Satisfied Simplexes in Frustrated Magnets Collin Broholm

1. Satisfied Simplexes in Frustrated Magnets Collin Broholm Johns Hopkins University and NIST Center for Neutron Research, USA • Introduction • Frustrated route to novel materials • Neutron Scattering • Weakly connected frustrated simplexes • Lattice of triangles • Kagome sandwich • Spinel AFM • Conclusions and Outlook

2. Thanks to Collaborators and Facilities M. Azuma Kyoto R. Bewley ISIS Facility W. J. L. Buyers Chalk River Y. Chen JHU S. W. Cheong Rutgers D. V. Ferraris JHU G. Gasparovic JHU Q. Huang NIST S. Ishiwata Kyoto M. Kibune Tokyo T. Lectka JHU S. H. Lee NIST M. Nohara Tokyo Y. Qiu JHU W. Ratcliff NIST D. H. Reich JHU J. Rittner JHU M. B. Stone JHU H. Takagi Tokyo M. Takano Kyoto H. Yardimci JHU I. A. Zaliznyak BNL ISIS facility, Rutherford Appleton Laboratory NIST Center for Neutron Research

3. Spin Versus Quantum Order 3D bi-partite AFM Weakly interacting spin pairs

4. Conceptual Phase Diagram for Quantum Magnets Chakravarty, Halperin, Nelson Sachdev T/J Quantum Critical 1/S, frustration, 1/z, H, P, x, …

5. A brief story of Antiferromangetism 1970 Nobel Prize in Physics to Hannes Alfvén and Louis Néel L. Néel 1904-2000 L. Néel From Nobel lecture (1970).

6. Staggered magnetization or QM singlet 1962 Nobel Prize in Physics L. D. Landau 1908-1968 L. D. Landau from Phys. Zs. UdSSR (1933).

7. Conceptual Phase Diagram for Quantum Magnets T/J Heavy Fermion behavior Non-fermi-liquids High TC Superconductivity Quantum Critical 1/S, frustration, 1/z, H, P, x, …

8. Approach QCP from Neel state 1. Assume Neel order, derive spin wave dispersion relation 2. Calculate the reduction in staggered magnetization due to quantum fluctuations 3. If then Neel order is untenable • diverges if through a surface • Soft points in D=1 magnets • Soft lines in D=2 magnets • Soft planes in D=3 magnets

9. Geometrical Interpretation Weak connectivity: Order in one part of lattice does not constrain surroundings This can occur in higher dimensions when there is frustration and/or low coordination number, z.

10. Nuclear scattering Magnetic scattering Inelastic Neutron Scattering

11. NIST Center for Neutron Research

12. ki kf Q SPINS cold neutron spectrometer at NCNR

13. Focusing analyzer system on SPINS

14. MAPS Spectrometer at ISIS in UK

15. Progression of near quantum critical models Spinel AFM La4Cu3MoO12 Kagome Slab

16. La4Cu3MoO12: A lattice of spin-1/2 trimers z=3/4 CuMoO plane Magnetic susceptibility Crystal Structure (Azuma et. al., PRB 62 R3588)

17. Frustrated quantum spin triangles J2 J1 J J J3 J Yiming Qiu et al. cond-mat/0205018

18. Spectroscopy of spin trimers 0.2 10 K Transition to quartet 0.1 0.0 70 K 0.1 0.0 Phonons HET/ISIS Yiming Qiu et al. cond-mat/0205018

19. Ordering of Composite spin-1/2 BT2/NIST

20. Spin trimer Antiferromagnetism 300K 2.6K Yiming Qiu et al. cond-mat/0205018

21. AFM on 2D kagome’ sandwich I. S. Hagemann et al. PRL (2001) TC/QCW << 1 ⇒Near Quantum Critical

22. Nano-scale order Satisfied Simplexes DCS/NIST

23. Relieving Frustration in a tetrahedron Available from for $700

24. Slowing local spin fluctuations at QCP SCGO QS-Ferrite DCS/NIST

25. Anomalous Freezing with minimal disorder Order-parameter-like Development of Small “frozen” moment IRIS/ISIS Low T specific heat indicates spin-wave-like normal modes A “simplex glass”

26. AFM on lattice of corner-sharing tetrahedra

27. TN<T<|QCW| : Dynamic Short Range Order • Points of interest: • 2p/Qr0=1.4 • ⇒ nn. AFM correlations • No scattering at low Q • ⇒ satisfied tetrahedra SPINS/NIST S.-H. Lee et al. PRL (2000)

28. T<TN : Resonant mode and spin waves • Points of interest: • 2p/Qr0=1.4 • ⇒ nn. AFM correlations • No scattering at low Q • ⇒ satisfied tetrahedra • Resonance for ħw ≈ J • Low energy spin waves SPINS/NIST S.-H. Lee et al. PRL (2000)

29. Q-dependence of inelastic scattering SPINS/NIST

30. Interpretation of Resilient Q-dependence • Same Q-dependence through range E, T • The corresponding correlations set by physics at higher energy scale (QCW) • Compare to the atomic magnetic form-factor set by eV scale physics • Interpret Q-dependence as form-factor for effective low E degrees that remain after satisfying simplexes • For ZnCr2O4 we might interpret robust Q-dependence as “inter-atomic form factor”. the “parts” would be Cr3+ spin-3/2.

31. Average form factor for AFM hexagons + ▬ + ▬ ▬ + Tchernyshyov et al. PRL (2001) S.-H. Lee et al. Nature (2002)

32. Interpretation of the fit • Physics at the scale of |QCW| orders spins antiferromagnetically on hexagons • Staggered magnetization of hexagons is effective low energy degree of freedom • ZnCr2O4 is transformed from strongly correlated spins to weakly correlated “hexagon directors” • Neutrons scatter from hexagon directors not individual spins S. H. Lee et al. Nature (2002)

33. Why AFM hexagons? • Low energy manifold has zero spin tetrahedra • Spins on tetrahdra form hinged parallelograms • Spins on hexagons form cart-wheel • Hexagons decouple when Antiferromagnetic • AFM hexagons account for 1/6 of spin entropy

34. Magneto-elastic first order transition SPINS/NIST

35. Straining to order Edge sharing n-n exchange in ZnCr2O4 depends on Cr-Cr distance,r. The implication is that there are forces between Cr3+ atoms Cr3+ Cr3+ O2- O2- These magneto-elastic interactions destabilize QC spin system on compliant lattice Tchernyshyov et al. PRL (2001) and PRB (2002)

36. Analysis of spin and lattice energies at TC Cubic paramagnet Ftet, Fcub TC T Tetrag. AFM From first moment sum-rule Based on scattering data above and below TC and assuming that nearest neighbor exchange dominates

37. Change in lattice energy change at TC Free energy of the two phases coincide at TC From this we derive increase in lattice energy at transition Compare to tetragonal strain energy Discrepancy calls for additional lattice changes at TC

38. Sensitivity to impurities near QCP TN Tf Ratcliff et al. PRB (2002)

39. Conclusions • Frustration and weak connectivity can greatly suppress TN in real materials leading to novel cooperative behaviors of matter • AFM interactions define a low energy manifold of states where near neighbor interactions are satisfied to the extent possible • A description in terms of fluctuating composite degrees of freedom appears to be productive • The local spin fluctuation rate, G, tends to zero close to the QCP • There is high sensitivity to perturbations at the QCP • Impurities yield “simplex glass” state • Lattice compliance can yield Neel order: “Straining to order”

40. Experimental Evidence for key quantum ordered phases Deconfined spinons outside spin-1/2 chain Spontaneous Gap formation for D>1 Extensive critical phases outside spin-1/2 chain Bond ordered phases Non-Fermi-Liquid in metallic quantum magnet at QCP Evidence for random singlet phase Impurity dynamics in gapped and critical phases Evidence for Orbitons and their dispersion Frustration mediated superconductivity MACS High Efficiency Spectrometer at NCNR Goals in Quantum Magnetism Design by T. D. Pike http://www.pha.jhu.edu/~broholm/MACS