Neutron scattering as a tool to studyquantum magnetism

Radu Coldea

Oxford

Outline

- principles of (magnetic) neutron scattering

- spin waves in a Heisenberg ferromagnet

- spin waves in square-lattice AFM La2CuO4

- quantum renormalization, spinons and method to determine

Hamiltonian – triangular AFM S = 1/2 Cs2CuCl4

- quantum phase transition in the Ising chain CoNb2O6 in

transverse field

Neutron reactors(Institute Laue Langevin)- nuclear fission of Uranium

Three-axis neutron spectrometer

neutronguide

spectrometers

ki – kf = QEi – Ef = E

ki

kf

Q

CliffordShull

BertramBrockhouse

Spallation neutron sources (ISIS, SNS …)

proton synchrotronaccelerator ~800MeV

p+

- “evaporation” when fast protonshit a heavy nucleus (Ta)

neutronguide

time-of-flightspectrometer

~40,000 detector elementscount simultaneously (time-stamp each arriving neutronLET@ISIS

Magnetic neutron diffraction

Neutrons have- no charge- spin-1/2 moment

ki

dipole moment M

kf

- Intensity ~

Periodic magnetic order=> magnetic Bragg peaksat Q = t ± q

magneticunit cell

Fourier transform of magneticmoment density (perp toscattering wavevector Q)

q

magneticstructural

Bragg peak t

t

Inelastic magnetic neutron scattering

matrix element for transition

ground state

excitedstateki

kf

sample

polarization factorFourier transform ofmagnetic e- density

Fourier transform of magneticmoment density M a

Single crystals for inelastic neutron scattering

7 single-crystal mount ~50 g(flux growth)

2.5 cm

Cs2CuCl4(solution growth)

La2CuO4

4 cm

CoNb2O6

(mirror furnace growth)

floating-zone mirror furnace

S- =- neutrons flip over one spin

magnon

H = J Si·Sj - h Siz

Jij

ij ij

ij

i

= (Si Sj+

+ Si+Sj ) + Si

zSjz - h Si

z_2

- -ij

Jij i

B

hopping

coherent propagation of spin-flip states(if Hamiltonian conservs Sz)

One spin flipmanifold

Wavevector (q)

(q) = J(q) - J(0) + h

(q)

Ene

rgy

ZeemanenergyFourier transform

of magnetic couplings

S-

magnonenergy

0

Gap

exactresult

J(q) = ijJ

ije

iqrij1

2_

Dispersion images exchange Hamiltonian

|q> = e iqri | i>i

1

N

Spin waves in a Heisenberg ferromagnet

-

--

- (0) +

H = - J Si·Sjij ijif all Jij > 0 T=0 ground state is ↑↑↑↑↑↑

Wavevector k

L (rlu)

incidentscattered

Magnon

Sz = -1

ki

kf

k

neutron

S z = -1/2

S z = + 1/2

ki - kf = k

Ei - Ef = ħ w(k)

Neutron scattering by ferromagnetic magnons

Spin waves in the square-lattice anti-ferromagnetGround state has Neel order(<S> reduced by quantum fluctuations)

Keimer at al, PRB (1992)

Magnetic Bragg peak (1/2,1/2,0)

J antiferromagnetic Spin wave excitations(approximate eigenstates)

La2CuO4

- insulating parent ofhigh-TC cuprates

- square-lattice of CuO2

planes, Cu2+ S=1/2

Neutron scattering experiments on La2CuO4

7 single crystal mount, ~50 g

ki

kf

magnonS = 1

RC et al, PRL 86, 5377 (2001)

Magnetic excitations in La2CuO4Collect maps of magnetic scattering in the whole2D Brillouin zone (h,k) at increasing energies E

325 meV

275 meV

75 meV

Spin-wave dispersion surface

Energy

RC et al, PRL 86, 5377 (2001)

Dispersion relations

La2CuO4, T=293 K

AntiferromagneticBragg peak

position

RC et al, PRL 86, 5377 (2001)

Dispersion relation and interactions

- “wiggle” in high-energy dispersion is evidencefor a cyclic-exchange between the 4 spins on aplaquette in addition to the main exchange J

- dispersion shape is a direct fingerprint of the magnetic interactions

RC et al, PRL 86, 5377 (2001)

Ring exchange in the Hubbard model

Expand up to 4electron hops

A.H. MacDonald (1990),Takahashi (1977)

- dispersions and intensitieswell described by linear spin-wave theory

t = 0.30(2) eV, U = 2.2 (4) eV

U/t = 7.3 ± 1.3 (10 K)

Hubbard model parameters for La2CuO4

Intensity renormalization factorZc = 0.51 +/- 0.13 (predicted 0.61)

RC et al, PRL 86, 5377 (2001)

Magnetic excitations in S =1/2 triangular lattice AFM Cs2CuCl4

JJ' - sharp spin-wave mode only very small

weight

- dominant scattering continuum withstrongly-dispersive boundaries

=> Quantum fluctuations very strong, spin-wave theory inadequate

1.5 1.25 1(0,0.5,l)

Energy (meV)0.0 0.2 0.4 0.6 0.8 1.0 1.2

Inte

nsi

ty(a

rb.

un

its)

0

2

4

6

E (meV)

Inte

nsi

ty(a

.u.)

0.1

1

0.2 0.8

RC et al, PRB 68, 134424 (2003)

Experimental method to determine Hamiltonianvia spin waves in the fully-polarized state in high field

BC

AppliedField

Fully-polarizedstate↑↑↑↑↑

gappedmagnons

k

ω

Δ

J antiferromagnetic

B=0 B >> J

State with AFcorrelations

Dispersion is the Fouriertransform of exchange

couplings

(k) = J (k) - J (0) + h

J (k) = S Jd exp( i k · d)d

12

Magneticallyordered, spin

liquid etc.

Exact eigenstate ifHamiltonianconserves SzBose condensation

of magnons at Bc

RC et al, PRL 88,137203 (2002)

Fourier transform ofcouplings J(q)

J=0.374(5) meVJ’=0.128(5) meV

2D Hamiltonian

Excitations in the saturated ferromagnetic phase at B=12 T

[0,k,0]1.0 1.5 2.0

[0,2,l]

0.0 0.5 1.0

B = 12 T || aT < 0.2 K

[0,,-1]

1.01.52.0

Ene

rgy

(meV

)

0.0

0.5

1.0

1.5

(010) (020) (021)(021)

D

b

JJ'

c

J’’= 0.017(2) meV interlayer couplingDa = 0.020(0) meV DM anisotropy | bc plane

Energy (meV)0.4 0.5 0.6

0

10

20

30 (0,1.447,0)

Lineshapes show sharpmagnon excitations

J’J_ ~

13_

_

Field // a (Tesla)0 2 4 6 8 10

Mag

net

izat

ion

(

)

0.0

0.2

0.4

0.6

0.8

1.0T=60 mK

Bc = 8.42 T

Field // a (Tesla)0 2 4 6 8 10

Sus

cep

tib

ilit

yd

M/d

B(

/T

)

0.0

0.1

0.2

0.3

0.4 T=60 mK

RC et al, PRL 88,137203 (2002)

FERROCONE

B (T) || a0 2 4 6 8 10 12

Ene

rgy

(meV

)

0.00

0.25

0.50

0.75

1.00

1.25

1.50(020)

(010)

(0,.447,1)

-(0,.4

47,1)+

B (T) || a0 2 4 6 8 10 12

Pea

kin

tens

ity

(arb

.uni

ts)

0.00

0.01

0.02

0.03

0.04(b)

(a)

B>Bc magnondispersion

Gap

B=BcB<Bc

Q

Magnon condensation below critical field induces transverse order

[0,k,0]

1.00 1.25 1.50 1.75 2.00

En

erg

y(m

eV)

0.0

0.5

1.0

1.5

D

Bragg peaks appearwhere the gap closes

K (rlu)

1.40 1.45 1.50In

ten

sity

a.u

.)

0

50

100

150 B > BcB < Bc

RC et al, PRL 88,137203 (2002)

Link magnetic order with magnon wavefunctions

(000)

[0 k 0]-0.5 0.0 0.5

Ene

rgy

(meV

)

0.0

0.5

1.0

+Q-Q

2 condensates at+Q and -Q

Alternating layerscounter-rotate

+D

Senseselected by

DM couplings

D (Si x Sj)

-D

B (T) || a0 1 2 6 8A

sym

met

ry<

Sb

>/<

Sc>

1.0

1.2

1.4

1.6

predicted asymmetry usingmagnon wavefunctions

[0,k,0]

1.00 1.25 1.50 1.75 2.00

Ene

rgy

(meV

)

0.0

0.5

1.0

1.5

RC et al, PRL 88,137203 (2002)

·

Quantum renormalization of incommensurate ordering wavevector

B=Bc

B<Bc

QBc= 8.44(1)T

c=0.0536(5)

B (T) || a0 2 4 6 8 10 12

Inco

mm

ensu

rati

on

(2

/b)

0.02

0.03

0.04

0.05

(0,1.5-,0)

(0,0.5-,1)

mean-fieldresult

Strong quantumrenormalization

e0/ecl=0.5620°10°

Classicalinstability

large-S

Wavevector k

QuantumSpin-1/2

Collinearspins

Energy

RC et al, PRL 88,137203 (2002)

Magnetic excitations at zero field: spin-waves fractionalizeinto pairs of S=1/2 spinons

JJ'

S=1/2Spinons

Kohno, Starykh, Balents, Nat. Phys. (07)

J’/J ~ 1/3, J=0.37 meV

S+ S- + S- S+i j j i

- anisotropic triangular lattice S=1/2 AFM Cs2CuCl4 hasspiral order coexisting with strong quantum fluctuations

- renormalization of Q-vector and zone-boundary energymeasured by quenching quantum fluctuations via field andrevealing “classical” behaviour

- dominant continuum scattering (spin-waves fractionalizeinto pairs of spinons)

Summary on anisotropic triangular AFM Cs2CuCl4

JJ'

La2CuO4

Headings, Hayden, RC …, PRL (2010)

Ising magnets and phase transitions

Temperature0 TC

<Sz>

add transverse field- B Sx

- B (S++S-) / 2

i,jH = - J Si Sj +

z z

quantum fluctuations

quantum tunneling

- 2D model Onsager exact solution (1944)

- classical Ising modelClassical

thermally-drivencontinuous phase

transition

B field0 BC

<Sz>Quantum fluctuations

driven continuousphase transition

T=0 “quantum melting of order”

An Ising ferromagnet in transverse field

- transverse field- B Sx = - B (S++S-) / 2

generates quantum fluctuationsthat “melt” the spontaneousmagnetic order at BC ~ J/2

iH = - J Si Si+1 +

z z

Field0

Ferromagnet

↑ ↑ ↑ ↑BC

TC Quantumparamagnet

z

T

continuous quantum phase transition

→→→→

Magnetization

Orderedmoment

- B S ix

x

Quantum criticality: experimental challengesand opportunities

-what is microscopic mechanism of transition, can one observe thequantum fluctuations that drive transition ?

- how quasiparticles evolve near critical point ?

- what are the fundamental symmetries that govern physics of QCP ?

- what are finite-T properties (interplay of thermal and quantumfluctuations, under what conditions universal scaling ?

Phase 1

TemperatureCritical

Phase 2

BC

0Tuning

parameter

iH = - J Si Si+1 +

z z

1D Ising chain in transverse field

↑↑↑↓↓↓↓↓↓↑↑• •

- transverse field B Sx ~ - B (S++S-)flips spins

=> propagating solitons(Jordan Wigner fermions)

Bx

2 ground states: or↑↑↑↑↑↑

domain walls (solitons)

↑↑↑↑↑↑

zJ

↑↑↑↑↑↑

↑↑↑↓↓↓↓↓↓↑↑• •

↑↑↑↓↑↑↑↑↑↑↑…

Ferro-magnet

QuantumparamagnetBC

Gap D

solitonsspin-flip

quasiparticle↑↑↑↓↓↓↓↓↓↑↑• •

classical soliton(large-S limit) →→←→→

Ising chain at criticality

- w/T scaling expected, special “conformal” symmetry

- different universality class from Luttinger liquids (1D Heisenberg and XYAFM chains)

gapless linear (Dirac) spectrum

w = c|k|

for critical solitons

1) good 1D character to see solitons2) low-exchange J ~ 1 meV to access critical field BC ~ J/2 < 10 T3) strong uniaxial anisotropy (Ising character) but not perfect to stillhave transverse g-factor

- best Ising magnets are based on Co2+ 3d7

lowest Kramers doublet effective spin-1/2 Ising-like

2D Ising AF K2CoF4 (Birgeneau ‘73, Cowley ‘84)1D Ising AF CsCoCl3 (Goff ’95) also CsCoBr3 (Nagler)

J ~ 12 meV BC > 50 T not accessible

Experimental requirements

Strong Crystal field + Spin Orbitl L•S

Quasi-1D Ising ferromagnet CoNb2O6

Single crystalof CoNb2O6

(Oxfordimage

furnace)

4 cm

Co2+

Oxygen

Ferromagneticsuperexchange

~ 90° bond Co-O-Co~ 20K ~ 2meV

Ferromagnetic order along chainStrong easy-axis (Ising) in ac plane

zig-zag Co2+ spin chain along c

T= 5 K, 1D phase above TN

Magnetic excitations in 1D phase seen byneutron scattering

↑↑↑↑↑↑↑↑↑↑↑

↑↑↑↑↑↓↑↑↑↑↑

↑↑↑↑↓↓↓↑↑↑↑

incident ↑↓

••

••- gapped continua characteristic of

2-soliton excitations

k1 k2

scattered

k1 k2RC et al, Science 327, 177 (2010)

Magnetic excitations in zero field

3D phase1D phase

T= 5 K, 1D phase above TN T= 0.04 K, deep in 3D phase

0

ordered↑ ↑ ↑ ↑

BC

TC

z

- rich structure : continuum as characteristic of 2-soliton excitations

+ sharp modes (bound states)

RC et al, Science 327, 177 (2010)

Excitations have 1D character –no measurable dispersion ┴ chains

1.1 1.2 1.3 1.4 1.50

0.1

0.2

0.3

0.4

Energy (meV)

Inte

nsi

ty(a

rb,u

nits

)

H=-2.5

H=-1.5

H=-2.0

|| chain direction

┴ chain direction

Energy (meV)1.0 1.2 1.4 1.6

Inte

nsit

y(a

rbun

its)

0.0

0.1

0.2

0.3

0.04 K

5 K

L = 0.00(5)

1

2

34

5

Zeeman ladder of bound states in 3D ordered phase

Bound states inconfining potential

↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑

↑ ↑ ↓ ↓ ↓ ↓ ↑ ↑ ↑x x x xJ’

E

Zeeman ladder of bound statesT =40 mKT = 5 K

12

… V(x) = l x

Continuum offree 2-soliton

statesSoliton separation

costs energy

l ~ J’ Sz

Longitudinal mean-field -hSz , l=2 h Sz

Energy (meV)1.0 1.2 1.4 1.6

Inte

nsit

y(a

rbun

its)

0.0

0.1

0.2

0.3

0.04 K

5 K

L = 0.00(5)

1

2

34

5

Zeeman ladder of bound states in 3D ordered phase

Bound states inconfining potential

↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑

↑ ↑ ↓ ↓ ↓ ↓ ↓ ↑ ↑x x x xJ’

E

Zeeman ladder of bound statesT =40 mKT = 5 K

12

…

Continuum offree 2-soliton

statesx V(x) = l x

Soliton separationcosts energy

l ~ J’ Sz

Longitudinal mean-field -hSz , l=2 h Sz

Energy (meV)1.0 1.2 1.4 1.6

Inte

nsit

y(a

rbun

its)

0.0

0.1

0.2

0.3

0.04 K

5 K

L = 0.00(5)

1

2

34

5

Zeeman ladder of bound states in 3D ordered phase

Bound states inconfining potential

↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑

↑ ↑ ↓ ↓ ↓ ↓ ↓ ↓ ↑ x x x xJ’

E

Zeeman ladder of bound statesT =40 mKT = 5 K

12

…

Continuum offree 2-soliton

statesx x V(x) = l x

Soliton separationcosts energy

l ~ J’ Sz

Longitudinal mean-field -hSz , l=2 h Sz

-10 -5 0 5

-0.5

0

0.5

z

Ai(

z)Soliton confinement

Schrödinger’s equation m m

McCoy&Wu (‘78)

=> bound states

kinetic energy string tension

zn = 2.33, 4.08, 5.52, 6.78…Airy function

Energy (meV)1.0 1.2 1.4 1.6

Inte

nsit

y(a

rbu

nits

)

0.0

0.1

0.2

0.3

0.04 K

5 K

L = 0.00(5)

1

2

34

5

Phenomenological model of soliton gas

J

↑↑↑↑↑↑

↑↑↑↓↓↓↑↑• •

- work perturbatively around the Ising limit

(spin clusters)2-soliton states

a 2 soliton state↑↑↑↓↓↓↓↓↓↑↑

= J ↑↑↑↓↓↓↓↓↓↑↑

-a ↑↑↑↓↓↓↓↓↓↓↑ +…

-b d n,1

from Ising J Sz Sz gap

soliton hopping

kinetic bound state

- Jxy ( Sx Sx + Sy Sy )i i+1

i i+1

from XY term↑↑↓↑↑ ↑↑↓↑↑+i+1 i-1

i i+1~ S+ S- + S- S+

i i+1 i i+1

H

RC et al, Science 327, 177 (2010)

Phenomenological model of soliton gas describes full spectrum

↑↑↑↓↓↓↓↓↓↑↑

T = 40 mK

Gap: J ~ 1.94 meV from Ising zz exchange

Bandwidth a = 0.12 J domain-wall hopping term[ microscopic origin SzSx … ?]

Kinetic bound state : transverse couplings for nn bond SxSx + SySy, J

┴/ J z = 0.24

and 2-nd neighbour AFM along chain J z' = - 0.15 J z

Weak confinement term: hz ~ 0.02 J longitudinal field includes interchain mean-field

numerical calculation agrees with exact analytic solution of effective HamiltonianS.B. Rutkevich, J. Stat. Phys (2010)

Experiments in applied transverse field

t

4 T3.25 T

↑↑↑↓↓↓↓↓↓↑↑• • BxSx = (S++S-)/2

Place crystal in metallic cage toprevent movement under high torque

CoNb2O6

crystal

Bz

torque

0 T

Gap decreaseswith field

wavevector

En

erg

y

- field tunes quasiparticle dispersion

Field ~ kinetic energy

Energy(meV)0.5 1.0 1.5 2.0

Inte

nsi

ty(a

.u.)

0

200

400

600

800

1000

Excitations as a function of transverse field

Single sharpmode

6 T4 T

Two-soliton states

7 T

→→←→→

Ordered Paramagnetic

Excitations changecharacter above

critical field

↑↑↑↓↓↓↓↓↓↑↑

Magnetic 3D LROBragg peak

Excitations intransverse field

2-soliton continuumRC et al, Science 327, 177 (2010)

Summary

- observed transmutation of quasiparticles at critical point

spin-flip2-soliton

states

- realized experimentally field-tuned quantum phase transition inquasi 1D Ising magnet CoNb2O6

ordered paramagnet

→←→

- neutron scattering is a very versatile probe of magnetic ordering anddynamics (meV->eV, magnetic field (15 T->25 T), low T (mK)- quantitative: probe dispersions of excitations through well-understood matrix element (quantitative comparison with theories)- sample size limited, large crystals needed (advanced crystal growth)- new neutron sources (ISIS 2nd target station, USA + JAPAN, ESS) willbring new opportunities: higher flux, higher resolution, wider coverage

Conclusions

50 detectors,2D data set

40,000 detectors,4D data set, 4energies measuredsimultaneously,

- complementary to resonant x-ray diffraction and inelastic (RIXS)(samples 10’s of mm, resolution >30 meV, T > few K – beam heating)

D.A. Tennant, (HZB Berlin)

Elisa Wheeler (Oxford)

Ewa Wawrzynska (Bristol)

M. Telling (ISIS)

K. Habicht, P. Smeibidl, K.Kieffer (HZB)

D. Prabhakaran (Oxford)

Ivelisse Cabrera (Oxford)

Jordan Thompson (Oxford)

R. Bewley, T. Guidi (ISIS)

C. Stock, J. Rodriguez-Rivera (NIST)

F. H.L. Essler, N. Robinson (Oxford)

Collaborators

CoNb2O6

Cs2CuCl4

S.M. Hayden (Bristol),

G. Aeppli (UCL)

T.G. Perring(ISIS)

C.D. Frost (ISIS)

T.E. Mason (Oak Ridge)

S.W. Cheong (Rutgers)

Z. Fisk (Florida)

La2CuO4

D. A. Tennant (HZB)

A.M. Tsvelik (Brookhaven)

K. Habicht, P. Smeibidl (HZB)

Z. Tylczynski (Poland)

Y. Tokiwa, F. Steglich (Dresden)