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Quantum effects in Magnetic Salts

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) Harrison (Edinburgh) et al. . outline.

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Quantum effects in Magnetic Salts

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  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) 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. Scattering experiments kf,Ef,sf ki,Ei,si Q=ki-kf hw=Ei-Ef Measure differential cross-section=ratio of outgoing flux per unit solid angle and energy to ingoing flux=d2s/dWdw

  6. inelastic neutron scatteringFermi’s Golden Rule • at T=0, • d2s/dWdw=Sf|<f|S(Q)+|0>|2d(w-E0+Ef) where S(Q)+ =SmSm+expiq.rm • for finite T • d2s/dWdw= kf/kiS(Q,w) where S(Q,w)=(n(w)+1)Imc(Q,w) • S(Q,w)=Fourier transform in space and time of 2-spin correlation function • <Si(0)Sj(t)> • =Int dt Sij expiQ(ri-rj)expiwt <Si(0)Sj(t)>

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

  8. ISIS Spallation Neutron Source

  9. ISIS - UK Pulsed Neutron Source

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

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

  12. The Samples

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

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

  15. H. Ronnow et al. Physical Review Letters87(3), pp. 037202/1, (2001)

  16. Copper formate tetradeuterate

  17. Christensen et al, unpub (2006)

  18. (p,0) (3p/2,p/2) (p,p) Christensen et al, unpub (2006)

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

  20. |0> = |Neel> + |correction> whereas flips along (0,p) and (p,0) cost 4J,2J or 0 e.g. - All diagonal flips along diagonal still cost 4J |SW> SW energy lower for (p,0) than for (3p/2,p) 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)

  21. How to verify? • Need to look at wavefunctions • info contained in matrix elements <k|S+k|0> measured directly by neutrons

  22. Christensen et al, unpub (2006)

  23. Spin wave theory predicts not only energies, but also <k|Sk+|0> Christensen et al, unpub (2006)

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

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

  26. Christensen et al, unpub (2006)

  27. Christensen et al, unpub (2006)

  28. Christensen et al, unpub (2006)

  29. 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

  30. 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?

  31. 2d Hubbard model at half filling non-zero t/U, so charges can move around still antiferromagnetic… why?

  32. > +... + > t2/U=J > t=0 t nonzero FM and AFM degenerate FM and AFM degeneracy split by t

  33. consider case of La2CuO4 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 Letters86(23), pp. 5377-5380, (2001)

  34. (p,p) (p,0) (3p/2,p/2) (3p/2,p/2) (p,0) (p,p)

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

  36. 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=SJSiSj+JcSiSjSkSl Sl Si Sj Sk

  37. R.Coldea et al., Physical Review Letters86(23), pp. 5377-5380, (2001)

  38. Where can Jc come from? Girvin, Mcdonald et al, PRB From our NS expmts-

  39. 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)

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

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

  42. summary • For most FM, QM hardly matters when we go much beyond ao, • QM does matter for real FM, LiHoF4 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 ~ |<f|S(Q)+|0>|2 where S(Q)+ =SmSm+expiq.rm

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