conservation laws and magnon decay in quantum spin liquids

Spin Waves - 2007, St PetersburgSpin Waves - 2007, St Petersburg

Conservation laws and magnon Conservation laws and magnon decay in quantum spin liquidsdecay in quantum spin liquids

Igor ZaliznyakIgor Zaliznyak

Neutron Scattering Group, Brookhaven National Laboratory


CollaboratorsCollaborators

• M. B. Stone

• C. Broholm, D. Reich, T. Hong

• S.-H. Lee

• S. V. Petrov

/ U. Virginia/ U. Virginia

OAK RIDGE NATIONAL LABORATORY


Particles in the UniverseParticles in the Universe

MeVMeV GeVGeV


Quasiparticles in condensed matterQuasiparticles in condensed matter

neutron out

neutron outkkff

meV, meV, μμeVeV

Quasiparticle:Quasiparticle:

phonon, magnonphonon, magnon

q = kq = kii - k - kff

neutron in

neutron in

kk ii

1 meV = 11.6 K1 meV = 11.6 K


Neutron scattering: how neutrons measure Neutron scattering: how neutrons measure quasiparticles.quasiparticles.

fi

fiffiii

zif

zf

i

f

m

k

m

kηEηEηSηS

k

k

dEdΩ

Ed

,

,b,,

22

2222 q

q

fi kkq m

k

m

kηEηEE fi

iiff 22

22

, ,2

22

20

,

dt

tMMeeq

qqr

k

k

dEdΩ

Ed

jjjj

iEti

mi

fmag jj RRqq

magnetic scattering length, rm = -5.39*10-13 cm

jj

tiiEti

jji

fnuc dteeebb

k

k

dEdΩ

Edjj

,

*,

20

2RqRqq

nuclear scattering length, b ~ 10-13 cm qqq

ESdEdΩ

Ed~

,2

Long-lived quasiparticle (magnon)

delta-function singularity in cross-section


What is quantum liquid?What is quantum liquid?

• What is liquid?− no shear modulus− no elastic scattering = no static correlation of density fluctuations

‹ρ(r1,0)ρ (r2,t)› → 0t → ∞

• What is quantum liquid? − all of the above at T → 0 (i.e. at temperatures much lower than inter-particle interactions in the system)

• Elemental quantum liquids:− H, He and their isotopes− made of light atoms strong quantum fluctuations


ε(q

) (K

elvi

n)

q (Å-1)

phonon

roton

maxonwhatsgoingon?

Excitations in quantum Bose liquid: Excitations in quantum Bose liquid: superfluid superfluid 44HeHe

Woods & Cowley, Rep. Prog. Phys. 36 (1973)


The “cutoff point” of the quasiparticle The “cutoff point” of the quasiparticle spectrum in the quantum Bose-liquidspectrum in the quantum Bose-liquid


Breakdown of the excitations in Breakdown of the excitations in 44He: He: experimentexperiment

H = qε (q) aq+

aq + q,q′ Vq,q′(aqa+q′a+

q-q′ + H.c.) + …


Roton decays and conservation lawsRoton decays and conservation laws

• Breakdown of roton quasiparticle spectrum at E > 2 due to pair decays satisfies:

– Particle non-conservation: cubic terms in the boson Hamiltonian

=> Vq,q′(aqa+q′a+

q-q′ + H.c.)

– Energy-momentum conservation

qq’

q”

q = q’ + q”

(q) = (q’) + (q”)


Quantum spin liquid: what is it?Quantum spin liquid: what is it?

• Quantum liquid state for a system of Heisenberg spins

H = J|| SiSi+||+ JSiSi

• Exchange couplings J||, J through orbital overlaps may be different

− J||/J >> 1 (<<1) parameterize quasi-1D (quasi-2D) case

Coupled chains

J||/J>> 1Coupled planes

J||/J<<1• no static spin correlations

‹Siα (0)Sj

β (t)› → 0, i.e. ‹Si

α (0)Sjβ (t)› = 0

• hence, no elastic scattering (e.g. no magnetic Bragg peaks)

t → ∞


Simple example: coupled S=1/2 dimersSimple example: coupled S=1/2 dimers

H = J0 S1S2J0/2 (S1 + S2)2 + const.

Single dimer: antiferromagnetically coupled S=1/2 pair

J0 > 0

0 = J0

singlet

triplet


Simple example: coupled S=1/2 dimersSimple example: coupled S=1/2 dimers(

q)

q/(2)

0 = J0

H = J0 S2iS2i+1J1 (S2i S2i+2)

Chain of weakly coupled dimers

Dispersion (q) ~ J0 + J1cos(q)

J0

J1

triplet


1D array of dimers (aka alternating chain)1D array of dimers (aka alternating chain)

Chains of weakly interacting dimers in

Cu(NO3)2x2.5D2O

CuCu2+2+ 3d9

S=1/2

E (

me

V)


Weakly interacting dimers in Weakly interacting dimers in Cu(NOCu(NO33))22x2.5Dx2.5D22OO

D. A. Tennant, C. Broholm, et. al. PRB 67, 054414 (2003)

Spin excitations never cross into 2-particle continuum and

live happily ever after


0.0 0.2 0.4 0.6 0.8 1.0

(q)

− quasiparticles with a gap ≈ 0.4J at q =

2 (q) = 2 + (cq)2

q/(2)

2

1D quantum spin liquid: Haldane spin chain1D quantum spin liquid: Haldane spin chain

− short-range-correlated “spin liquid” Haldane ground state

• Heisenberg antiferromagnetic chain with S = 1S = 1

Quantum Monte-Carlo for 128 spins.

Regnault, Zaliznyak & Meshkov, J. Phys. C (1993)


Spin-quasiparticles in Haldane chains in Spin-quasiparticles in Haldane chains in CsNiClCsNiCl33

NiNi2+2+ 3d8

J = 2.3 meV = 26 K J = 0.03 meV = 0.37 K = 0.014 J

D = 0.002 meV = 0.023 K = 0.0009 J

3D magnetic order below TN = 4.84 Kunimportant for high energies

S=1 S=1 chains


Spin-quasiparticles in Haldane chains in Spin-quasiparticles in Haldane chains in CsNiClCsNiCl33


Magnon quasiparticle breakdown in CsNiClMagnon quasiparticle breakdown in CsNiCl33

I. A. Zaliznyak, S.-H. Lee, S. V. Petrov, PRL 017202 (2001)


Spectrum termination in the dimer-chain Spectrum termination in the dimer-chain material IPA-CuClmaterial IPA-CuCl33

T. Masuda, A. Zheludev, et. al., PRL 96 047210 (2006)


weak interaction

2D quantum spin liquid: a lattice of 2D quantum spin liquid: a lattice of frustrated dimersfrustrated dimers

M. B. Stone, I. Zaliznyak, et. al. PRB (2001)

(C4H12N2)Cu2Cl6 (PHCC)

− singlet disordered ground state

− gapped triplet spin excitation

strong interaction

CuCu2+2+ 3d9

S=1/2

h

l


Magnon spectrum termination line in PHCCMagnon spectrum termination line in PHCC

max{E2-particle (q)}

min{E2-particle (q)}

E1-particle(q)

Spectrum termination line


PHCC: dispersion along the diagonalPHCC: dispersion along the diagonal800

600

400

200

0

Q = (0.5,0,-1.5) resolution-corrected fit

400

300

200

100

0

Q = (0.25,0,-1.25)resolution-corrected fit

200

150

100

50

0

7654321


Inte

nsity

(co

unts

in 1

m

in)

200

150

100

50

0


150

100

50

0


120

80

40

0

7654321

Q = (0,0,1) resolution-corrected fit

E (meV) E (meV)


2D map of the spectrum along both 2D map of the spectrum along both directionsdirections

7

6

5

4

3

2

1

0

E (

meV

)

0.4 0.3 0.2 0.1 0

89

100

2

3

4

5

6

Inte

grat

ed in

t (ar

b.)

0.50.40.30.20.10

Total Triplon Continuum

3.02.52.01.51.0 log(intensity)

(0.5,0,-1-l) (h,0,-1-h)

0.20

0.15

0.10

0.05

0

(

meV

)0.5 0.4 0.3 0.2 0.1 0

(h 0 -1-h)

•a

M. B. Stone, I. Zaliznyak,

T. Hong, C. L. Broholm, D. H. Reich, Nature 440 (2006)


Magnon breakdown: theoryMagnon breakdown: theory

Kolezhuk and Sachdev, PRL 96 087203 (2006)

Zhitomirsky, PRB 73 100404R (2006)

Coherent magnon disappearsWidth appears at the crossing point


Spectrum end point in helium-4 and Spectrum end point in helium-4 and quantum spin liquid in PHCCquantum spin liquid in PHCC

4

3

2

1

0

(meV

)3210

Q (Å-1)

a

2

qc

1.0

0.8

0.6

0.4

0.2

0

S(Q

,

) (1

/meV

)

0.150

S(Q

,

)

6420 (meV)

0.4

0.2

00.15

0

2.6 Å-1b1.3 K

1.85 K

2.25 K

M. B. Stone, I. Zaliznyak, T. Hong, C. L. Broholm, D. H. Reich, Nature 440 (2006)


Spectrum breakdown in quantum spin liquid Spectrum breakdown in quantum spin liquid in PHCC in magnetic fieldin PHCC in magnetic field

I. Zaliznyak, T. Hong, M. B. Stone, C. L. Broholm, D. H. Reich, unpublished

gB

gB Sz=+1

Sz=-1Sz=0


Spectrum end point in PHCC in magnetic Spectrum end point in PHCC in magnetic field: spin conservationfield: spin conservation


SummarySummary

• Quasiparticle spectrum breakdown at E > 2 is a generic property of quantum Bose (spin) fluids

• Governed by conservation laws

• Roton breakdown in He-4

– particle non-conservation

– energy-momentum conservation

• Magnon breakdown in quantum magnets

– particle non-conservation

– energy-momentum conservation

– spin angular momentum conservation => apparent in magnetic field


How do neutrons measure excitations.How do neutrons measure excitations.


Breakdown of the roton excitation in Breakdown of the roton excitation in 44He: He: early experimentsearly experiments

Graf, Minkiewicz, Bjerum Moller & Passell, Phys. Rev. A (1974)Fak & Bossy, J. Low Temp. Phys. (1998)

Montfrooij & Svensson, J. Low Temp. Phys. (2000)

H = qε (q) aq+

aq + q,q′ Vq,q′(aqa+q′a+

q-q′ + H.c.) + …


What would be a “spin solid”?What would be a “spin solid”?

• Heisenberg antiferromagnet with classical spins, S >> 1S >> 1

− ground state has static Neel order (spin density wave with propagation vector q = )

− elastic magnetic Bragg scattering at q =

n n+1

SSnn = S = S0 0 cos(cos(n)n)

− quasiparticles are gapless Goldstone magnons

(q) ~ sin(q)

(q)

q/(2)

0.0 0.2 0.4 0.6 0.8 1.0


Temperature dependence in copper nitrateTemperature dependence in copper nitrate


Temperature dependence in PHCCTemperature dependence in PHCC

40

20

0

6420 (meV)

60

30

0

Inte

nsi

ty (

cou

nts

/ 2

min

.)

180

120

60

0180

120

60

0

(0.5 0 -1)

a

6420 (meV)

(0.15 0 -1.15)

c T = 1.5 K T = 10 K T = 15 K T = 20 K

6420 (meV)

(0.5 0 -1.5)

b 800

400

0420

400

200

0420


PHCC: a two-dimensional quantum spin PHCC: a two-dimensional quantum spin liquidliquid

• gap = 1 meV• bandwidth = 1.8 meV

• Single dispersive mode along h

• Single dispersive mode along l

• Non-dispersive mode along k


Dispersion along the side (Dispersion along the side (ll) in PHCC) in PHCC800

600

400

200

0


300

200

100

0


400

300

200

100

0


400

300

200

100

0

7654321

Q = (0.5 0 -1) resolution-corrected fit

Inte

nsity

(co

unts

in 1

m

in)

E (meV)


PHCC: a two-dimensional quantum spin PHCC: a two-dimensional quantum spin liquidliquid

• = 1 meV, bandwidth = 1.8 meV

• Single dispersive mode along L

• Non-dispersive mode along K40

20

0

40

20

04.03.02.01.0

(meV)

40

20

0

T = 1.8 KT = 50 K

(0, k, 0.5)

k = 0.5

k = 0.75

k = 1.0

Inte

nsity

(co

unts

/min

)

T=1.4K30

20

10

0

30

20

10

03.02.01.00

(meV)

30

20

10

0

Inte

nsity

(C

ount

s / m

in)

(h, 0, 1.5)

h = 0.6

h = 0.7

h = 0.8

• Single dispersive mode along H

80

40

03.02.01.00

(meV)

80

40

0

80

40

0

(0.5, 0, l)

l = 1.5

l = 1.6

l = 1.8

Inte

nsity

(co

unts

/min

)


Neutron scattering cross-sectionNeutron scattering cross-section

fi

fiffiii

zif

zf

i

f

m

k

m

kηEηEηSηS

k

k

dEdΩ

Ed

,

,b,,

22

2222 q

q

fi kkq m

k

m

kηEηEE fi

iiff 22

22

, ,

,

22

22

dttMMee

q

qqr

k

k

dEdΩ

Ed

jjjj

iEti

mi

fmag jj RRqq

magnetic scattering length, rm = -5.39*10-13 cm

jj

tiiEti

jji

fnuc dteeebb

k

k

dEdΩ

Edjj

,

*,

20

2RqRqq

nuclear scattering length, b ~ 10-13 cm

m

k

m

kEEηSηS fi

fiiziif

zfffi 22

2222

,,T,, kk


Quasiparticle cross-sectionQuasiparticle cross-section

qqq

ESdEdΩ

Ed~

,2

Quasiparticle (undamped)

singularity in cross-section (delta-function)


How do neutrons measure quasiparticles.How do neutrons measure quasiparticles.

I. A. Zaliznyak and S.-H. Lee,

in Modern Techniques for Characterizing Magnetic Materials, Ed. Y. Zhu, Springer (2005)

M o n o ch ro m a to r (2 ) s

F o cu s in g an a ly ze r

S am p le

D e tec to r

B. Brokhouse (1961)


How neutrons measure excitations now.How neutrons measure excitations now.

B. Brokhouse (1961) Gain up to factor 10

M o n o ch ro m a to r (2 ) s

F o cu s in g an a ly ze r

S am p le

D e tec to r

Gain up to factor 5

I. A. Zaliznyak and S.-H. Lee,

in Modern Techniques for Characterizing Magnetic Materials, Ed. Y. Zhu, Springer (2005)

M o n o ch ro m a to r

A n a ly ze r

S am p le

D e tec to r

conservation laws and magnon decay in quantum spin liquids

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