1. Quantum effects in Magnetic Salts G. Aeppli (LCN) N-B. Christensen (PSI) H. Ronnow (PSI) D. McMorrow (LCN) S.M. Hayden (Bristol) R. Coldea (Bristol) T.G. Perring (RAL) Z.Fisk (UC) S-W. Cheong (Rutgers) A.Harrison (Edinburgh) et al.

2. outline Introduction – salts  quantum mechanics  classical magnetism RE fluoride magnet LiHoF4 – model quantum phase transition 1d model magnets 2d model magnets – Heisenberg & Hubbard models

3. Experimental program Observe dynamics– Is there anything other than Neel state and spin waves? Over what length scale do quantum degrees of freedom matter?

4. Pictures are essential – can’t understand nor use what we can’t visualize- difficulty is that antiferromagnet has no external field- need atomic-scale object which interacts with spins Subatomic bar magnet – neutron Atomic scale light – X-rays

5. k i,E i,  i k f,E f,  f Q=k i -k f h  =E i -E f Scattering experiments Measure differential cross-section=ratio of outgoing flux per unit solid angle and energy to ingoing flux=   

6. inelastic neutron scattering Fermi’s Golden Rule at T=0,    =  f | | 2  -E 0 +E f ) where S(Q) + =  m S m + expiq.r m for finite T    k f /k i  S(Q,  ) where S(Q,  )=(n(  +1)Im  (Q,  ) S(Q,  )=Fourier transform in space and time of 2-spin correlation function =Int dt  ij expiQ(r i -r j )expi  t

7. ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ Original Nucleus Proton Recoiling particles remaining in nucleus EpEp Emerging “Cascade” Particles (high energy, E < E p ) ~ (n, p. π, …) (These may collide with other nuclei with effects similar to that of the original proton collision.) Excited Nucleus ~10 –20 sec Evaporating Particles (Low energy, E ~ 1–10 MeV); (n, p, d, t, … (mostly n) and  rays and electrons.)   e Residual Radioactive Nucleus Electrons (usually e + ) and gamma rays due to radioactive decay. e  > 1 sec ~

8. ISIS Spallation Neutron Source

9. ISIS - UK Pulsed Neutron Source

10. MAPS Anatomy Sample Fermi Chopper Low Angle 3º-20º High Angle 20º-60º Moderator t=0 ‘Nimonic’ Chopper

11. Information 576 detectors 147,456 total pixels 36,864 spectra 0.5Gb Typically collect 100 million data points

12. The Samples

14. Two-dimensional Heisenberg AFM is stable for S=1/2 & square lattice

16. Copper formate tetrahydrate Crystallites (copper carbonate + formic acid) 2D XRD mapping (still some texture present because crystals have not been crushed fully)

17. H. Ronnow et al. Physical Review Letters 87(3), pp. 037202/1, (2001)

18.  Copper formate tetradeuterate

19. Christensen et al, unpub (2006)

20.  Christensen et al, unpub (2006)

21. Neel state is not a good eigenstate |0>=|Neel> +  i |Neel states with 1 spin flipped> +  i |Neel states with 2 spins flipped>+… [real space basis] entanglement |0>=|Neel>+  k a k |spin wave with momentum k>+… [momentum space basis] What are consequences for spin waves? Why is there softening of the mode at ( ,0) ZB relative to (3  /2,  /2) ?

22. |Neel> +|correction>|0> = |SW> All diagonal flips along diagonal still cost 4J whereas flips along (0,  ) and ( ,0) cost 4J,2J or 0 e.g. -  SW energy lower for ( ,0) than for (3  ) C. Broholm and G. Aeppli, Chapter 2 in "Strong Interactions in Low Dimensions (Physics and Chemistry of Materials With Low Dimensional Structures)", D. Baeriswyl and L. Degiorgi,Eds. Kluwer ISBN: 1402017987 (2004)

23. How to verify? Need to look at wavefunctions  info contained in matrix elements measured directly by neutrons

24. Christensen et al, unpub (2006)

25. Spin wave theory predicts not only energies, but also Christensen et al, unpub (2006)

26. Discrepancies exactly where dispersion deviates the most! Christensen et al, unpub (2006)

27. Another consequence of mixing of classical eigenstates to form quantum states- ‘multimagnon’ continuum S k +|0>=  k’ a k’ S k +|k’> =  k’ a k’ |k-k’>  many magnons produced by S+ k  multimagnon continuum Can we see?

28. Christensen et al, unpub (2006)

31. 2-d Heisenberg model Ordered AFM moment Propagating spin waves Corrections to Neel state (aka RVB, entanglement) seen explicitly in Zone boundary dispersion Single particle pole(spin wave amplitude) Multiparticle continuum Theory – Singh et al, Anderson et al

32. Now add carriers … but still keep it insulating Is the parent of the hi-Tc materials really a S=1/2 AFM on a square lattice?

33. 2d Hubbard model at half filling non-zero t/U, so charges can move around still antiferromagnetic… why?

34. > > t 2 /U=J t=0 t nonzero + > +... FM and AFM degenerateFM and AFM degeneracy split by t

35. consider case of La 2 CuO 4 for which t~0.3eV and U~3eV from electron spectroscopy, but ordered moment is as expected for 2D Heisenberg model R.Coldea, S. M. Hayden, G. Aeppli, T. G. Perring, C. D. Frost, T. E. Mason, S.-W. Cheong, Z. Fisk, Physical Review Letters 86(23), pp. 5377-5380, (2001)

38.  

39. Why? Try simple AFM model with nnn interactions-  Most probable fits have ferromagnetic J’

40. ferromagnetic next nearest neighbor coupling not expected based on quantum chemistry are we using the wrong Hamiltonian? consider ring exchange terms which provide much better fit to small cluster calculations and explain light scattering anomalies, i.e. H=SJS i S j +J c S i S j S k S l SiSi SjSj SlSl SkSk

41. R.Coldea et al., Physical Review Letters 86(23), pp. 5377-5380, (2001)

42. Where can J c come from? Girvin, Mcdonald et al, PRB From our NS expmts-

43. Is there intuitive way to see where ZB dispersion comes from? C. Broholm and G. Aeppli, Chapter 2 in "Strong Interactions in Low Dimensions (Physics and Chemistry of Materials With Low Dimensional Structures)", D. Baeriswyl and L. Degiorgi,Eds. Kluwer ISBN: 1402017987 (2004)

44. For Heisenberg AFM, there was softening of the mode at (1/2,0) ZB relative to (1/4,1/4) |Neel> +|correction> |0> = |SW> All diagonal flips along diagonal still cost 4J whereas flips along (0,1) and (1,0) cost 4J,2J or 0 e.g. -

45. Hubbard model- hardening of the mode at (1/2,0) ZB relative to (1/4,1/4) |Neel> +|correction> |0> = |SW> flips along diagonal away from doubly occupied site cost <3J whereas flips along (0,1) cost 3J or more because of electron confinement

46. summary For most FM, QM hardly matters when we go much beyond a o, QM does matter for real FM, LiHoF 4 in a transverse field For AFM, QM can matter hugely and create new & interesting composite degrees of freedom – 1d physics especially interesting 2d Heisenberg AFM is more interesting than we thought, & different from Hubbard model IENS basic probe of entanglement and quantum coherence because x-section ~ | | 2 where S(Q) + =  m S m + expiq.r m