1. Magnetic Monopoles In Spin Ice
Sinéad Griffin
May 28, 2010

2. Magnetic Monopoles in Spin Ice
What are the features of these magnetic excitations?
How do the material properties cause them?
Castelnovo et al (2008), Morris et al (2009)

3. What are Magnetic Monopoles?
Benjamin Franklin
- electric fields caused by + and – charge
- magnetic fields caused by N and S poles
Maxwell
- 1800s: link between magnetism and electricity
- Maxwell’s Equations for Electromagnetism (1873):
Faraday
- suggested idea of magnetic charges as a computational tool
Without Monopoles
Jackson (1975), Maxwell (1873)

4. What are Magnetic Monopoles?
Benjamin Franklin
- electric fields caused by + and – charge
- magnetic fields caused by N and S poles
Maxwell
- 1800s: link between magnetism and electricity
- Maxwell’s Equations for Electromagnetism (1873):
Faraday
- suggested idea of magnetic charges as a computational tool
Without Monopoles
With Monopoles
Jackson (1975), Maxwell (1873)

5. Magnetic Monopoles: A Physical Reality?
Pierre Curie
- first to suggest the existence of isolated magnetic monopoles
Paul Dirac
- firm theoretical grounding of magnetic monopoles in 1931 paper
- quantization of electric charge is the result of the existence of isolated electric and magnetic charges.
DIRAC MONOPOLE: A magnetic monopole is an isolated particle that acts as a source of a magnetic field (i.e. non-zero divergence).
DIRAC STRING: Infinitely thin flux tubes that connect magnetic monopoles.
Many Grand Unified Theories predict the existence of magnetic monopoles that were created in the beginning of the universe.
Curie (1894), Dirac (1931), Polchinski (2003)

6. Spin Ice Pyrochlores R2Ti2O7 or R2Sn2O7 Bramwell & Gringas (2001)
R: rare earth with a large magnetic moment
Bramwell & Gringas (2001)

7. Spin Ice Pyrochlores R2Ti2O7 or R2Sn2O7 Greedan (2005)
R: rare earth with a large magnetic moment
Gd2Ti2O7 Phase transition to unusual long range order at 1K
Tb2Ti2O7 Cooperative paramagnetic/ spin liquid
Dy2Ti2O7 Spin Ice
Ho2Ti2O7 Spin Ice
Er2Ti2O7 Long range magnetic order
Greedan (2005)

8. Pyrochlore Lattice = R2Ti2O7 → R2Ti2O6O’
Two sublattices: Ti2O6 Corner-sharing octahedra
R2O’ Corner-sharing tetrahedra =

9. Geometrical Frustration in Pyrochlores
Geometric Frustration: Local ordering rules are incompatible with long-range ordering
2D Frustration: AFM ordering on a triangle
1/3 of spins frustrated
3D Frustration: FM ordering in tetrahedron (A2O’)
½ of spins frustrated
Ising spin axis is along diamond bonds
Snyder et al (2001)

10. Ice Rules ↔ Hydrogen Positions In Ice-Ih Magnetic Tetrahedra
Phil Anderson pointed out a connection in 1956
Hydrogen Positions
In Ice-Ih ↔
Magnetic Tetrahedra
in Pyrochlores
H2O
Wurtzite structure
Two H close, two H far
R2O’
Pyrochlore sublattice
Two spins in, two spins out
Anderson (1956), Pauling (1945)

11. Ground State Entropy at 0K
Apply spin ice rules (two in, two out) to a 2D lattice
Choice of which spins point in and which spins point out:
Degenerate ground state
Residual Entropy
S=kBlnW
W= 3/2 for Ice-Ih or spin ice.
Giaque & Stout (1936)

12. Creating Monopoles in Spin Ice
T > 0K Thermal fluctuations cause disorder in the lattice
Image from

13. Creating Monopoles in Spin Ice
T > 0K Thermal fluctuations cause disorder in the lattice
Some spins will flip

14. Creating Monopoles in Spin Ice
T > 0K Thermal fluctuations cause disorder in the lattice
Some spins will flip
Defect propagates through the lattice to preserve the ice rules

15. Are these Dirac Monopoles/Strings?
Packets of magnetic charge
Separated by ‘string’ of flipped spins
Move independently - don’t keep their partners
- many ways of connecting a + and –
Always occur in pairs
Dipole model describes the interactions (not point charges, but oriented dipoles)
Dirac string not infinitely thin, rather an observable flux tube
Divergence is always zero
So what are these magnetic excitations?
Not fundamental particles but emergent quasi-particles
Have many characteristics of isolated magnetic charges
Example of 3D fractionalization of magnetic charge
Castelnovo et al (2008), Jaubert et al (2009)

16. Uses for Quasi Magnetic Monopoles
Magnetricity:
Magnetic quasiparticles are charge-like and have an associated current
Influence of applying magnetic and electric fields
Model for understanding frustrated magnetic systems
Study of ground state degeneracy related to other systems such as protein folding and relaxor ferroelectrics
Bramwell et. al (2009),

17. References Magnetic monopoles in spin ice Nature 451 42 (2008)
C. Castelnovo, R. Moessner & S. L. Sondhi
Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7
Science (2009)
D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner,
K. C. Rule, J. U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, R. S. Perry
Classical Electrodynamics
Wiley (1975)
J. D. Jackson
Sur la possibilité d’existence de la conductibilité magnétique et du magnétisme libre
Séances de la Société Français de Physique, p 76 (1984)
P. Curie
Quantised singularities in the electromagnetic field
Proc. R. Soc. A (1931)
P. A. M. Dirac
Monopoles, Duality and String Theory
hep-th/ (2003)
J. Polchinski
Spin Ice State in Frustrated Magnetic Pyrochlore Materials
Science (2001)
S. T. Bramwell & M. J. P. Gringas
Frustrated rare earth magnetism: Spin glasses, spin liquids and spin ices in pyrochlore oxides
Journal of Alloys and Compounds (2005)
J. E. Greedan

18. References How ‘spin ice’ freezes Nature 413 48 (2001)
J. Snyder, J. S. Slusky, R. J. Cava & P. Schiffer
Ordering and Antiferromagnetism in Ferrites
Physical Review (1956)
P. W. Anderson
The Nature of the Chemical Bond
Cornell Univ. Press, Ithaca, New York (1945)
L. Pauling
The entropy of water and the third law of thermodynamics The heat capacity of ice from 15K to 273K.
J. Am. Chem. Soc (1936)
W. F. Giaque & J. W. Stout
Signature of magnetic monopole and Dirac string dynamics in spin ice
Nature Physics (2009)
L. D. C. Jaubert and P. C. W. Holdsworth
Measurement of the charge and current of magnetic monopoles in spin ice
Nature (2009)
S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran & T. Fennell