1. Classical Antiferromagnets On The Pyrochlore Lattice S. 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)

2. 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

3. Pyrochlore lattice Lattice of corner sharing tetrahedra Tetrahedra 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

4. 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)

5. Correlations? However, correlations for T ¿ J have sharp features (Zinkin et al, 1997) indicative of long ranged correlations, albeit no divergences in S(q) “bowties” in [hhk] plane These arise from dipolar correlations.

6. 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 B av = 0

7. 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

8. 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:

9. N = 1 formulae accurate to 2% at all distances! (Data for [101] and [211] directions for L=8, 16, 32, 48) (correlator) £ distance 3 distance

10. Spin Ice Harris et al, 1997 Compounds (Ho 2 Ti 2 O 7, Dy 2 Ti 2 O 7 ) in which dipolar interactions and single ion Anisotropy 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

11. Spin Ice in a [111] magnetic field Matsuhira et al, 2002 Two magnetization plateaux and a non-trivial ground state entropy curve

12. 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-dimer problem

13. 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 Ewald summation work of Gingras and collaborators

14. 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.