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.