Pyrochlore spin ice

Roderich Moessner

CNRS and LPT-ENS

-2 Π0

2 Π

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0

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1

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Pyrochlore spin ice

Overview

• The history of (nearest-neighbour) spin ice– Pauling’s entropy– Anderson’s mapping– Experimental discovery of Ho2Ti2O7 by Bramwell+Harris– Ramirez’ entropy experiment

• Spin ice: the real Hamiltonian– Dipolar spin ice (Bangalore group)– Self-screening (Waterloo group)

• Why spin ice obeys the ice rules– projective equivalence and the model dipoles– emergent vs. intrinsic gauge structure– dipolar spins are ice because ice is dipolar

Pyrochlore spin ice

From a tetrahedron to the Pauling entropy S0

Consider Ising spins σi = ±1 with antiferromagnetic J > 0:

Htet = J∑

〈ij〉

σiσj =J

2

(

4∑

i=1

σi

)2

+ const

• Number of ground states: Ngs =(

42

)

= 6 for one tetrahedron

Hpyro =∑

tet

Htet =J

2

∑

tet

(

∑

i∈tet

σi

)2

• Pauling estimate: ground-state constraints independent

Ngs = 2n(6/16)n/2 = (3/2)n/2 ⇒ S0 = 12ln 3

2

Pyrochlore spin ice

Spin ice Anderson 1956

• Ho2Ti2O7 (and Dy2Ti2O7) are pyrochlore Ising magnetswhich do not order at T ≪ ΘW Bramwell+Harris

• Residual low-T entropy: Pauling entropy for water iceS0 = (1/2) ln(3/2) Ramirez et al.:

Pyrochlore spin ice

Spin ice and the ice rules

H = −E∑

i

(

d̂κ(i) · Si

)2

+ J∑

〈ij〉

Si · Sj = −(J/3)∑

〈ij〉

σiσj

• Spins, Si, are forced to point along local [111] axes, d̂κ(i)

⇒ anisotropy, E, generates Ising pseudospins: Si = σid̂κ(i)

• there are four sublattices, κ, with d̂κ · d̂κ′ = −1/3⇒J for pseudospins changes sign!

• Ground states have∑

σi = 0 for eachtetrahedron

• These are the two-intwo-out states (Bernal-Fowler ice rules)

Pyrochlore spin ice

Dipolar spin ice: the real Hamiltonian

• The ice model is not the correct microscopic starting point• Real interactions, D, are mainly dipolar: long-ranged!

Dij =Si · Sj − 3(Si · r̂ij)(Si · r̂ij)

r3ij

• Question: Why is Pauling entropy (from nearest-neighbourmodel) measured in dipolar spin ice??? Siddharthan+Shastry

• Simulations including truncated Bangalore group and especiallyEwald-summed Waterloo group dipolar interactions givereasonable agreement with experimental data.

Pyrochlore spin ice

‘Self-screening’ of dipolar interaction Gingras et al.

• Crucial observation: Spectra of n.n. and dipolar interactionsare very similar in mean-field

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2 Π

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0

2 Π

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-1

0

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0

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0

2 Π

-5

0

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0

“In other words, there must be an almost exact symmetryfulfilled in this system when long-range dipolar interactions aretaken into account. However, the same long-range nature ofthese interactions renders it difficult to construct a simple andintuitive picutre of their effects ...” den Hertog+Gingras, PRL 2000

Pyrochlore spin ice

Enforcing the ice rules by projection: P

• Degeneracy ⇒ local zero-energy modes ⇒ flat bands

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2 Π

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0

2 Π

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-3

-2

-1

0

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0

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0

1

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• At low T , only modes in flat bands are populated– deforming finite-energy bands without consequence– can replace J by a projector, P

=⇒ all non-zero modes cost the same energy– must keep eigenvectors the same!

Pyrochlore spin ice

Our model dipolar interaction: P

D =8π

3P + ∆ with ∆ ∼ O(1/r5)

• Ice rules imply conservation law for Si

=⇒ ‘Emergent gauge structure’∑

i∈tet

σi = 0 ⇔ ∇ · S = 0 =⇒ S = ∇× A

• Ground states differ byreversing spins around closedloop, for which average 〈S〉 = 0

• upon coarse-graining: low avera-ge 〈S〉 preferred ⇒ E ∼ (∇×A)2

=⇒ artificial magnetostatics

Pyrochlore spin ice

Intrinsic vs. emergent gauge structure

• Magnetostatic interaction, D, between real dipoles also

∇ · B = 0 and E ∼ (∇× A)2

• Nonetheless: geometric coincidence that D ∝ P (only Ising)• almost flat lines for Gd2Ti2O7 probably similar origin• Ising ground states of J and P the same ⇒ Ramirez expt!

|0〉 =∑

µ=1,2

∑

q

aµ(q)|vµ(q)〉

– only made out of modes |vµ(q)〉 in flat bands µ = 1, 2

– |S| = 1=⇒ nonlinear constraints on amplitudes aµ(q),which need not be resolved (we know the ground states!)

• D and P differ at lowest T ; J and P at any T > 0

Pyrochlore spin ice

Reconstructing D =8π

3P + ∆: note the gap

-2 Π0

2 Π

-2 Π

0

2 Π

-4

-3

-2

-1

0

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0

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0

2 Π

0

1

2

3

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0

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0

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0

spectra of

J PP + ∆nn P + ∆nn

+∆fn

Pyrochlore spin ice

How to distinguish D, P and J

D vs. P:• ∆ gives weak dispersion to flat bands

=⇒ ordering expected at low TBangalore+Waterloo groups

• ordering not observed experimentally-2 Π

0

2 Π-2 Π

0

2 Π

3.75

3.8

3.85

Π

0

J vs. P:• Long-range P – long-range correlations even at high T :

〈SiSj〉 ∝ −Pij

T∼

1

Tr3

• Short-range J – short-range correlations at any T > 0:

ξ ∝ exp(2J/3T )

Pyrochlore spin ice

Why spin ice obeys the ice rules

Projective equivalence of emergent and intrinsic gauge structure• Ground states of n.n. J and model dipole P identical• Real dipole D deviates from P only at short distances• Finite-T properties differ

Dipolar spins are ice because ice is dipolar

Collaborators:

• K. Gregor (Princeton)• S. V. Isakov (Toronto)• S. L. Sondhi (Princeton)

Pyrochlore spin ice

Some other issues in spin ice

• Liquid-gas transition in magnetic field• Decoupled 1d chains in magnetic field• Magnetisation plateau in [111] field

– dimensional reduction to kagome ice with Skag > 0– giant entropy peak near saturation

• Dynamics: how spin ice freezes• Artificial electrodynamics

– bow-tie correlations: algebraic but not critical– quantum problem: pyrochlore photons

• Real (water) ice: emergent, not intrinsic, gauge structure

Pyrochlore spin ice