Frustration and Field-driven Quantum Criticality Collin Broholm

1. Frustration and Field-driven Quantum Criticality Collin Broholm Johns Hopkins University and NIST Center for Neutron Research, USA • Frustrated magnets close to a QCP • Lattice of triangles • Kagome sandwich • Spinel AFM • Field driven QCP in Gapped Spin Systems • Spin-1 chain • Frustrated AFM bi-layer

2. Thanks to many Collaborators (yesterday’s talk) G. Aeppli UCL Y. Chen LANL P. Hammar formerly at JHU M. Kenzelmann JHU & NIST C. P. Landee Clarke University Seunghun Lee NIST K. Lefmann Risø National Lab Y. Qiu NIST & Univ. Maryland D. H. Reich JHU C. Rische Univ. of Copenhagen M. B. Stone Penn State University H. Takagi ISSP M. M. Turnbull Clarke University G. Xu BNL

3. Thanks to many Collaborators (today’s talk) 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

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

5. Magnetic Frustration Interacting spin pairs cannot simultaneously be in their lowest energy configuration Frustrated

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

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

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

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

10. Magnetic Ordering of Composite spin-1/2

11. Strongly fluctuating spin trimer AFM 300K 2.6K Yiming Qiu et al. cond-mat/0205018

12. AFM on kagome’ sandwich I. S. Hagemann et al. PRL (2001) QCW=-500 K but no phase transition for T>4 K

13. Decoupled Q- and w-dependence

14. AFM interactions satisfied w/o LRO

15. Local Spin relaxation rate →0 SCGO QS Ferrite SCGO: a=0.71(2) QS-Ferrite: a=0.67(2)

16. What side of the QCP? I. S. Hagemann et al. PRL (2001) C~T2 Linearly Dispersive mode in 2D Clandestine Long Range Order?

17. Low T Spin freezing DC susceptibility “AC susceptibility” ~250 GHz ~250 MHz C~T2 may come from “spin-waves” formed by twisting frozen spins of satisfied tetrahedra

18. AFM on lattice of corner-sharing tetrahedra

19. Interactions satisfied w/o LRO

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

21. A possible interpretation • Physics at the scale of |QCW| order spins antiferromagnetically on hexagons • Staggered magnetization of hexagons is effective low energy degree of freedom • System is transformed from strongly correlated spins to weakly correlated “hexagon directors” • Neutrons scatter from hexagon directors not individual spins

22. 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

23. Instabilities close to QCP T/J 1/S, frustration, H, P, x, … ?

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

25. 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 S.-H. Lee et al. PRL (2000)

26. Magneto-elastic first order transition

27. 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)

28. Sensitivity to impurities near quantum criticality TN Tf Ratcliff et al. PRB (2002)

29. Conclusions part#1 • Frustration and weak connectivity can greatly suppress TN in real materials • AFM interactions satisfied to the extent possible without LRO • The local spin relaxation • A description in terms of fluctuating composite degrees of freedom appears to be relevant • High sensitivity to various perturbations: • Impurities yield spin glass type state • Lattice distortions can induce Neel order

30. Gapped phases in isotropic spin systems? n = number of spins per primitive unit cell S = the spin quantum number m = the magnetization per spin n(S-m) = Oshikawa, Yamanaka, and Affleck (1997) and Oshikawa (2000) • gaps in non-magnetized spin chains? • Uniform spin ½ chain 1.½= ½ no gap • Alternating spin ½ chain 2.½= 1 perhaps • (2n+1) leg spin ½ ladder (2n+1).½ = n+½ no gap • 2n leg spin ½ ladder 2n.½ = n perhaps • Uniform spin 1 chain 1.1= 1 perhaps Integer: gap possible Non-Integer: gap impossible

31. Haldane gap in Y2BaNiO5 MAPS (ISIS) 60 hw(meV) 40 Impure 20 Pure 0 1.5 0 2 0.5 1 q (p) 1-cosq S(qw)~d(w-e(q)) e(q)

32. Two length scales in a quantum magnet Equal time correlation length 60 hw(meV) 40 20 0 1.5 0 2 0.5 1 q (p) Triplet Coherence length : length of coherent triplet wave packet

33. Coherence in a fluctuating system w ³ D h w = D h Short range G.S. spin correlations Coherent triplet propagation

34. Coherence and correlation lengths versus T Damle and Sachdev theory of triplon scattering Including impurity scattering Jolicoeur and Golinelly Quantum non-linear s model

35. Macroscopic singlet ground state of S=1 chain • Magnets with 2S=nz have a nearest neighbor singlet covering • with full lattice symmetry. • This is exact ground state for spin projection Hamiltonian • Excited states are propagating bond triplets separated from • the ground state by an energy gap Haldane PRL 1983 Affleck, Kennedy, Lieb, and Tasaki PRL 1987

36. Form factor for chain-end spin Kenzelmann et al. PRL (2003)

37. Spin-1 chains a’la carte Intra-chain exchange Anisotropy Inter-chain exchange

38. Spin-1 chains that can be magnetized b c No staggered g-tensor Staggered g-tensor NENP=Ni(C2H8N2)2NO2ClO4 NDMAP=Ni(C5H14N2)2N3(PF6) [ClO4]- Ni(en)2 NO2

39. q=p excitations versus H in NENP Enderle et al. Physica B (2000) NENP T=35 mK 0 T 12 T 13 T 14.5 T Data from BENSC, Hahn-Meittner Institute

40. NENP with staggered g-tensor: Statics 3 (110) B=2 T 2 Intensity (103 cts/min.) 1 0 0 2 4 6 8 10 T (K) Applied field breaks translational symmetry Cross-over instead of Quantum Phase Transition

41. True Critical Point in NDMAP

42. NDMAP without staggered field: Statics H||a Quasi 2D H||b 3D LRO Haldane Singlet Chen et al., PRL (2001) Data from NIST Center for neutron research

43. Singlet Ground state in PHCC J1=12.5 K a=0.6 c/cmax Daoud et al., PRB (1986).

44. Structure of C4H12N2Cu2Cl6=PHCC N Cl c c C Cu a b

45. 2D dispersion relation hw(meV) 1 0 l h 0 1

46. Zeeman splitting of cooperative triplet PHCC T=60 mK GS-level crossing for H8 T Quantum phase transition

47. Non-linear Magnetization Curve

48. H-T Phase Diagram from Magnetization

49. Field-induced AFM Order T=1.77 K H=14.5 T Intensity x>300 c

50. Order Parameter Critical Exponent Bragg Intensity  M2 b=0.4 (1) Compare to b=0.355 for 3D X-Y model