Scaling and the Crossover Diagram of a Quantum Antiferromagnet

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 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. detector banks velocity selector Scattered neutrons sample direction of incoming neutrons magnetic chains 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.

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

8. KCuF3 - a Quasi-One-Dimensional Antiferromagnet J1 J0 • 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. Paramagnetic region for kT>J Effect of discrete lattice 1D Quantum critical region Crossover region 3D Non-linear Sigma model Crossover phase diagram T=6K-ground state

17. KCuF3 at T=6K(<TN=39K) KCuF3 at T=50K(>TN=39K) Three-dimensional Non-linear Sigma Model 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, 11 001 (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 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. T=6K T=200K 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

24. Paramagnetic Phase T=50K T=6K • 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 Paramagnetic region Discrete lattice Quantum critical region Crossover region 3D Non-linear Sigma model T=300K T=6K-ground state

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.