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Quantum Effects in Magnetic Salts: Insights from Neutron Scattering and Theoretical Models

This study explores the interplay between quantum mechanics and classical magnetism in magnetic salts, focusing on the model quantum phase transition in the rare earth fluoride magnet LiHoF4. It covers one-dimensional (1D) and two-dimensional (2D) model magnets, utilizing Heisenberg and Hubbard models. An experimental program is devised to investigate the dynamics beyond the Neel state, emphasizing the need for atomic-scale objects to interact with spins. Neutron scattering and X-ray experiments elucidate spin wave phenomena and quantum effects critical to understanding magnetic materials.

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Quantum Effects in Magnetic Salts: Insights from Neutron Scattering and Theoretical Models

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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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