Classical Antiferromagnets On The Pyrochlore Lattice

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. Pyrochlorelattice 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 Bav = 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 2004Analytically 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! (correlator) £ distance3 distance (Data for [101] and [211] directions for L=8, 16, 32, 48)

10. Spin Ice Harris et al, 1997 Compounds (Ho2Ti2O7, Dy2Ti2O7) 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, 2002Two 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.