Experimentally Motivated Problems in Some Highly...

KITP KITP –– Moments and Multiplets in Mott Materials (August 13 - December 14, 2007)

Michel GingrasDepartment of Physics & Astronomy, University of Waterloo, Waterloo, Ontario, CanadaandCanadian Institute for Advanced Research/Quantum Materials Program

Experimentally Motivated Problems Experimentally Motivated Problems in Some Highly Frustrated Rarein Some Highly Frustrated Rare--

Earth Magnetic Materials Earth Magnetic Materials

Collaborators• Ali Tabei (Waterloo)• Pawel Stasiak (Waterloo)• Taras Yavors’kii (Waterloo)• Hamid Molavian (Waterloo) • Adrian del Maestro (Harvard)• Francois Vernay (Waterloo PSI)• Matt Enjalran (USC)• Ying-Jer Kao (Nat. U. Taiwan,Tapei)• Jean-Yves Fortin (CNRS, Strasbourg)• Benjamin Canals (CNRS, Grenoble)

• Steve Bramwell (U.Coll. London)• Tom Fennell (U. Coll. London)• Jan Kycia (Waterloo)• Jeff Quilliam (Waterloo)• Kate Ross (Waterloo)• Linton Corruccini (UC Davis)• Oleg Petrenko (Warwick)

Theory

Experiment

Insulating Rare-Earth Magnetic Materials

• 4f orbitals are burried under 5s, 6s, 5d orbitals: exchange interactions Jij are “small”: θCW ~ 100 —101 K

• Re3+ can have large magnetic moments µ ~ 100 —101 µB• Magnetic dipolar interactions, D ~ 10-1 — 100 K ~ θCW

GENERAL SPIN HAMILTONIAN

20B 3 5

C BF

J J 3

(

(J )(J )

J J

J B

(

)

)4

J

ij i jj i

ii

i j i ij j ij

j i ij ij

i

H

r rg

r r

V

J

gα

µ µπ

µ

>

>

=

+

−

−

−

+

∑

∑

∑

i i

i

i

i

Crystal field part of H. This is a single-particle part of the Hamiltonian.it describes how the local electrostatic/chemical environment lifts the otherwise (2J+1) degeneracy of the otherwise free rare-earth ion.

Pyrochlore lattice:(Gd,Tb,Ho,Dy)2(Ti,Sn)2O7

Body-centered tetragonal lattice:LiHoxY1-xF4

Garnet lattice:Gd3Ga5O12

Spin IcePyrochlore lattice:(Ho,Dy)2(Ti,Sn)2O7

Ho2Ti2O7, Dy2Ti2O7, Ho2Sn2O7, Dy2Sn2O7,

Real materials show manifestations of Pauling’s ground state entropy magnetic analogues of water ice

2

12 1

( )( ) ( )T

T

C TS T S T dTT

− = ∫

Ramirez et. al, Nature 399, 333 (1999)

Dipolar Ising Spin Ice Model

Moved from J=15/2 S=1/2 effective classical Ising spin

( ),

3 5

ˆ ˆ

ˆ ˆ ˆ ˆ3( )( )

ji

ji

zzi j i j

i j

zi j i ij j ij zi j

i j ij ij

H z z

z z z r z r

rD

J

r

σ σ

σ σ>

= − +

• •−

∑

∑

i

iNot known

Known

den Hertog and Gingras, Phys. Rev. Lett. 84, 3430 (2000).

Real materials show manifestations of Pauling’s ground state entropy magnetic analogues of water ice

den Hertog and Gingras, Phys. Rev. Lett. 84, 3430 (2000).

Neutron scattering in Ho2Ti2O7

(h,h,0)

(0,0,l)

nn-spin ice

dipolar spin iceS. T. Bramwell, et al., Phys. Rev. Lett. 87, 047205 (2001).

Monte Carlo Phase Diagram of the Dipolar Spin Ice ModelMonte Carlo Phase Diagram of the Dipolar Spin Ice Model

Ho2Ti2O7

Jnn/Dnn ≈ - 0.22Dy2Ti2O7

Jnn/Dnn ≈ - 0.52

Tb2Ti2O7

Jnn/Dnn ≈ - 1.1

PM

2nd

OPT • LRO

• 1st OPT

910 .DJ

nn

nn −≈

C

TNo PT

C

T

Neutron Scattering in Dy2Ti2O7

T=400 mK

Experiment Monte Carlo

Gives exchange Jnn

??

Brillouin zone boundary scattering

ZnCr2O4 spinelθCW ~ -390 KTc =12.5 K

Experiment Cluster

Monte Carlo on dipolar spin ice model

Cluster

Fit to many experiments: M(H), Cv(T,H)

Consistent determination

0.7K<T<1.49K 1.5<T<2.8K

( ),

3 5

ˆ ˆ

ˆ ˆ ˆ ˆ3( )( )

ji

ji

zzi j i j

i j

zi j i ij

i

j ij zi j

i j ij ij

jH z z

z z z r z rD

r r

J σ σ

σ σ>

=− +

• •−

∑

∑

i

i

Conclusion about Dy2Ti2O7 Spin Ice

• At least in this system, it appears that “cluster-like” / “composite-spin” – like correlations are not emergent.

• They are accidentally fined-tuned by Nature.• They are the signature of the development of

“superstructure” in S(q) arising from ancillary perturbations to H0 in the “spin liquid” state.

Yavorsk’ii, Fennell, Gingras and Bramwell; arXiv:0707.3477

Gd3Ga5O12 Garnet (GGG)

•Complex lattice with 24 atoms in conventional cubic unit cell

Yavors’kii, Enjalran and Gingras; Phys. Rev. Lett. 97, 267203 [2006]

AFM

Re-entrance,4He-like P-T phase

diagram

Fiel

d (T

esla

)

0.40.2

1.0

0.5PM - SL

Temperature (K)

Gd3Ga5O12 garnet

Spin Glass or Extended SRO

• H~0T, T< 180mK, spin glass phase or extended SR correlations, ξ=100A? • No LRO at H=0.• Spin Liquid phase at intermediate fields.• AFM phase at stronger fields.• Reentrance resembles 4He melting.

1 K

Zero field experimental picture of: neutrons

Isotopically enriched 160Gd

Petrenko et al., Phys. Rev. Lett. 80, 4570 (1998)Pict: Roger Melko

Zero field experimental picture of : neutrons

1. How can the existence of sharp peaks in neutron scattering be understood in the context of the bulk measurements finding glassy behavior and muon spin relaxation finding persistent spin dynamics down to 20 mK?

2. There has been some skepticism in the field as to the correctness/accuracy/intrinsicness of those neutron data.

1) Determine the ordering wavevector using Gaussian approximation(i.e.) mean-field theory

20B 3 5

J J

J J 3(J )(J ) ( )4

ij i jj i

i j i ij j ij

j i ij ij

H J

r rg

r r

µ µπ

>

>

=−

+ −

∑

∑

i

i i i

2) Determine the right J2 and J3 by comparing the experimental vs theoretical critical neutron scattering calculated using MFT[similar conclusion reached by considering paramagnetic S(q)]

√

XX

X

Theoretical CalculationsBy allowing a fine-tuning of the exchange interactions beyond nearest neighbors, we are able to suggest that the incommensurate ordering seen in Gd3Ga5O12 garnet (GGG) may be “intrinsic” to the would-have been “idealized/disorder-free system”.

( )ord2 0.29,0.29,0qaπ

≅

Similar results reached with MCTc

MC ~0.2K / Tcexp ~0.14K

What does that mean?What kind of state is GGG in at low temperatures?

T. Yavors’kii, M. Enjalran, and M. J. P. Gingras Physical Review Letters 97, 267203 (2006)

Fluctuating “spin liquid”

regions

Ordered/ “spin solid” regions with glassy

“components”

Conclusions as per Gd3Ga5O12 (GGG)

Conclusion about Gd3Ga5O12

• Gd3Ga5O12 is displaying incommensurate order – system is on the verge …

• Bulk is akin to a spin slush…

Transverse Field LiHoxY1-xF4

Bx

The Transverse-Field Ising Model

Pierre-Gilles de GennesParis, France 1932 PGdG introduced the TFIM to describe

collective low energy proton excitations & tunneling effects in ferroelectrics:

∑∑ Γ−=i,

- xi

ji

zj

ziij SSSJH

Quantum mechanics (tunneling between the “up” and “down” direction) is introduced by the transverse field term (proportional to Г)

[ ] 0, ≠zi

xi SSP.G. de Gennes,

Collective Motions of Hydrogen BondsSolid. State. Comm. 1, 132 (1963).

Essentials of Γ-T Phase Diagrammfz

iS Γ< Γc

TTc

T

Γ

Γc

Tc(Γ=0)

Paramagnetic/disordered

Ordered/Frozen

Ordered/Frozen

Paramagnetic/disordered

R. J. Elliott, P. Pfeuty, and C. Wood,Ising Model with a Transverse FieldPhys. Rev. Lett. 25, 443(1970).

x-T Phase Diagram of LiHoxY1-xF4

x

41 FYLiHo xx −

Tc(x)PM

FM

SG

Г~ħ disorder- Г plane

AG

“ferroglass”composition

x

41 FYLiHo xx −Tc(x)PM

FMSG

Transverse field (Γ) vs temperature (T) phase diagram of LiHoxY1-xF4; x=0.44

Г

J. Brooke, U. Chicago; Ph.DThesis.

Transverse field (Transverse field (ΓΓ)) vsvs temperature (T) phase diagram of temperature (T) phase diagram of LiHoLiHoxxYY11--xxFF44; x=0.167; x=0.167

Spin glass composition

M T H T H T H( , ) ( ) ( ) ...= − +χ χ1 33

x

41 FYLiHo xx −

Tc(x)PM

FMSG

Microscopic origin of transverse field Ising model in LiHoxY1-xF4

e x

2

3 5

3 ( )1

z zi j

i j

zi j z z

i ji j i j i j

xi

i

H J S S

rD S S

r r

S

>

=

⎛ ⎞⎜ ⎟+ −⎜ ⎟⎝ ⎠

− Γ

∑

∑

∑

CF ex,

3 5

B x

({J }) J J

J • J 3(J )(J )

B

i i ji j

i j i ij j ij

i j ij ij

xi

i

H V J

r rD

r r

g J

α

µ

>

= + •

• •+ −

−

∑

∑

∑Remember: moved from J=8 S=1/2 effective spin

!!!!!!

Leading Interactions in LiHoxY1-xF4: long-range magnetic dipole-dipole REVISITED:

( )∑>

−••−•=ji

ijjijijijijinn rJrrJJJDRH 33 ˆˆ3εε

• In presence of the randomness, there will be random terms of the form:

zi

xi

zij

xijji JJrrεε

• Since there is broken symmetry along thex direction by the applied transverse field, the x-component of Ji will be nonzero:

zi

zi

j

zi

xj

zij

xijj JhJJrr ≈∑ ε

• This means that there is a random field along the z direction introduced by the combination of (i) the applied transverse field along the x direction and (ii) the random dilution of Ho3+ Y3+.

Generation of Longitudinal Random Fieldsa) Pure case z

x

Bx=0T>Tc

mirror plane

Bx≠0T>Tc

average Jx component

No local magnetic fieldalong z at reference point Bx

Generation of Longitudinal Random Fieldsb) Diluted case

z

Bx≠0T>Tc

average Jx component

x

A component of thelocal (magnetic) field is inducedalong z at reference point

Hence, the minimal model that we have is:

• Not only random Ising systems only in a transverse field:

∑∑ Γ−−=i,

xi

ji

zj

ziijJH σσσ

• But random Ising systems in a transverse field andcorrelated random longitudinal field:

, i, - z z x

ij iz

j ii j

i z ii

H hJ σσ σ σ=− − Γ∑∑ ∑

Effective Hamiltonian MethodEffective Hamiltonian Method

( ) 3 5

J J (J )(J )( 3 ) c

i j i ij j iji j

j i ij ij

R RH D

R RVε ε

>

• • •= − +∑

∑∉ −

=P EE

Rβ

βα

ββ

00

αα= ∑∈α P

P

effH PHP PHRHP= + + ⋅ ⋅ ⋅

eff, ,

0

1 12 2

z z x zij i j ij i j

i j i j

x z zrii i

ii

iz

ih S

H J S S K S S

S h S

= − −

−Γ −−

∑ ∑

∑ ∑ ∑

Mean-Field Phase Diagram

Toy Model

∑∑∑∑∑ −−Γ−−−=i

ziri

i

ziz

i

xi

zj

xi

jiij

zj

zi

jiij ShShSSSKSSJH 0

,, 21

21

( )2221

2 2exp2

)( JNJJ

NJP ijij −⎟⎠⎞

⎜⎝⎛=π

( )2221

2 2exp2

)( KNKKNKP ijij −⎟

⎠⎞

⎜⎝⎛=π

( )∆−⎟⎠⎞

⎜⎝⎛

∆= 2exp

21)( 2

21

riri hhPπ

field) magnetic e transvers(external of functions are J

and J

, 2 xBJK Γ∆

Tabei, Gingras, Kao et al., Phys. Rev. Lett., 97, 237203 (2006).

Conclusion about LiHoxY1-xF4 in a Transverse Field

• This system is a new realization in a magneticcontext of the random field Ising model.

Yavors’kii et al., Phys. Rev. Lett. 97, 267203 [2006]

• Similar conclusion reached by Schechter et al.M. Schechter and N. Laflorencie; Phys. Rev. Lett. 97, 137204 (2006)

• Some evidence from experiment that RFIM is at work in LiHoxY1-xF4 in FM regime of x: D. M. Silevitch et al.; Nature 448, 567-570 (Aug. 2007)

Tb2Ti2O7 Pyrochlore•Complex lattice with 16 atoms in conventional cubic unit cell

,1 ,2 ,3 ,4 0S S S S∆ ∆ ∆ ∆+ + + =

However, crystalline field effects and dipolar interactions playan important role in Re2Ti2O7 pyrochlores.

H. Molavian, B. Canals and M.J.P. Gingras , Phys. Rev. Lett. 98, 157204 (2007).

Monte Carlo Phase Diagram of the Dipolar Spin Ice ModelMonte Carlo Phase Diagram of the Dipolar Spin Ice Model

Ho2Ti2O7

Jnn/Dnn ≈ - 0.22Dy2Ti2O7

Jnn/Dnn ≈ - 0.52

Tb2Ti2O7

Jnn/Dnn ≈ - 1.1

PM

2nd

OPT • LRO

• 1st OPT

910 .DJ

nn

nn −≈

C

TNo PT

C

T

1/T 1

(µs-1

)

Tf

Muon Spin Relaxation Study of Tb2Ti2O7

J.S. Gardner et al. Phys. Rev. Lett. 82, 1012 (1999)

Exp

erim

ent

MonteCarlo

M.C. of Tb2Ti2O7 with <111> Ising spins at T=5K

⎥mJ⟩ wavefunction decomposition

Dy3+ (J=15/2)Ho3+ (J=8)

⎥±15/2⟩ + O(10-1)

~ 350 K

⎥±8⟩ + O(10-1)

~ 280 K

Tb3+ (J=6)(should be a nonmagnetic singlet, as Tm3+ !)

⎥±4⟩ + εg ⎥±1⟩; εg ~ O(10-1)

~ 20 K⎥±5⟩ + εe⎥±2⟩; εe ~ O(10-1)

~ 20 K

{⎥-3⟩,⎥+6⟩, -6⟩}

{⎥+3⟩, ⎥-3⟩,⎥+6⟩, -6⟩}

~ 80 K

Gingras et al., Phys. Rev. B 61, 6496 (2000)

⎥±4⟩ + εg ⎥±1⟩; εg ~ O(10-1)

~ 20 K⎥±5⟩ + εe⎥±2⟩; εe ~ O(10-1)

~ 80 K

~ 20 K

{⎥-3⟩,⎥+6⟩, -6⟩}

{⎥+3⟩, ⎥-3⟩,⎥+6⟩, -6⟩}

• The issue is that this gap is not very much larger than the exchangeand dipolar interactions, Hint.

• Hence, Hint induces admixing via virtual transitions among CEF levels.• This problem is not unrelated to the one of multispin (ring-exchange)

interactions generated via double-occupancy in the Hubbard model.

Gingras et al., Phys. Rev. B 61, 6496 (2000)

Quantum Fluctuations via Excited StatesQuantum Fluctuations via Excited States

e 15 4ψ β≈ + − e 2

5 4ψ β≈ − − +

0 14 5ψ α≈ − − 0 2

4 5ψ α≈ − +

J+ Jz

There is a mechanism for spin flips, and some isotropy can be restored.

Is Tb2Ti2O7 a spin liquid down to T=0with nearby quantum critical points?

Temperature

Jnn/Dnn

Paramagnetic

Q=0Néel Spin Ice

~ 20 K

~ 80 K

~ 20 K

The “third axis” (e.g. finite anisotropy {crystal field gap ∆} controls quantum fluctuations)

Exact Diagonalization of Single Tetrahedron

⎥±4⟩ + εg ⎥±1⟩; εg ~ O(10-1)

~ 20 K⎥±5⟩ + εe⎥±2⟩; εe ~ O(10-1)

~ 80 K

~ 20 K

{⎥-3⟩,⎥+6⟩, -6⟩}

{⎥+3⟩, ⎥-3⟩,⎥+6⟩, -6⟩}

Shed some light on quantum fluctuation channels

Single Tetrahedron Exact Diagonalization

0 0.01 0.02 0.03 0.04 0.05 0.060.15

0.16

0.17

0.18

0.19

0.2

0.21

Singlet

Doublet

1/∆ (K−1)

Jc (

K)

0 0.02 0.04 0.06

0.15

0.16

0.17

0.18

0.19

0.2

Doublet

Sextet

Effective Hamiltonian MethodEffective Hamiltonian Method

3 5,

J J (J )(J )J J ( 3 )i j i ij j ij

i ji j j i ij ij

R RH J D

R R>

• • •= • + −∑ ∑

∑∉ −

=P EE

Rβ

βα

ββ

00

αα= ∑∈α P

P

effH P H P P H R H P= + + ⋅ ⋅ ⋅

ex ex

e

eff

x dip dip ex dip dip

ex dip

( )

PH RH P

PH RH P PH RH P P

H

H

PH P

RH P

PH P

+ + +

= + +

0 0.01 0.02 0.03 0.04 0.05 0.060.15

0.16

0.17

0.18

0.19

0.2

0.21

Singlet

Doublet

1/∆ (K−1)

Jc (

K)

0 0.02 0.04 0.06

0.15

0.16

0.17

0.18

0.19

0.2

Doublet

Sextet

N-body quantum effectsplay an important rolein the renormalizationof the Néel – spin-ice boundary.

+ + + +=

+ + …+ + ++

Is Tb2Ti2O7 a spin liquid down to T=0with nearby quantum critical points?

Temperature

Jnn/Dnn

The “third axis” (e.g. finite anisotropy {1 over the crystal field gap∆} controls quantum fluctuations)

Paramagnetic

Q=0Néel Spin Ice

~ 20 K

~ 80 K

~ 20 K

Conclusion about Tb2Ti2O7

• Tb2Ti2O7 is an exotic material system – akin to a genuine 3D spin liquid. Why? Perhaps is pushed into a quantum disordered “quantum spin ice” regime. More experiments and calculations are needed.

Gd2Sn2O7 Pyrochlore

• A. Del Maestro, M. J. P. Gingras; Phys. Rev. B 76, 064418 (2007).• J.A. Quilliam, K.A. Ross, A.G. Del Maestro, M.J.P. Gingras, L.R. Corrucciniand J.B. Kycia; Phys. Rev. Lett. 99, 097201 (2007).

The neutron diffraction below T=1 K is well described by the so-called Palmer-Chalker ground states (3 discrete g.s. (×2))

S. E. Palmer and J. T. Chalker, Phys. Rev. B 62, 488 (2000).

Top view

“top” (001) view

NMR/muSR Relaxation Regimes

[ ] [ ] ( )B B

2 2ex ex ex

2( / ) ( / ) 1

~

For , Raman process

Conv

; ~ exp( / ) ;

~ / ; ~ ( 1)

ersely

(

/ ;

)

,

c

c

c

n k T n k T d

T T TT T

J zS S T T

T T

gµ

λ ε ε

λ

ω

εε

λ

λ ω

∞

∆× +

∆

−∆ ∆

Λ +

∫∼

ω0

bω aω

Y2Mo2O7

22 K

No order seen via any techniques down to ~ 50 mK

• Two-step transition to long-range order.• Mechanisms and ground state not fully understood.• Yet, neutron sees long-range order below 700 mK.

But neutron scattering sees a long-range ordered spin ice state: Mirebeau et al., Phys. Rev. Lett. 94, 246402 (2005).

Summary

• (Tb,Gd,Er,Yb)2(Ti/Sn)2O7 pyrochlores display λ>0 down to the lowest temperature.

• This persistent spin dynamics suggests unconventional ground state and/or excitations.

• “Psychologically”, one may be able to accept λ>0 in the previous materials since the ground state is all of them lacks a complete understanding.

• Not so for the next material (Gd2Sn2O7) since neutron scattering reveals long range order.

MuSR on Gd2Sn2O7

Sharp transition

P. Dalmas de Réotier, P. C. M. Gubbens and A. YaouancJ. Phys.: Condens. Matter 16, S4687(2004).

10-1 100 101

T [K]

100

101

102

Cp

[Jm

ol−

1 K−

1 ]

T 2 fit

0.2 0.4 0.60.8T [K]

10-1

100

101

Cp

[Jm

ol−

1K

−1]

Interestingly … the specific heat has also been interpreted as unconventional

2~C T

In conventional three-dimensionallong-range ordered antiferromagnets one expects T:

3~ ; ~ exp( / ) ;

cC T T TC T T

∆

−∆ ∆

P. Bonville et al, J. Phys. Condens. Matter 15, 7777 (2003).

But neutron scattering sees long-range order below 1 K: T=0.1 K

The neutron diffraction below T=1 K is well described by the so-called Palmer-Chalker ground states (3 discrete g.s. (×2))

S. E. Palmer and J. T. Chalker, Phys. Rev. B 62, 488 (2000).

Top view

“top” (001) view

This is “interesting” …

The identification of the PC ground state via neutron diffraction motivates a microscopic description.

A. G. del Maestro and M. J. P. Gingras; “Low temperature specific heat andpossible gap to magnetic excitations in the Heisenberg pyrochloreantiferromagnet Gd2Sn207”; Phys. Rev. B 76, 064418 (2007).

10-1 100 101

T [K]

100

101

102C

p[J

mol

−1 K

−1 ]

T 2 fit

0.2 0.4 0.60.8T [K]

10-1

100

101

Cp

[Jm

ol−

1K

−1]

Maybe a gap is opening up

Magnetic Interactions Involved

20B 3 5

CF

(| |) J J

J J 3(J )(J ) ( )

4

(J )

ij ij i jj i

i j i ij j ij

j i ij ij

i

H J r

r rg

r r

V α

µ µπ

>

>

= −

+ −

+

∑

∑

i

i i i

Crystal field part of H. This is a single-particle part of the Hamiltonian.it describes how the local electrostatic/chemical environment lifts the otherwise (2J+1) degeneracy of the otherwise free rare-earth ion.

Knowing the classical ground state of H, one can do 1/S spin-wave expansion away from it. After some work … one can recast H as:

†0 , ,( ) 1/ 2k k

k

H H k a aα α αα

ω ⎡ ⎤= + +⎣ ⎦∑∑

Γ X W L K Γ

0.0

1.0

2.0

3.0

4.0

ε(k

)[K

]

J2/|J1| = 0.00, J31/|J1| = 0.00

J2/|J1| = 0.03, J31/|J1| = −0.03

J2/|J1| = −0.03, J31/|J1| = 0.03

2

31

32J

J

J

J1

J2/|J1| = 0.02, J31/|J1| = 0.0

Exp. data

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

T [K]

10-1

100

Cv

[Jm

ol−

1 K−

1 ]

Spin Wave TheoryExp. data

0.2 0.4 0.6 0.8T [K]

5.5

6.0

6.5

7.0

m[µ

B/G

d3+

]

“Evidence for gapped spin-wave excitations in the frustrated Gd2Sn2O7pyrochlore antiferromagnet from low-temperature specific heat measurements”J. A. Quilliam, K. A. Ross, A. G. Del Maestro, M. J. P. Gingras, L. R. Corruccini, and J. B. Kycia (Phys. Rev. Lett. 99 097201 [2007]).

Pushing the study further …

350 mK

Gap-like behavior, C~ exp(-∆/T) Fit to spin-wave theory

0.01

0.1

1

10

100

Spe

cifi

c H

eat (

J / K

mol

Gd)

0.12 3 4 5 6 7 8 9

12 3

Temperature (K)

Total C (this work) C (from Bonville et al.)

T 2

power law

T 3

power law Nuclear contribution

Conclusions as per Gd2Sn2O7 pyrochlore- Neutron seemingly “sees” long range order.- That order is semi-classical (very weak quantum fluctuations ~ 3%).- That order agrees with simplest theoretical expectations.- Specific heat reveals gapped magnetic excitations below ~350 mK- The magnetic excitations are spin-wave like. - These quantitatively characterize the experimental specific heat.- Yet muSR “sees” persistent spin dynamics down to the lowest

temperature.- What is it that gives rise to this persistent spin dynamics in this

material?- Most/more interestingly, what is the influence of this “cause” in

Gd2Sn2O7 on the observed PSD in other (rare-earth oxide) HFMs?

THE END