Scaling and the Crossover Diagram of a Quantum

Antiferromagnet

B. Lake, Oxford University

D.A. Tennant, St Andrew’s University S.E. Nagler, Oak Ridge National Lab C.D. Frost, Rutherford Appleton Lab

Outline

• Magnetic excitations, spinons and spin-waves• Neutron scattering and the MAPS instrument at the ISIS

neutron spallation source• KCuF3 – quasi-one-dimensional, spin-1/2, Heisenberg

antiferromagnet• MAPS measurements for KCuF3, full S(Q,) and sum

rules• Quantum critical scaling in KCuF3 and the crossover

phase diagram.• Departures from scaling 3D nonlinear sigma model are

identified as crossover phase, lattice effect and the paramagnetic phase

Three-Dimensional Spin-1/2 Antiferromagnet

• The ground state has long-range Néel order.

• The excitations are spin-wave characterised by– Spin value of 1– Transverse oscillations– Well-defined energy

ordered spin moment

transverse oscillations

longitudinal oscillations

One-Dimensional Spin-1/2 Antiferromagnet

• The ground state has no long-range Néel order.

• The excitations are spinons characterised by– Spin value of 1/2– Rotationally invariant

oscillations– Spread out in energy

Neutron ScatteringNeutrons properties spin-1/2 no charge Neutron with velocity v de Broglie wavelength =h/mv wavevector k=2/ momentum p=hk kinetic energy E=mv/2=hk/2m Neutron sources Reactor HFIR(USA), ILL(FR) Spallation source

(accelerator) SNS(USA),ISIS(UK) (U.K.), SNS (ORNL)

dsin

Elastic scattering Bragg’s Law - 2dsin=n

dsin

Inelastic scattering

Conservation of momentumq = hki + hkf

Conservation of energyE=Ei-Ef

Ei,ki

Ei,ki

Ef,kf

Ef,kf

The ISIS Spallation Neutron Source

incoming neutrons

velocity selector

sampledirection of

magnetic chains

detector banks

Scattered neutrons

• The spallation source produces pulses of neutrons

• Incident neutron energy is selected by a chopper

• Final neutron energy is calculated by timing the neutrons

• The wavevector transfer is obtained from the position of the detector and the sample orientation.

•MAPS at ISIS•2D and 1D problems: “complete” spin-spin correlations

Qh

Qk

2D Heisenberg AF Rb2MnF4

Tom Hubermann, Thesis (2004)

Data from MAPS

• Time-of-flight techniques together with large PSD detector coverage allow simultaneous measurement of large expanses of wavevector and energy

• The complete S(Q,w) can be obtained and compared to theory.

|| , 1, , ,

,

ˆr l r l r l r l

r l

H J S S J S S

KCuF3 - a Quasi-One-Dimensional Antiferromagnet

• Cu2+ ions carry spin=1/2

• Chains parallel to c direction

Strong antiferromagnetic coupling along c, J0 = -34 meV

Weak ferromagnetic coupling along a and b, J1/J0 ~ 0.02

• Long-range antiferromagnetic order below TN ~ 39K

Ordered spin moment SZ=1/4

only 50% of each spin is ordered

J0

J1

Confinement of spinons by interchain coupling Spinons – short time/distance Spinwaves – long time/distance scales

1D wavevector

En

erg

y (

me

V)

+2D 1D

E

KCuF3 – Magnetic Excitation measured using MAPS

|| , 1, , ,,

ˆr l r l r l r l

r l

H J S S J S S

E

Correlation functions

Hohenberg and Brinkman

0 0

2

1( , ) exp( ) (0) ( )

2

1( , ) 1 ( ) '' ( , )

'' ( , ')2 ( , ')( , 0) ' 2 '(1 exp( ))

' '

( , ) ( ) ( )

( , ) [ , ( )]

Q Q

BZ BZ

q q

S q dt i t S S tN

S q n q

q S qq d d

dqd S q dqS q S

d S q i S H S t

Spin-Spin Correlation

Fluctuation-Dissipation

Kramers-Kronig

normalization

Equation of motion dSq/dt~[H,Sq]

Internal energy U for KCuF3

1ˆ

r rr

H J S S

( , ) , ( )

(1 cos( ))

q qd S q i S H S t

J q U

0 100 200 300 400

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Temperature (K)

En

erg

y-p

er-

spin

(u

nits

of

J)

The Muller Ansatz – Ground State Magnetic Excitations

• The Muller Ansatz gives the T=0K excitation spectrum

2 2( , ) ( )

sin , sin / 22

zzL U

L

L U

AS q q q

q

Jq J q

Muller et al PRB 24, 1429 (1981).

The One-Dimensional static correlations for T>TN in KCuF3

( ) ( , )

1 sin( / 2)ln ,( 0).

2 cos( / 2)

1/ ,( ~ )r r d

S q d S q

A qq

q

S S d q

Algebraically decaying “critical”correlations in the ground state as expected from Muller Ansatz.

Spinwave model result

The static correlations are given by

For the Muller ansatzMuller et al PRB 24, 1429 (1981)

'( ) tan

2 2xx A q

S q

T=6K

Quantum Critical Points

• The concept of quantum criticality was used to unify the physics of different quantum regions.

• A quantum critical point is a phase transition occuring at T=0, due to quantum fluctuations.

• It is characterised by E/T scaling.

• The ideal one-dimensional Spin=1/2 Heisenberg antiferromagnet is a Luttinger Liquid quantum critical point at T=0.

• Quantum criticality is implied by Shulz’s formula:

).4/3(/)4/1()(

;4

||4

||Im

1),(

/

/

ixixx

Tqv

Tqv

TA

ee

qSkT

kT

Important question:

How much influence does a QCP have on the physics of a system that is “nearby”? E.G the quasi-1D S=1/2 Heisenberg antiferromagnet

• The Q= data for each temperature is tested for E/T scaling as a function of energy.

• Scaling is obeyed over an extensive range of energies. • At low energies, and temperatures scaling breaks down as

interchain correlations become important. • At high energies scaling again breaks down due to lattice effects

Energy/Temperature Scaling in KCuF3

Crossover phase diagram

T=6K-ground state

Paramagnetic region for kT>J

1D Quantum critical region

Effect of discrete lattice

Crossover region

3D Non-linear Sigma model

Three-dimensional Non-linear Sigma Model

For T<TN=39K, there is long-range order, the excitations are

– Spin-waves at low energies

– Spinons at high energies.

KCuF3 at T=6K(<TN=39K)

KCuF3 at T=50K(>TN=39K)

TN=39K

• Spin-waves are long-lifetime modes (resolution limited)

• They exist for – energies below M=11meV (Zone

boundary energy to chains)– Temperatures below TN

Three-Dimensional Non-linear Sigma Model

M=11meV

TN=39K

Wavevector (0,0,L)

Crossover Phase

• For T<TN the crossover phase exists for energies between M and 2M.

• It is characterised by a lump of scattering between the spin-wave branches at an energy of ~16 meV.

T=6K T=6K

THEORY – F. H. L. Essler, A.M. Tsvelik, G. Delfino, Phys. Rev. B 56, 11 001 (1997).H. J. Schulz, Phys. Rev. Lett. 77, 2790 (1996).

Crossover Phase

• Predicts singly degenerate longitudinal mode with energy gap

• Theory accurately reproduces the energy gap and intensity, but requires a broadened mode (FWHM~5meV)

Crossover Phase

Polarised neutron scattering can separate transverse and longitudinal scattering

Spin-Flip (longitudinal) Scattering •Single peak at (0,0,1.5) and 15 meV

longitudinal mode

Non-Spin-Flip (transverse) Scattering•Two peaks around (0,0,1.5)

transverse modes

Crossover PhaseBaCu2Si2O7KCuF3 Transverse Scattering

KCuF3

Transervse mode

BaCu2Si2O7

Transverse mode and continuum

Longitudinal Scattering

KCuF3

Broadened longitudinal mode Possible longitudinal continuum

BaCu2Si2O7

Longitudinal mode and continuumOr

Longitudinal continuum

Zheludev et al

Lattice Effect

• The Schulz expression is a field theory and ignores the discrete nature of the lattice

• The Schulz expression and therefore scaling will not hold where the discreteness of the lattice is important

• Departures from linearity occur at a one-spinon energy of ~40meV and therefore a two-spinon energy of ~80meV

T=6K

T=200K

Paramagnetic Phase

• At sufficiently high temperatures the material becomes paramagnetic.

• The Curie-Weiss temperature is

– Tcurie-Weiss=216K (J=34meV)

• At T=300K, paramagnetism dominates over the critical scaling behaviour.

T=6K T=50K

T=200K T=300K

Conclusion

T=6K-ground state

3D Non-linear Sigma model

Crossover region

Quantum critical region

Discrete lattice

Paramagnetic region

T=300K

Conclusions

• Spallation source instruments such as MAPS at ISIS give the full S(Q,).

• S(Q,) means that sum rules can be calculated and compared to theory.

• The data can be used to test for critical scaling in a quantum magnet and determine the range in energy temperature etc where scaling occurs.

• For KCuF3 – a quasi-one-dimensional, spin-1/2 antiferromagnet - critical scaling exists over a large extent of energy and temperature space.

• Departures from scaling have been identified and used to construct the crossover diagram for the material.