Scaling and the Crossover Diagram of a Quantum Antiferromagnet
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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
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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
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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
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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
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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
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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.
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•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.
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|| , 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
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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
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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]
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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)
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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).
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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
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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
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• 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
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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
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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
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• 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)
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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
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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)
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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
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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
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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
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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
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Conclusion
T=6K-ground state
3D Non-linear Sigma model
Crossover region
Quantum critical region
Discrete lattice
Paramagnetic region
T=300K
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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.