DYNAMICAL PROPERTIES OF THE ANISOTROPIC TRIANGULAR QUANTUM ANTIFERROMAGNET WITH DZYALOSHINSKII-MORIYA INTERACTION March 16, 2006March Meeting 2006, Baltimore.

DYNAMICAL PROPERTIES OF THE DYNAMICAL PROPERTIES OF THE ANISOTROPIC TRIANGULAR QUANTUM ANISOTROPIC TRIANGULAR QUANTUM

ANTIFERROMAGNET WITH ANTIFERROMAGNET WITH DZYALOSHINSKII-MORIYA INTERACTIONDZYALOSHINSKII-MORIYA INTERACTION

March 16, 2006March Meeting 2006, Baltimore 1/13

Rastko SknepnekRastko Sknepnek, Denis Dalidovich, John Berlinsky, Junhua , Denis Dalidovich, John Berlinsky, Junhua Zhang, Catherine KallinZhang, Catherine Kallin

Department of Physics and AstronomyDepartment of Physics and AstronomyMcMaster UniversityMcMaster University

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OutlineOutline

• MotivationMotivation

• 1/S expansion1/S expansion

• Renormalized spectrumRenormalized spectrum

• Structure factorStructure factor

• SummarySummary

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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..

extended scattering continuum.extended 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?

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Microscopic HamiltonianMicroscopic Hamiltonian

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)

March 16, 2006March Meeting 2006, Baltimore

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

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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.

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Spin wave theorySpin wave theory

March 2, 2006Iowa State University 7/13

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

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

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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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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

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(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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G scanG scan

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

J = 0.374 meVJ = 0.374 meV

D = 0.01meVD = 0.01meV

J’ = 0.128 meVJ’ = 0.128 meV

March 16, 2006March Meeting 2006, Baltimore

1/S theory1/S theory

experimentexperiment

March 2, 2006Iowa State University 13/13

Summary and conclusionsSummary and conclusions

...to be continued...to be continued