Magnetic Monopoles
In Spin Ice
Sinéad Griffin
May 28, 2010
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)
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):
Without Monopoles
Faraday
- suggested idea of magnetic charges as a computational tool
Jackson (1975), Maxwell (1873)
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):
Without Monopoles
With Monopoles
Faraday
- suggested idea of magnetic charges as a computational tool
Jackson (1975), Maxwell (1873)
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)
Spin Ice Pyrochlores
R2Ti2O7 or R2Sn2O7
R: rare earth with a large magnetic moment
Bramwell & Gringas (2001)
Spin Ice Pyrochlores
R2Ti2O7 or R2Sn2O7
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)
Pyrochlore Lattice
R2Ti2O7 → R2Ti2O6O’
Two sublattices: Ti2O6
R2O’
=
Corner-sharing octahedra
Corner-sharing tetrahedra
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)
Ice Rules
Phil Anderson pointed out a connection in 1956
Hydrogen Positions
In Ice-Ih
↔
Magnetic Tetrahedra
in Pyrochlores
H2O
R2O’
Wurtzite structure
Pyrochlore sublattice
Two H close, two H far
Two spins in, two spins out
Anderson (1956), Pauling (1945)
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)
Creating Monopoles in Spin Ice
T > 0K
Thermal fluctuations cause disorder in the lattice
Image from www.backreaction.blogspot.com
Creating Monopoles in Spin Ice
T > 0K
Thermal fluctuations cause disorder in the lattice
Some spins will flip
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
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)
Uses for Quasi Magnetic Monopoles
Bramwell et. al (2009),
•
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
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 326 411 (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 133 60 (1931)
P. A. M. Dirac
Monopoles, Duality and String Theory
hep-th/0304042 (2003)
J. Polchinski
Spin Ice State in Frustrated Magnetic Pyrochlore Materials
Science 294 1495 (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 408-412 444 (2005)
J. E. Greedan
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 102 4 1008 (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. 58 1144 (1936)
W. F. Giaque & J. W. Stout
Signature of magnetic monopole and Dirac string dynamics in spin ice
Nature Physics 5 258 (2009)
L. D. C. Jaubert and P. C. W. Holdsworth
Measurement of the charge and current of magnetic monopoles in spin ice
Nature 461 956 (2009)
S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran & T. Fennell