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