spin structure factor of the frustrated quantum magnet cs 2 cucl 4

SPIN STRUCTURE FACTOR OF THESPIN STRUCTURE FACTOR OF THEFRUSTRATED QUANTUM MAGNET CsFRUSTRATED QUANTUM MAGNET Cs22CuClCuCl44

March 2, 2006Iowa State University 1/30

Rastko SknepnekRastko Sknepnek

Department of Physics and AstronomyDepartment of Physics and AstronomyMcMaster UniversityMcMaster University

In collaboration with:In collaboration with:

Denis DalidovichDenis DalidovichA.A. John BerlinskyJohn BerlinskyJunhua ZhangJunhua ZhangCatherine KallinCatherine Kallin

March 2, 2006Iowa State University 2/30

OutlineOutline

• MotivationMotivation

• Spin waves vs. spinonsSpin waves vs. spinons

• Experiment on CsExperiment on Cs22CuClCuCl44

• Nonlinear spin wave theory for CsNonlinear spin wave theory for Cs22CuClCuCl44

• SummarySummary

March 2, 2006Iowa State University 3/30

MotivationMotivation

(R. Coldea, (R. Coldea, et alet al., PRB ., PRB 6868, 134424 (2003)) , 134424 (2003))

Neutron scattering measurements on quantum magnet CsNeutron scattering measurements on quantum magnet Cs22CuClCuCl44..

dynamical correlations aredynamical correlations aredominated by an extendeddominated by an extendedscattering continuum.scattering continuum.

Signature of deconfined, fractionalizedSignature of deconfined, fractionalizedspin-1/2 (spinon) excitations? spin-1/2 (spinon) excitations?

Can this broad scattering continuum be explained within aCan this broad scattering continuum be explained within aconventional 1/S expansion?conventional 1/S expansion?

(Complementary work: M. Y. Veillette, et al., PRB (2005))

March 2, 2006Iowa State University 4/30

Spin WavesSpin Waves

Heisenberg Hamiltonian:Heisenberg Hamiltonian:

• J<0 – ferromagnetic ground stateJ<0 – ferromagnetic ground state• J>0 – antiferromagnet (Néel ground state)J>0 – antiferromagnet (Néel ground state)

• Spin waves are excitations of the (anti)-ferromagnetically ordered state.Spin waves are excitations of the (anti)-ferromagnetically ordered state.• Exciting a spin wave means creating a quasi-particle called Exciting a spin wave means creating a quasi-particle called magnonmagnon..• Magnons are S=1 bosons.Magnons are S=1 bosons.

Dispersion relations (kDispersion relations (k0):0):

2| | ( | |)k

S J a k

(ferromagnet)(ferromagnet)

| |k

JzSa k

(antiferromagnet)(antiferromagnet)

,

ˆ ˆˆi j

i j

H J S S

March 2, 2006Iowa State University 5/30

Ground state of an antiferromagnetGround state of an antiferromagnet

Antiferromagnetic Heisenberg Hamiltonian:Antiferromagnetic Heisenberg Hamiltonian:

, ,

ˆ ˆ ˆ ˆ ˆ ˆˆ2

z zi j i j i j

i j i j

JH J S S S S S S (J>0)

State State can not be the ground state - it can not be the ground state - it is notis not an eigenstate of the Hamiltonian. an eigenstate of the Hamiltonian.

• Antiparallel alignment gains energy only from the Antiparallel alignment gains energy only from the z-zz-z part of the Hamiltonian. part of the Hamiltonian.• True ground state - the spins fluctuate so the system gains energy from the True ground state - the spins fluctuate so the system gains energy from the spin-flipspin-flip terms. terms.

Ground state of the Heisenberg antiferromagnet shows quantum fluctuations.Ground state of the Heisenberg antiferromagnet shows quantum fluctuations.

How important is the quantum nature of the spin?

quantum correction

classical energy~

1

SReduction of the staggered magnetization due to quantum fluctuations:

March 2, 2006Iowa State University 6/30

Spin Liquid and Fractionalization in 1dSpin Liquid and Fractionalization in 1d

• In D=1 quantum fluctuations destroy In D=1 quantum fluctuations destroy long range orderlong range order..• Spin-spin correlation falls off as a power law.Spin-spin correlation falls off as a power law.• Ground state is a singlet with total spin SGround state is a singlet with total spin Stottot=0 (exactly found using Bethe ansatz). =0 (exactly found using Bethe ansatz).

Excitations are not spin-1 magnons but pairs of fractionalized spin-1/2 spinons.Excitations are not spin-1 magnons but pairs of fractionalized spin-1/2 spinons.

(D.A.Tennant, et al, PRL (1993))(D.A.Tennant, et al, PRL (1993))

Prototypical system KCuFPrototypical system KCuF33..

(half-integer spin)(half-integer spin)

• Spinons appear in pairs.Spinons appear in pairs.• Excitation spectrum is a continuum with two softExcitation spectrum is a continuum with two softpoints (0 and points (0 and ))

Fractionalization: Excitations have quantum numbers thatFractionalization: Excitations have quantum numbers thatare fractions of quantum numbers of the local degrees ofare fractions of quantum numbers of the local degrees offreedom.freedom.

Geometrical Frustration in 2dGeometrical Frustration in 2d

• Ising-like ground state is possible only on bipartite lattices.Ising-like ground state is possible only on bipartite lattices.

• Non-bipartite lattices (e.g., triangular) exhibit Non-bipartite lattices (e.g., triangular) exhibit geometrical geometrical frustrationfrustration..

• On an isotropic triangular lattice the ground state is a three sub-lattice Néel state. On an isotropic triangular lattice the ground state is a three sub-lattice Néel state.

March 2, 2006Iowa State University 7/30

March 2, 2006Iowa State University 8/30

Resonating valence bond (RVB)Resonating valence bond (RVB) (P.W. Anderson, Mater. Res. Bull. (1973))

• Ground state – linear superposition of disordered Ground state – linear superposition of disordered valence bondvalence bond configurations. configurations.• Each bond is formed by a pair of spins in a Each bond is formed by a pair of spins in a singlet statesinglet state..

RVB state has the following properties:RVB state has the following properties:

• spin rotation SU(2) symmetry is not broken.spin rotation SU(2) symmetry is not broken.• spin-spin, dimer-dimer, etc. correlations are spin-spin, dimer-dimer, etc. correlations are exponentially decaying – no LRO.exponentially decaying – no LRO.• excitations are gapped spin-1/2 deconfined spinons. excitations are gapped spin-1/2 deconfined spinons.

RVB state is an example of a two dimensional spin liquid.RVB state is an example of a two dimensional spin liquid.

Spin crystalSpin crystal Spin LiquidSpin Liquid

Ground stateGround state Semiclassical Néel orderSemiclassical Néel order Quantum LiquidQuantum Liquid

Order parameterOrder parameter Staggered magnetizationStaggered magnetization No local order parameterNo local order parameter

ExcitationsExcitations Gapless magnonsGapless magnons Gapped deconfined spinonsGapped deconfined spinons

Is there any experimental realization of Is there any experimental realization of two dimensional spin liquid?two dimensional spin liquid?

March 2, 2006Iowa State University 9/30

CsCs22CuClCuCl4 4 - a spin-1/2 frustrated quantum magnet.- a spin-1/2 frustrated quantum magnet.

Crystalline structure:Crystalline structure:

• Orthorhombic (Pnma) structure.Orthorhombic (Pnma) structure.• Lattice parameters (at T=0.3K)Lattice parameters (at T=0.3K) aa = 9.65Å = 9.65Å bb = 7.48Å = 7.48Å cc = 12.26Å. = 12.26Å.• CuClCuCl44

2-2- tetrahedra arranged in layers. tetrahedra arranged in layers.

((bc bc plane) separated along plane) separated along aa by Cs by Cs++ ions. ions.

March 2, 2006Iowa State University 10/30

CsCs22CuClCuCl4 4 is an insulator with each Cu is an insulator with each Cu2+2+ carrying a spin 1/2. 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 superexchangeSpins interact via antiferromagnetic superexchangecoupling.coupling.• Superexchange route is mediated by two nonmagneticSuperexchange route is mediated by two nonmagneticClCl-- ions. ions.• Superexchange is mainly restricted to the Superexchange is mainly restricted to the bcbc planes planes

March 2, 2006Iowa State University 11/30

Coupling constantsCoupling constants

Measurements in high magnetic field (12T):Measurements in high magnetic field (12T):

J = 0.374(5) meV J = 0.374(5) meV

J’ = 0.128(5) meVJ’ = 0.128(5) meV

J’’= 0.017(2) meVJ’’= 0.017(2) meV

JJJ’J’

J’’J’’

High magnetic field experiment also observe small splitting into two magnon branches.High magnetic field experiment also observe small splitting into two magnon branches.

17.7o

DD

Indication of a weak Dzyaloshinskii-Moriya (DM) interaction.Indication of a weak Dzyaloshinskii-Moriya (DM) interaction.

D = 0.020(2) meVD = 0.020(2) meV DM interaction creates an easy plane anisotropy. DM interaction creates an easy plane anisotropy.

Below TBelow TNN=0.62K the interlayer coupling=0.62K the interlayer coupling

J’’ stabilizesJ’’ stabilizes long range order. long range order. The order is an The order is an incommensurateincommensurate spin spinspiral in the (spiral in the (bcbc) plane.) plane.

0

12 ( )

2 bQ e

00=0.030(2)=0.030(2)

The HamiltonianThe Hamiltonian

Relatively large ratio Relatively large ratio J’/J≈1/3J’/J≈1/3 and and considerable dispersionconsiderable dispersion along both along both bb and and cc directions directionsindicate indicate two dimensionaltwo dimensional nature of the system. nature of the system.

Effective Hamiltonian:Effective Hamiltonian:

1 2 1 2 1 2' ( ) ( 1) ( )n

R R RR R R R RR

H JS S J S S S D S S S

March 2, 2006Iowa State University 12/30

March 2, 2006Iowa State University 13/30

A few remarks...A few remarks...

• A strong scattering continuum A strong scattering continuum does not does not automaticallyautomatically entail a spin liquid phase.entail a spin liquid phase.• Magnon-magnonMagnon-magnon interaction can cause a broad scattering interaction can cause a broad scattering continuum in a conventional magnetically ordered phase. continuum in a conventional magnetically ordered phase.

In CsIn Cs22CuClCuCl44 strong scattering continuum is expected because: strong scattering continuum is expected because:

• low (S=1/2) spinlow (S=1/2) spin and the and the frustrationfrustration lead to a small ordered moment and strong lead to a small ordered moment and strongquantum fluctuationsquantum fluctuations• the magnon interaction in non-collinear spin structures induces coupling betweenthe magnon interaction in non-collinear spin structures induces coupling betweentransversetransverse and and longitudinallongitudinal spin fluctuations spin fluctuations additional additional dampingdamping of the spin waves. of the spin waves.

It is necessary to go beyond linear spin wave theory by taking into account magnon-magnonIt is necessary to go beyond linear spin wave theory by taking into account magnon-magnoninteractions within a framework of 1/S expansion.interactions within a framework of 1/S expansion.

Spin wave theorySpin wave theory

March 2, 2006Iowa State University 14/30

Classical ground state is an Classical ground state is an incommensurate incommensurate spin-spiralspin-spiralalong strong-bond (along strong-bond (bb) direction with the ordering wave) direction with the ordering wavevector Q.vector Q.

12 ( )

2 bQ e

In order to find ground state energy we introduce a local reference frame:In order to find ground state energy we introduce a local reference frame:

cos( ) sin( )xR R R

S S Q R S Q R

sin( ) cos( )zR R R

S S Q R S Q R

yR R

S S

Classical ground state energy:Classical ground state energy:

2 (0) 2( ) TG Q

S E Q S J

TQ Q Q

J J iD

3cos 2 'cos cos

2 2yx

xk

kkJ J k J

32 sin cos

2 2yx

k

kkD iD

Ordering wave vector:Ordering wave vector:

(linear)(linear)

• D = 0D = 01 '

arcsin 0.05472

J

J

• D = 0.02meVD = 0.02meV 0.0533

1/S expansion1/S expansion

To go beyond linear spin-wave theory we employ Holstein-Primakoff transformation:To go beyond linear spin-wave theory we employ Holstein-Primakoff transformation:

† †1 12 1 2 1

2 4R R R R R R R R RS S iS S a a a S a a a

S S

† † † †1 12 1 2 1

2 4R R R R R R R R RS S iS Sa a a Sa a a

S S

†R R R

S S a a

†' , '

[ , ]R R R R

a a

Where Where aa’s are ’s are bosonicbosonic spin-wave creation and annihilation operators. spin-wave creation and annihilation operators.

'[ , ] 0

R Ra a † †

' '[ , ] 0

R Ra a

The Hamiltonian for the The Hamiltonian for the interacting magnonsinteracting magnons becomes: becomes:

2 (0) (2) (3) (4)( ) ( )GS E Q H H H H

March 2, 2006Iowa State University 15/30

March 2, 2006Iowa State University 16/30

(2) † † †22

kk k k k k k k

k

BS A a a a a a a

H

Quadratic part of the Hamiltonian:Quadratic part of the Hamiltonian:

1

4 2T T Tk

Qk Q k Q k

JA J J J

1

2 4T Tk

k Q k Q k

JB J J

Magnon-magnon interaction is described by:Magnon-magnon interaction is described by:

1 2 3 1 2 3 1 2 3

1 2 3

(3) † † †1 2 ,

, ,

,2 2 k k k k k k k k k

k k k

i Sf k k a a a a a a

N

H

1 2 3 4 1 2 3 4

1 2 3 41 2 3 4 1 2 3 4 1 2 3 4

† †1 1 2 3 4 ,(4)

† † † †, , , 2 1 2 3 ,

, , ,1

4 , ,

k k k k k k k k

k k k kk k k k k k k k k k k k

f k k k k a a a a

N f k k k a a a a a a a a

H

2 2

k k kA B

March 2, 2006Iowa State University 17/30

Ground state energy and ordering wave vectorGround state energy and ordering wave vector

0

2 (1) (2)2( ) ( ) ( )T

G G GQE Q S J SE Q E Q

Quantum corrections of the ordering wave-vector are:

In a 1/S expansion quantum corrections of the ground state energy are:

(1) (2)

0 22 2

Q QQ Q

S S

Where(1) 1

( ) TG Q k

k

E Q JN

0

12

(1)

2

1TT

Q Q kk k

k k Q

JJ A BQ

NQ Q

etc.

March 2, 2006Iowa State University 18/30

1/S theory D = 0 meVD = 0 meV D = 0.02 meVD = 0.02 meV

S2EG(0)/J -0.265-0.265 -0.291-0.291

SEG(1)/J -0.157-0.157 -0.138-0.138

EG(2)/J -0.0332-0.0332 -0.0256-0.0256

Emin/J -0.459-0.459 -0.454-0.454

Experiments (Cs2CuCl4)

EG/J - -0.5*-0.5*

*Y. Tokiwa, *Y. Tokiwa, et al.et al., cond-mat/0601272 (2006), cond-mat/0601272 (2006)

1/S theory D = 0 meVD = 0 meV D = 0.02 meVD = 0.02 meV

Q0/2 0.55470.5547 0.55330.5533

Q(1)/2 -0.0324-0.0324 -0.0228-0.0228

Q(2)/2 -0.011-0.011 --

Q/2 0.51130.5113 0.53080.5308

Experiments (Cs2CuCl4)

Q/2 - 0.530(2)**0.530(2)**** ** R. Coldea, R. Coldea, et alet al., PRB ., PRB 6868, 134424 (2003), 134424 (2003)

March 2, 2006Iowa State University 19/30

Green’s functionGreen’s function

To calculate physical observables we need Green’s function for magnons.To calculate physical observables we need Green’s function for magnons.

1 (0) 1ˆ ˆ ˆ( , ) ( , ) ( , )G k G k k

††

( )ˆ ˆ( , ) (0) ( )( )

i t kk k

k

a tiG k dte T a a t

a t

(0) 12 2ˆ ( , )2 2

k k

k k

SA i SBG k

SB SA i

(4) (3) (4) (3)11 11 12 12(4) (3)

(4) (3) (4) (3)21 21 22 22

( ) ( , ) ( ) ( , )ˆ ˆ ˆ( , ) ( ) ( , )( ) ( , ) ( ) ( , )

k k k kk k k

k k k k

’’s are the self-energies which we calculate to the order 1/S.s are the self-energies which we calculate to the order 1/S.

March 2, 2006Iowa State University 20/30

Sublattice magnetizationSublattice magnetization

Staggered magnetization:Staggered magnetization:

011

1,

2i

k

dM S G k e

N i

To the lowest order in 1/S:To the lowest order in 1/S:

(1) 11

2k

k k

AM

N

March 2, 2006Iowa State University 21/30

The second order correction has two contributions:The second order correction has two contributions:

(1)(2)

2I

QM

S

0(2) (0) (0)

11

1 ˆ ˆˆ, , ,2

iII

k

dM e G k k G k

N i

CsCs22CuClCuCl44

Numerical integration carried using DCUHRE method – Cuba 1.2 library, by T. Hahn)Numerical integration carried using DCUHRE method – Cuba 1.2 library, by T. Hahn)

March 2, 2006Iowa State University 22/30

Energy spectrumEnergy spectrum

The renormalized magnon energy spectrum is determined by poles of the Green’s function.The renormalized magnon energy spectrum is determined by poles of the Green’s function.

(0) 1ˆ ˆRe det , , 0k k

G k k

Which leads to the nonlinear self-consistency equation:Which leads to the nonlinear self-consistency equation:

, , ,k k k k

f A B k

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On renormalization of coupling constants.On renormalization of coupling constants.

(R. Coldea, R. Coldea, et alet al., PRB ., PRB 6868, 134424 (2003)), 134424 (2003))

In order to quantify the “quantum” renormalization of the magnon dispersion relationIn order to quantify the “quantum” renormalization of the magnon dispersion relationone fits the 1/S result to a linear spin-wave dispersion with “effective” coupling constants.one fits the 1/S result to a linear spin-wave dispersion with “effective” coupling constants.

1/S theory1/S theory Exp.Exp.

Jren/Jbare 1.1311.131 1.63(5)1.63(5)

J’ren/J’bare 0.6480.648 0.84(9)0.84(9)

Dren/Dbare 0.720.72 --

Spin structure factorSpin structure factor

( )1, (0) ( )

2ik i k i t k R

ROR

S k dt S S t e

Neutron scattering spectra is expressed in terms of Fourier-transformed real-timeNeutron scattering spectra is expressed in terms of Fourier-transformed real-timedynamical correlation function:dynamical correlation function:

Magnon-magnon interaction leads to the mixing of longitudinal (Magnon-magnon interaction leads to the mixing of longitudinal () and transversal () and transversal ())modes (detailed derivation in T. Ohyama&H. Shiba, J. Phys. Soc. Jpn. (1993))modes (detailed derivation in T. Ohyama&H. Shiba, J. Phys. Soc. Jpn. (1993))

, , ,tot xx yyx yS k p S k p S k

, ,yyS k S k

1, , , ,

4xxS k S k S k S k

March 2, 2006Iowa State University 24/30

March 2, 2006Iowa State University 25/30

(R. Coldea, R. Coldea, et alet al., PRB ., PRB 6868, 134424 (2003)), 134424 (2003))

G scanG scan

Scan along a path at the edge of the Brillouin zone. Scan along a path at the edge of the Brillouin zone. kkx x = = kkyy = 2 = 2(1.53-0.32(1.53-0.32-0.1-0.122) )

Energy resolution Energy resolution E=0.016meVE=0.016meV

Momentum resolution Momentum resolution k/2k/2 = 0.085 = 0.085

D = 0.02meV

linear SW theory k=0.22meV

linear SW theory k+/-Q= 0.28meV

two-magnon continuum

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Energy resolution Energy resolution E=0.016meVE=0.016meV

Momentum resolution Momentum resolution k/2k/2 = 0.085 = 0.085

Energy resolution Energy resolution E=0.002meVE=0.002meV

Momentum resolution Momentum resolution k/2k/2 = 0 = 0

Significant broadening due to finite momentum resolution.Significant broadening due to finite momentum resolution.

Near G point the dispersion relation has large modulation along Near G point the dispersion relation has large modulation along bb direction. direction.

March 2, 2006Iowa State University 27/30

G scan

exp.

What happens if we lower D?What happens if we lower D?

Energy resolution Energy resolution E=0.016meVE=0.016meV

Momentum resolution Momentum resolution k/2k/2 = 0.085 = 0.085D = 0.01meVD = 0.01meV

March 2, 2006Iowa State University 28/30

Energy resolution Energy resolution E=0.016meVE=0.016meV Momentum resolution Momentum resolution k/2k/2 = 0.085 = 0.085

Smaller value for D fits experiments better!Smaller value for D fits experiments better!

D=0.02meVD=0.02meV

experimentalexperimentalposition of the peak position of the peak = 0.10(1) meV= 0.10(1) meV

March 2, 2006Iowa State University 28/30

Summary and conclusionsSummary and conclusions

• derived derived non-linear spin wavenon-linear spin wave theory for the frustrated triangular magnet Cs theory for the frustrated triangular magnet Cs22CuClCuCl44. .

We have...We have...

• calculated quantum corrections to the calculated quantum corrections to the ground state energyground state energy and and sublattice magnetizationsublattice magnetizationto the 2to the 2ndnd order in 1/S. order in 1/S.

• calculated calculated spin structure factorspin structure factor and compared it to the recent inelastic neutron scattering and compared it to the recent inelastic neutron scatteringdata data

We find that 1/S theory:We find that 1/S theory:

• gives good prediction for the ground state energy and ordering wave vector.gives good prediction for the ground state energy and ordering wave vector.

• significantly underestimates renormalization of the coupling constants.significantly underestimates renormalization of the coupling constants.

• significant scattering weight is shifted toward higher energies, but not sufficient to significant scattering weight is shifted toward higher energies, but not sufficient to fully explain experiments. fully explain experiments.

March 2, 2006Iowa State University 29/30

Other approachesOther approaches

• 1d coupled chains1d coupled chains

M. Bocquet, M. Bocquet, et alet al., PRB (2001)., PRB (2001)O. Starykh, L. Balents, (unpublished) (2006)O. Starykh, L. Balents, (unpublished) (2006)

• Algebraic vortex liquidAlgebraic vortex liquid

J. Alicea, J. Alicea, et alet al., PRL (2005)., PRL (2005)J. Alicea, J. Alicea, et alet al., PRB (2005)., PRB (2005)

• High-T expansionHigh-T expansion

W.Zheng, W.Zheng, et alet al., PRB (2005)., PRB (2005)

• Proximity of a spin liquid quantum critical pointProximity of a spin liquid quantum critical point

S.V. Isakov, et al., PRB (2005)S.V. Isakov, et al., PRB (2005)

March 2, 2006Iowa State University 30/30

Thank You!Thank You!