 − σx)Gk,ω
]+ 2P
(1)k,ωP
(2)k,ωTr
[σzGk,ω
].(59)
71
The (unpolarized) inelastic neutron scattering cross section is
d2σ
dωdΩ= |fk|2
∑µν
(δµν − kµkν
)Sµνk,ω,
= |fk|2(
1− k2a
)Saak,ω +
(1 + k
2a
)Sbbk,ω
. (60)
72
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