# Presentation: 466428

## 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)

## 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 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

##  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) indicative
- of 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 **B**av = 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

- 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

## 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-dimer
- problem

## 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

## 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.