1. 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

2. 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

3. 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

4. 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

5. Neutron Scattering Ei,ki Ef,kf Ei,ki Ef,kf Elastic scattering
Bragg’s Law - 2dsin=n
Inelastic scattering
Conservation of momentum
q = hki + hkf
Conservation of energy
E=Ei-Ef
Ei,ki
Ef,kf
Ei,ki
Ef,kf    
dsin
dsin

6. The ISIS Spallation Neutron Source
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.
incoming neutrons
velocity selector
sample
direction of
magnetic chains
detector banks
Scattered
neutrons
The ISIS neutron spallation source allows large regions of reciprocal space and energy to be measured simultaneously. So that you obtain the

7. Data from MAPS MAPS at ISIS
2D and 1D problems: “complete” spin-spin correlations Qk
2D Heisenberg AF Rb2MnF4
Tom Hubermann, Thesis (2004) Qh
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.

8. KCuF3 - a Quasi-One-Dimensional AntiferromagnetJ0 J1
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

9. KCuF3 – Magnetic Excitation measured using MAPS
Confinement of spinons by interchain coupling
Spinons – short time/distance
Spinwaves – long time/distance scales
Energy (meV)
1D wavevector E E
+2D 1D

10. Correlation functions
Spin-Spin Correlation
Fluctuation-Dissipation
Kramers-Kronig
normalization
Equation of motion dSq/dt~[H,Sq]
Hohenberg and Brinkman

11. Internal energy U for KCuF3

12. The Muller Ansatz – Ground State Magnetic Excitations
The Muller Ansatz gives the T=0K excitation spectrum
Muller et al PRB 24, 1429 (1981).

13. The One-Dimensional static correlations for T>TN in KCuF3
T=6K
The static correlations are given by
For the Muller ansatz
Muller et al PRB 24, 1429 (1981)
Algebraically decaying “critical”
correlations in the ground state as expected from Muller Ansatz.
Spinwave model result

14. 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:
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

15. Energy/Temperature Scaling in KCuF3
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

16. Crossover phase diagram
Paramagnetic region for kT>J
Effect of discrete lattice
T=6K-ground state
3D Non-linear Sigma model
Crossover region
1D Quantum critical region

17. Three-dimensional Non-linear Sigma Model
KCuF3 at T=6K(<TN=39K)
TN=39K
KCuF3 at T=50K(>TN=39K)
For T<TN=39K, there is long-range order, the excitations are
Spin-waves at low energies
Spinons at high energies.

18. Three-Dimensional Non-linear Sigma Model
Spin-waves are long-lifetime modes (resolution limited)
They exist for
energies below M=11meV (Zone boundary energy  to chains)
Temperatures below TN
M=11meV
TN=39K
Wavevector (0,0,L)

19. Crossover Phase
T=6K
T=6K
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.

20. 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)
THEORY –
F. H. L. Essler, A.M. Tsvelik, G. Delfino,
Phys. Rev. B 56, (1997).
H. J. Schulz, Phys. Rev. Lett. 77, 2790 (1996).

21. 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

22. Crossover Phase Transverse Scattering Longitudinal Scattering KCuF3
BaCu2Si2O7
Transverse Scattering
KCuF3
Transervse mode
BaCu2Si2O7
Transverse mode and continuum
Longitudinal Scattering
KCuF3
Broadened longitudinal mode
Possible longitudinal continuum
BaCu2Si2O7
Longitudinal mode and continuum Or
Longitudinal continuum
Zheludev et al

23. Lattice Effect
T=6K
T=200K
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

24. Paramagnetic Phase
T=6K
T=50K
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=200K
T=300K

25. Conclusion T=300K Paramagnetic region Discrete lattice
T=6K-ground state
Quantum critical region
3D Non-linear Sigma model
Crossover region

26. 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.