Classical Antiferromagnets On The Pyrochlore Lattice S. L. Sondhi (Princeton) with R. Moessner, S....

Post on 30-Dec-2015

222 views 2 download

Tags:

Transcript of Classical Antiferromagnets On The Pyrochlore Lattice S. L. Sondhi (Princeton) with R. Moessner, S....

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.