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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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**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**• Introduction – saltsquantum mechanicsclassical magnetism • RE fluoride magnet LiHoF4 – model quantum phase transition • 1d model magnets • 2d model magnets – Heisenberg & Hubbard models**Experimental program**• Observe dynamics– • Is there anything other than Neel state • and spin waves? • Over what length scale do quantum degrees of freedom matter?**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**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**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)>**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**MAPS Anatomy**Moderator t=0 ‘Nimonic’ Chopper Sample Low Angle 3º-20º Fermi Chopper High Angle 20º-60º**Information**576 detectors 147,456 total pixels 36,864 spectra 0.5Gb Typically collect 100 million data points**Two-dimensional Heisenberg AFM is stable for S=1/2 & square**lattice**Copper formate tetrahydrate**2D XRD mapping (still some texture present because crystals have not been crushed fully) Crystallites (copper carbonate + formic acid)**H. Ronnow et al. Physical Review Letters87(3), pp. 037202/1,**(2001)**(p,0) (3p/2,p/2) (p,p)**Christensen et al, unpub (2006)**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?**|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)**How to verify?**• Need to look at wavefunctions • info contained in matrix elements <k|S+k|0> measured directly by neutrons**Spin wave theory predicts not only energies, but also**<k|Sk+|0> Christensen et al, unpub (2006)**Discrepancies exactly where dispersion deviates the most!**Christensen et al, unpub (2006)**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?**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**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?**2d Hubbard model at half filling**non-zero t/U, so charges can move around still antiferromagnetic… why?**>**+... + > t2/U=J > t=0 t nonzero FM and AFM degenerate FM and AFM degeneracy split by t**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)**(p,p) (p,0) (3p/2,p/2)**(3p/2,p/2) (p,0) (p,p)**Why? Try simple AFM model with nnn interactions-** Most probable fits have ferromagnetic J’**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**R.Coldea et al., Physical Review Letters86(23), pp.**5377-5380, (2001)**Where can Jc come from?**Girvin, Mcdonald et al, PRB From our NS expmts-**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)**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>**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>**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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