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

Frustrated Magnetic Pyrochlores – Frustrated Magnetic Pyrochlores – a Rich Playground for the Study of a Rich Playground for the Study of Exotic Collective Phenomena: Exotic Collective Phenomena:

the Case of Spin Ice the Case of Spin Ice

AlbaNova and Nordita Physics Colloqium, Stockholm, September 24, 2009

Collaborators• Ali Tabei (WaterlooUChicago)• Pawel Stasiak (Waterloo)• Taras Yavors’kii (Waterloo)• Hamid Molavian (Waterloo) • Adrian del Maestro (HarvardUBC)• Jacob Cosman (UBC)• 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)• Bruce Gaulin (McMaster)• John Greedan (McMaster)• Jason Gardner (NIST)• Rob Hill (Waterloo)• Rob Kiefl (UBC/TRIUMF)• Jan Kycia (Waterloo)• Jeff Quilliam (Waterloo)• Kate Ross (WaterlooMcMaster)• Linton Corruccini (UC Davis)• Oleg Petrenko (Warwick)

Theory

Experiment

FundingOperating

Infrastructure

Outline• PART I – Introduction: Magnetism, frustration and

frustrated magnetism– Magnetism– Frustration– Why study frustrated magnetism

• PART II – Three examples:– Spin ice phenomenology– “Monopoles” in spin ice– Emergence of frustration from quantum fluctuations

• Conclusion

PART I

Introduction:

Geometric frustration and frustrated magnetism

Frustration is ubiquitous incondensed matter physics –even in other correlated states of matter

What is Frustration?Inability of a system to minimize its total classical ground state energy by minimizing the energy of all pairwise interactions, pair by pair.

Arises in:• molecular and ionic crystals (e.g. N2, H2 , KCN)• superconducting Josephson junctions in a B field• liquid crystalline materials (e.g. nematic/smectic)• proteins (e.g. protein folding problem)• random disordered magnets (e.g. spin glasses)• modulated nuclear matter in neutron stars (a.k.a. “nuclear pasta”)

Néel state(two sublattices)

Louis Néel1904-2000; Physics Nobel Prize 1970

,

; 1z z zi j i

i j

H J S S S

The non-frustrated case first

or ?

Despite frustration, Heisenberg and XY spins display a (three sublattices) Néel ordered state but Th and Qu fluctuations are enhanced

Frustrated plaquette / Ising spins

120º

O(N) spins can somewhatrelieve the frustration

Not all frustrated condensedmatter systems are magnetic

The Protein Folding ProblemThe Protein Folding Problem

is a frustration problem too … is a frustration problem too …

The structure of hydrogen/protons in Ice

O

H

is a geometrical frustration problem too … is a geometrical frustration problem too …

Ice Water

Covalent bond

Hydrogen bond

H

O

H

Pressure-Temperature Phase Diagram of Water

Hexagonal Ice

Crystal structure:

• O2- ordered; H+ disordered

• Bernal & Fowler Ice rules:

[1] 1 H on each O-O bond

[2] “2-H close, 2-H far”

• Lowest energy state not obtained.

• S(T0) ≠ 0.

• Hexagonal ice (Ih) is the most common form of ice.

O

H

Molecular Rotation in Ice at 10oK, Free Energy of Formation and Entropy of Water W. Giauque and M. Ashley, Phys. Rev. 43, 81 (1933)

0 45.1 44.23 0.87S S R R R

1895-1982Chem. NobelPrize, 1949

J. Am. Chem. Soc. 57, 2680 (1935) exp

0 0~ log(3/ 2) vs ~ 0.87S R S R

• Somehow … ice is frustrated.

• That is, its microscopic Hamiltonian (even with restricted oxygen positions, and focusing only on the protons) must be frustrated.

• Perhaps surprisingly, after 75 years, such microscopic understanding of ice water in terms of frustration is still incomplete.

• Following the discovery of high-temperature superconductivity in a research theme that developed in the late 1980s and early1990s, and which still commanding increasing attention within condensed matter physics, has been to search for novel strongly correlated classical and quantum mechanical low-temperature magnetic states.

• One highly successful approach has been the study of highly frustrated magnetic (HMF) systems.

• These are very interesting in that they do not admit a long-range ordered state at zero temperature even at the classical level.

• They are therefore extremely sensitive to small perturbations and quantum mechanical effects, hence their propensity to exhibit exotic behaviours.

Why Study Highly Frustrated Magnetic Systems?Why Study Highly Frustrated Magnetic Systems?

kagome

hyperkagome

pyrochlore garnet

kagome

hyperkagome

pyrochlore garnet

Example: Kagome Lattice

120 degrees between spins

Ground State of the Kagome Lattice Antiferromagnet

kagome

hyperkagome

pyrochlore garnet

kagomehyperkagome

pyrochlore garnet

,

1, 2, 3, 4, 0

1, 2, 3, 4,

2~

2

0

i ji j

H J S S

JH S S S S E

S S S S

1 23 4

Collective Paramagnetism on the Pyrochlore Lattice

• Lots and lots of phenomena have been observed in HFMs over the past twenty years

• In that context, the magnetic pyrochlore oxides of generic formula A2B2O7 are particularly interesting:

(A3+)2(B) 4+) 2(O2-)7

What is possible in terms of A2B2O7 pyrochlores:

What is possible in terms of A2B2O7 pyrochlores:

What is possible

Conditional possibility

Pyrochlore oxides have been shown to exhibit a multitude of phenomena:

- conventional long range order: Gd2Sn2O7 - disorder-free spin glass freezing: Y2Mo2O7

- thermal and/or quantum order-by-disorder: Er2Ti2O7

- anomalous Hall effect: Nd2Mo2O7

- persistent T→0 spin dynamics seen in muSR: all of them- field-driven transitions of various kinds- spin ice phenomenology: (Ho,Dy)2(Ti,Sn)2O7

- spin liquid/collective paramagnetic behavior: Tb2Ti2O7

- exotic deconfined topological excitations in spin ices

Pyrochlore oxides have been shown to exhibit a multitude of phenomena:

- conventional long range order: Gd2Sn2O7 - disorder-free spin glass freezing: Y2Mo2O7

- thermal and/or quantum order-by-disorder: Er2Ti2O7

- anomalous Hall effect: Nd2Mo2O7

- persistent T→0 spin dynamics seen in muSR: all of them- field-driven transitions of various kinds- spin ice phenomenology: (Ho,Dy)2(Ti,Sn)2O7

- spin liquid/collective paramagnetic behavior: Tb2Ti2O7

- exotic deconfined topological excitations in spin ices

PART II: Three examples

• Three examples:

– Spin ice phenomenology

– Monopoles in spin ice

– Emergence of frustration from quantum fluctuations

Lack of Order in a Ferromagnet

M. J. Harris, S. T. Bramwell,. F. McMorrow, T. Zeiske and K. W. Godfrey, Phys. Rev. Lett. 79, 2554 (1997)

- No order down to T~20mK despite CW ~ + 2K!- f CW/Tmin> 100

• Ho2Ti2O7

• CW ~ + 2K! (i.e. ferromagnetic interactions)

111 Single Ion Anisotropy in pyrochlore oxides

Ho,Dy crystal electric field effect

AF interaction

FM interaction

(111) Ising(Non-Frustrated)

Heisenberg(Frustrated Coop. PM/ Spin Liquid )

(111) Ising(Frustrated Spin Ice)

Heisenberg(Non-Frustrated)

2ˆ

i iD S z

D

iz

FM and AFMswap

frustration!

Local 111 Ising anisotropy

6 ground states for a tetrahedron

• Tetrahedral proton coordination• Bernal-Fowler ice rules

Proton displacement: vector at mid-point

Ferromagnetically coupled Ising magnetic moments on pyrochlore lattice are ice-like!!

Proton Position in Ice vs Spin Orientation in FM Ising Pyrochlore

S. T. Bramwell and M. J. Harris, J. Phys. Condens. Matter 10, L215 (1998)

O2-

H+

Degeneracy of Ising Pyrochlore Ferromagnet

6N/4(6/16)N/4

Hence, at T=0, the entropy per spin is: 2

3ln

2

1B0 kS T

Same (Pauling) entropy at T=0 as water ice!!! hence the name “spin ice”

M. J. Harris, S. T. Bramwell, D. F. McMorrow, T. Zeiske and K. W. Godfrey, Phys. Rev. Lett. 79, 2554 (1997)

For N moments on the pyrochlore lattice with n.n.ferromagnetic Jij

exchange, there are

equivalent groundstates of the system!

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

Ramirez et. al, “Zero Point Entropy in Spin Ice”, Nature 399, 333 (1999)

2

12 1

( )( ) ( )

T

T

C TS T S T dT

T

Difference is Pauling’s S0 !!

This means, the value of S(T=0 )Should have beeen set to S0 not 0

Insulating Rare-Earth Magnetic Materials

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

• R.E.3+ can have large magnetic moments ~ 100 —101 B

• Magnetic dipolar interactions, D ~ 10-1 — 100 K ~ θCW

Dipolar Ising Spin Ice Model

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

den Hertog and Gingras, Phys. Rev. Lett. 84, 3430 (2000)Bramwell and Gingras, Science 294, 1495 (2001)Yavors’kii et al. Phys. Rev. Lett. 101, 037204 (2008)

,

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 jij ij

H z z

z z z r z r

rD

J

r

Not 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

S(q) obtained from Monte Carlo results on a generalized dipolar spin ice model: Jij beyond n.n.

ExperimentMonte Carlo

Yavors’kii et al. Phys. Rev. Lett. 101, 037204 (2008)

Some of what I have not talked about re: spin ice

• Spin ice in a field (topological {Kastelyn} transitions)

• Where do the ice rules really come from?• Magnetic monopole-like topological defects• Artificial ice, stuffed ice• Dynamics of spin ice • Structural and spin fluctuations in spin ice• How is the 3rd law of thermodynamic (S0≠0)

resolved?• How to put back quantum mechanics? –

Relevant to the Tb2Ti2O7 spin liquid material

Some of what I have not talked about re: spin ice

• Spin ice in a field (topological {Kastelyn} transitions)

• Where do the ice rules really come from?• Magnetic monopole-like topological defects• Artificial ice, stuffed ice, other ice-like materials• Dynamics of spin ice • Structural and spin fluctuations in spin ice• How is the seemingly violation of the 3rd law of

thermodynamic (S0≠0) ultimately resolved?• How to put back quantum mechanics? –

Relevant to the Tb2Ti2O7 spin liquid material

PART II: Three examples

• Three examples:

– Spin ice phenomenology

– Monopoles in spin ice

– Emergence of frustration from quantum fluctuations

Singular correlations in spin ice state• The spin ice rule can be mapped in the long length scale

limit to a non-divergent field

• The excitations (spin flip) that break the ice rule create effective “magnetic charges”, or monopoles in that field.

• The system obeys a “magnetic Gauss’ Law” which relates the density of poles to the divergence of the field

0

monopoles

Singular correlations in spin ice state• Because of the ice rule, hence the divergence free-

condition of the field • The spin-spin correlations in real-space decay as an

effective dipolar type

• Very different than the exponential decay of the spin-spin correlations in a thermally disordered paramagnet

• As a result, the Fourier transform, hence the neutron scattering, show singular behaviors, “pinch points”, at specific reciprocal lattice points.

2

' 5

1 3( ) ( ') ~ ~

'u v uv

r r

r r rS r S r

r r r

Pinch points

Contour plot of neutron scattering intensity

Neutron scattering of Ho2Ti2O7 spin ice

Fennell et al., to appear in Science, Oct. 2009

002 point

Fennell et al., to appear in Science, Oct. 2009

vicinity of 002 point

transverse scan through pinch point at 002

Monopoles and Dirac strings

Castelnovo et al., Nature 451, 41 (2008).

• Thermal excitations lead to deconfined monopoles• This is because the “Dirac string” of reversed magnetic moments• left behind is tensionless, thanks to the nature of the spin ice state

Dirac strings in a magnetic field

Dirac strings in a magnetic field

Morris et al., to appear in Science, Oct. 2009

• Excitations in spin ices consists of deconfined monopole-like topological defects

• These have been recently observed and reported in 2 neutron scattering papers in Science and 1 muon spin relaxation paper in Nature.

• There is currently efforts in understanding the role of these objects on the spin dynamics in spin ice. 1theory paper addressing this issue was published in Nature Physics in April 2009.

• There are bound to be more interesting effects tied to monopoles and “Dirac strings” in spin ice to be reported over the next couple of years.

“Conclusion” about Monopoles in Spin ice

PART II: Three examples

• Three examples:

– Spin ice phenomenology

– Monopoles in spin ice

– Emergence of frustration from quantum fluctuations

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

Ho2Ti2O7

Jnn/Dnn - 0.22

Dy2Ti2O7

Jnn/Dnn - 0.52

Tb2Ti2O7

Jnn/Dnn - 1.1

PM

2nd OPT • LRO

• 1st OPT

910.D

J

nn

nn

C

TNo PT

C

T

1/T

1 (

s-1)

Tf

Muon Spin Relaxation Study of Tb2Ti2O7

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

θCW ~ -20 KTc < 20 mK

Dy3+ (J=15/2)

15/2 + O(10-1)

~ 350 K

8 + O(10-1)

Ho3+ (J=8)

~ 280 K

mJ wavefunction decomposition

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

4 + g 1; g ~ O(10-1)

~ 20 K

5 + e2; e ~ O(10-1)

~ 80 K

~ 20 K

{-3,+6, -6

+3, -3,+6, -6

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

Effective Hamiltonian MethodEffective Hamiltonian Method

P

P

P EE

R

00

effH PHP PHRHP

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

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

= + + + +

+ + + +

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

+ + …

• Perhaps Tb2Ti2O7 admits a truly novel/exotic quantum mechanical ground state!

• It is possible that the effective (field theory) describing this material is akin to 3+1 QED

• This would mean that this system may exhibit (i) emerging (gauge) “photon” (ii) fractionalized & deconfined excitations

• It is not clear at this time what is the role of the lattice vibration in this material on the magnetic phenomena observed

• More theoretical and experimental work is needed.

“Conclusion” about Tb2Ti2O7

• Frustrated magnetic materials are presenting themselves as a rich source of new phenomena in condensed matter physics.

• Among them, spin ices have proven to be particularly interesting – I have discussed only a few topics about them.

• There is currently active worldwide research efforts in this field with networks in Japan and Europe.

• Other interesting phenomena are bound to be discovered in the next few years.

Conclusion

The end