Transcript p,0
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. 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 visualizedifficulty is that antiferromagnet has no external fieldneed 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 scattering Fermi’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/ki S(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)> Original Nucleus Recoiling particles remaining in nucleus ‘ ‘ ‘ Ep Emerging “Cascade” Particles (high energy,~E < Ep) (n, p. π, …) (These may collide with other nuclei with effects similar to that of the original proton collision.) ‘ Proton ‘ Excited Nucleus ‘ ‘ ‘ ~10–20 sec g ‘ Residual Radioactive Nucleus > 1 sec ~ ‘ e g Evaporating Particles (Low energy, E ~ 1–10 MeV); (n, p, d, t, … (mostly n) and g rays and electrons.) g Electrons (usually e+) and gamma rays due to radioactive decay. ‘ e ISIS Spallation Neutron Source ISIS - UK Pulsed Neutron Source MAPS Anatomy Moderator t=0 ‘Nimonic’ Chopper Sample Fermi Chopper High Angle 20º-60º Low Angle 3º-20º Information 576 detectors 147,456 total pixels 36,864 spectra 0.5Gb Typically collect 100 million data points The Samples Two-dimensional Heisenberg AFM is stable for S=1/2 & square lattice Copper formate tetrahydrate Crystallites (copper carbonate + formic acid) 2D XRD mapping (still some texture present because crystals have not been crushed fully) H. Ronnow et al. Physical Review Letters 87(3), pp. 037202/1, (2001) Copper formate tetradeuterate Christensen et al, unpub (2006) (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> = |SW> |Neel> All diagonal flips along diagonal still cost 4J + |correction> whereas flips along (0,p) and (p,0) cost 4J,2J or 0 e.g. - 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 Christensen et al, unpub (2006) 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? Christensen et al, unpub (2006) Christensen et al, unpub (2006) Christensen et al, unpub (2006) 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 H = t (c s i , j , = † is ) c js H.c. U ni ni i non-zero t/U, so charges can move around still antiferromagnetic… why? H = t (c i , j ,s = † is > ) c js H.c. U ni ni i + > +... t2/U=J > t=0 FM and AFM degenerate t nonzero 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 Letters 86(23), pp. 5377-5380, (2001) (p,p) (3p/2,p/2) (p,0) (p,0) (p,p) (3p/2,p/2) 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 Sk Sj J=143.35 meV and J c =48.49 meV. R.Coldea et al., Physical Review Letters 86(23), pp. 5377-5380, (2001) Where can Jc come from? (c H = t i , j ,s = † is ) c js H.c. U ni ni i H = J S i S j J S i S i J S i S i i ,i i, j Jc (S S )(S i j k i ,i S l ) (S i S l )(S k S j ) (S i S k )(S j S l ) i , j , k ,l J = 4t 2 U 24 t 4 U 3 , J c = 80 t 4 U 3 and J = J = 4 t 4 U 3 Girvin, Mcdonald et al, PRB From our NS expmtst=0.320.02 eV and U=2.60.3 eV 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) |0> |SW> = |Neel> All diagonal flips along diagonal still cost 4J + |correction> whereas flips along (0,1) and (1,0) cost 4J,2J or 0 e.g. - Hubbard model- hardening of the mode at (1/2,0) ZB relative to (1/4,1/4) |0> |SW> = |Neel> flips along diagonal away from doubly occupied site cost <3J + |correction> whereas flips along (0,1) cost 3J or more because of electron confinement 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