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

outline Introduction – salts quantum mechanics classical magnetism RE fluoride magnet Li. Ho. F outline Introduction – salts quantum mechanics classical magnetism RE fluoride magnet Li. Ho. F 4 – model quantum phase transition 1 d model magnets 2 d model magnets – Heisenberg & Hubbard models

Experimental program Observe dynamics– Is there anything other than Neel state and spin waves? 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 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 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=d 2 s/d. Wdw

inelastic neutron scattering Fermi’s Golden Rule at T=0, d 2 s/d. Wdw=Sf|<f|S(Q)+|0>|2 d(w-E 0+Ef) inelastic neutron scattering Fermi’s Golden Rule at T=0, d 2 s/d. Wdw=Sf||2 d(w-E 0+Ef) where S(Q)+ =Sm. Sm+expiq. rm for finite T d 2 s/d. Wdw= 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 =Int dt Sij expi. Q(ri-rj)expiwt

Original Nucleus Recoiling particles remaining in nucleus ‘ ‘ ‘ Ep Emerging “Cascade” Particles 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 Me. V); (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 Spallation Neutron Source

ISIS - UK Pulsed Neutron Source ISIS - UK Pulsed Neutron Source

MAPS Anatomy Moderator t=0 ‘Nimonic’ Chopper Sample Fermi Chopper High Angle 20º-60º Low Angle 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. 5 Gb Typically Information 576 detectors 147, 456 total pixels 36, 864 spectra 0. 5 Gb Typically collect 100 million data points

The Samples The Samples

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

Copper formate tetrahydrate Crystallites (copper carbonate + formic acid) 2 D XRD mapping (still Copper formate tetrahydrate Crystallites (copper carbonate + formic acid) 2 D 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) H. Ronnow et al. Physical Review Letters 87(3), pp. 037202/1, (2001)

w Copper formate tetradeuterate w Copper formate tetradeuterate

Christensen et al, unpub (2006) Christensen et al, unpub (2006)

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

Why is there softening of the mode at (p, 0) ZB relative to (3 Why is there softening of the mode at (p, 0) ZB relative to (3 p/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 4 J + |0> = |SW> |Neel> All diagonal flips along diagonal still cost 4 J + |correction> whereas flips along (0, p) and (p, 0) cost 4 J, 2 J or 0 e. g. - SW energy lower for (p, 0) than for (3 p/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> How to verify? Need to look at wavefunctions info contained in matrix elements measured directly by neutrons

Christensen et al, unpub (2006) Christensen et al, unpub (2006)

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

Discrepancies exactly where dispersion deviates the most! 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’> 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) 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 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 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?

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

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

consider case of La 2 Cu. O 4 for which t~0. 3 e. V consider case of La 2 Cu. O 4 for which t~0. 3 e. V and U~3 e. V from electron spectroscopy, but ordered moment is as expected for 2 D 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) (3 p/2, p/2) (p, 0) (p, p) (3 p/2, p/2) (p, p) (3 p/2, p/2) (p, 0) (p, p) (3 p/2, p/2)

Why? Try simple AFM model with nnn interactions- Most probable fits have ferromagnetic J’ 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 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=SJSi. Sj+Jc. Si. Sj. Sk. Sl Sl Si Sk Sj

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

Where can Jc come from? Girvin, Mcdonald et al, PRB From our NS expmts- 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 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 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 4 J + |correction> whereas flips along (0, 1) and (1, 0) cost 4 J, 2 J or 0 e. g. -

Hubbard model- hardening of the mode at (1/2, 0) ZB relative to (1/4, 1/4) 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 <3 J + |correction> whereas flips along (0, 1) cost 3 J or more because of electron confinement

summary For most FM, QM hardly matters when we go much beyond ao, QM summary For most FM, QM hardly matters when we go much beyond ao, QM does matter for real FM, Li. Ho. F 4 in a transverse field For AFM, QM can matter hugely and create new & interesting composite degrees of freedom – 1 d physics especially interesting 2 d 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)+ =Sm. Sm+expiq. rm