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

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

ki,Ei,i

kf,Ef,f

Q=ki-kf

h=Ei-Ef

Scattering experiments

Measure differential cross-section=ratio of outgoing fluxper unit solid angle and energy to ingoing flux=

inelastic neutron scatteringFermi’s Golden Rule

at T=0, =f|<f|S(Q)+|0>|2-E0+Ef) where S(Q)+

=mSm+expiq.rm

for finite Tkf/kiS(Q,) where S(Q,)=(n(+1)Im(Q,)

S(Q,)=Fourier transform in space and time of 2-spin correlation function

<Si(0)Sj(t)>

=Int dt ij expiQ(ri-rj)expit <Si(0)Sj(t)>

‘‘

‘

‘

‘

‘

‘

‘

‘

‘

‘

OriginalNucleus

Proton

Recoiling particlesremaining in nucleus

Ep Emerging “Cascade” Particles(high energy, E < Ep)~ (n, p. π, …)

(These may collide with othernuclei with effects similar tothat of the original protoncollision.)

ExcitedNucleus

~10–20 secEvaporating Particles(Low energy, E ~ 1–10 MeV);(n, p, d, t, … (mostly n)and rays and electrons.)

e

ResidualRadioactiveNucleus Electrons (usually e+)

and gamma rays due toradioactive decay.

e> 1 sec~

ISIS Spallation Neutron Source

ISIS - UK Pulsed Neutron Source

MAPS Anatomy

Sample

Fermi ChopperLow Angle 3º-20º

High Angle 20º-60º

Moderator

t=0 ‘Nimonic’ Chopper

Information

576 detectors 147,456 total pixels36,864 spectra0.5Gb

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)

Christensen et al, unpub (2006)

Neel state is not a good eigenstate

|0>=|Neel> + i|Neel states with 1 spin flipped> +i|Neel states with 2 spins flipped>+…[real space basis] entanglement

|0>=|Neel>+kak|spin wave with momentum k>+…[momentum space basis]

What are consequences for spin waves?

Why is there softening of the mode at (,0) ZB relative to (3/2,/2) ?

|Neel> + |correction>|0> =

|SW> All diagonalflips alongdiagonal still cost 4J

whereas flips along (0,) and (,0) cost 4J,2J or 0e.g. -

SW energy lower for (,0) than for (3)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’ continuumSk+|0>=k’ak’Sk+|k’>

= k’ak’|k-k’>

many magnons produced by S+k

multimagnon continuumCan we see?

Christensen et al, unpub (2006)

Christensen et al, unpub (2006)

Christensen et al, unpub (2006)

2-d Heisenberg model

Ordered AFM momentPropagating spin wavesCorrections to Neel state (aka RVB, entanglement) seen explicitly inZone boundary dispersionSingle 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

i

ii

jiji nnUcctH

,,

† H.c.

non-zero t/U, so charges can move around

still antiferromagnetic… why?

ii

iji

ji nnUcctH

,,

† H.c.

>

>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 Letters 86(23), pp. 5377-5380, (2001)

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

Si

Sj

Sl

Sk

J=143.35 meV and cJ=48.49 meV.R.Coldea et al., Physical Review Letters 86(23), pp. 5377-5380, (2001)

Where can Jc come from?

ii

iji

ji nnUcctH

,,

† H.c.

lkjiljkijklilkjic

iiii

iiii

jiji

J

JJJH

,,,

,,,

SSSSSSSSSSSS

SSSSSS

342 244 UtUtJ , 3480UtJc and 344UtJJ

t=0.320.02 eV and U=2.60.3 eV

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> =

|SW> All diagonalflips alongdiagonal still cost 4J

whereas flips along (0,1) and (1,0) cost 4J,2J or 0e.g. -

Hubbard model- hardening of the mode at (1/2,0) ZB relative to (1/4,1/4)

|Neel> + |correction>|0> =

|SW> flips alongdiagonal away from doubly occupied site

cost <3J

whereas flips along (0,1) cost 3J or more because of electron confinement

summaryFor 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 Hubbardmodel

IENS basic probe of entanglement and quantum coherencebecause x-section ~ |<f|S(Q)+|0>|2 where S(Q)+ =mSm

+expiq.rm