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Classical Antiferromagnets On
The Pyrochlore LatticeS. L. Sondhi (Princeton)
with R. Moessner, S. Isakov, K. Raman, K. Gregor
[1] R. Moessner and S. L. Sondhi, Phys. Rev. B 68, 064411 (2003)[2] S. V. Isakov, K. S. Raman, R. Moessner and S. L. Sondhi, cond-mat/0404417(to appear in PRB)[3] S. V. Isakov, K. Gregor, R. Moessner and S. L. Sondhi, cond-mat/0407004 (to appear in PRL); (C. L. Henley, cond-mat/0407005)
Outline
• O(N) antiferromagnets on the pyrochlore: generalities
• T ! 0 (dipolar) correlations
• N=1: Spin Ice
• Spin Ice in an [111] magnetic field
• Why Spin Ice obeys the ice rule
Pyrochlore lattice
Lattice of corner sharing tetrahedraTetrahedra live on an FCC lattice
This talk
Consider classical statistical mechanics with
Highly frustrated: ground state manifold with 2N -4 d.o.f per tetrahedron
Neel ordering frustrated, but order by disorder possible.Are there phase transitions for T > 0?
Answered by Moessner and Chalker (1998)
• For N=1 (Ising) not an option• For N=2 collinear ordering, maybe Neel eventually• For N ¸ 3 no phase transition
i.e. N=1, 3 1 are cooperative paramagnets
Thermodynamics
Can be well approximated locally, e.g.
Pauling estimate for S(T=0) at N=1 (entropy of ice)
T), U(T) for N=3 via single tetrahedron (Moessner and Berlinsky, 1999)
Correlations?
However, correlations for T ¿ J have sharp features (Zinkin et al, 1997) indicativeof long ranged correlations, albeit no divergences in S(q)
“bowties” in [hhk] plane
These arise from dipolar correlations.
Conservation law
Orient bonds on the bipartite dual (diamond) lattice from one sublattice to the other
Define N vector fields on each bond
on each tetrahedron in grounds states, implies
at each dual site
Second ingredient: rotation of closed loops of B connects ground states) large density of states near Bav = 0
Using these “magnetic” fields we can construct a coarse grained partition function
Solve constraint B = r £ A to get Maxwell theory for N gauge fields
which leads to
and thence to the spin correlators
1/N Expansion Garanin and Canals 1999, 2001 Isakov et al 2004
Analytically soluble N = 1 yields dipolar correlations
Dipolar correlations persist to all orders in 1/N. Quantitatively:
N = 1 formulae accurate to 2% at all distances!
(Data for [101] and [211] directions for L=8, 16, 32, 48)
(correlator) £ distance3
distance
Spin Ice Harris et al, 1997
Compounds (Ho2Ti2O7, Dy2Ti2O7) in which dipolar interactions and single ionAnisotropy lead to ice rules (Bernal-Fowler rules): “two in, two out”
S ! B (N=1)
) Dipolar correlations Youngblood and Axe, 1981 Hermele et al, 2003
Also for protons in ice Hamilton and Axe, 1972
Spin Ice in a [111] magnetic field Matsuhira et al, 2002
Two magnetization plateaux and a non-trivial ground state entropy curve
Freeze triangular layers first – still leaves extensive entropy in the Kagome layers
Maps to honeycomb dimer problem• Exact entropy• Correlations• Dynamics via height representation• Kasteleyn transition
Second crossover is monomer-dimerproblem
Why spin ice obeys the ice rules
Q: Why doesn’t the long range of the dipolar interaction invalidate the local ice rule?
A: Ice rules and dipolar interactions both produce dipolar correlations!
Technically
G-1 and G can be diagonalized by the same matrix! This explains the Ewaldsummation work of Gingras and collaborators
Summary
• Nearest neighbor O(N) antiferromagnets on the pyrochlore lattice are cooperative paramagnets for N 2 and do not exhibit finite temperature phase transitions.
• However, the ground state constraint leads to a diverging correlation length as T ! 0 and “universal” dipolar correlations which reflect an underlying set of massless gauge fields.
• These can be accurately computed in the 1/N expansion.
• Spin ice in a [111] magnetic field undergoes a non trivial magnetization process about which much is known for the nearest neighbor model.
• Dipolar spin ice is ice because ice is dipoles.