Geometry, topology and frustration:the physics of spin ice

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Roderich Moessner

CNRS and LPT-ENS

9 March 2005, Magdeburg

Geometry, topology and frustration:the physics of spin ice

Overview

• Spin ice: experimental discovery and basic model• Spin ice in a field — dimensional reduction to “kagome ice”:

– magnetisation plateau: entropy, correlations– tilted field: Kasteleyn transition– termination of plateau: entropy peak

• Conservation law and gauge theory:– algebraic dipolar correlations– exact N = ∞ solution

• Why spin ice obeys the ice rules• Conclusions and outlook• Collaborators:

K. Gregor, S. V. Isakov, K. Raman, S. L. Sondhi

Geometry, topology and frustration:the physics of spin ice

Spin ice

• Experimental discovery of afrustrated pyrochlore Isingferromagnet by Harris et al.:Ho2Ti2O7, followed byDy2Ti2O7.

• Missing entropy (Ramirez et al.)

at low T is precisely the Pau-ling entropy for cubic ice Ic,SPauling = (1/2) ln(3/2):

Geometry, topology and frustration:the physics of spin ice

Crystal structure of spin ice

• Spins reside on pyrochlore lattice• Pyrochlore lattice consists of corner-sharing tetrahedra

• All sites and bonds are symmetry-equivalent• not a Bravais lattice

Geometry, topology and frustration:the physics of spin ice

Spin ice, a frustrated ferromagnet ( ΘW > 0)

• Ising spins are forced topoint along local [111] axes

• there are four sublattices, κ

• anisotropy generates Isingpseudospins: Si = σidκ(i)

• J changes sign• exchange favours two-in

two-out states (Bernal-Fowler ice rules):

∑

i σi = 0

(a) (b) (c)1

H = J∑

〈ij〉

Si · Sj − E∑

i

(

dκ(i) · Si

)2

= −(J/3)∑

<ij>

σiσj

Geometry, topology and frustration:the physics of spin ice

The Pauling entropy

• An anisotropic pyrochlore ferromagnet leads to apseudospin antiferromagnet

• Pyrochlore antiferromagnets are highly frustrated.• Frustration leads to a large ground-state degeneracy.• Degeneracy of single tetrahedron:

– ground-state configurations: ng =(

42

)

= 6 (ice rules)– fraction of ground-state configurations: f = ng/4

2 = 3/8

• Pauling estimate of entropy for full lattice:– total number of ground states: 2nsfnt = 2nsfns/2

– ground state entropy per spin S0 = 12ln 3

2

• Connection between ice and pyrochlores due to Anderson

Geometry, topology and frustration:the physics of spin ice

Spin ice in a field

• Ising spins are forced topoint along local [111] axes

• there are four sublattices, κ

• anisotropy generates Isingpseudospins: Si = σidκ(i)

• J changes sign• B becomes staggered

(a)

[100] [110] [111]

(b) (c)

(d)

<110>

<100> <111>

H = J∑

〈ij〉

Si · Sj − E∑

i

(

dκ(i) · Si

)2

− gµBJ∑

i

B · Si

= −(J/3)∑

<ij>

σiσj − gµBJ∑

i

B · dκ(i)σi

Geometry, topology and frustration:the physics of spin ice

Magnetisation plateau in a [111] field

• exchange favours two-intwo-out states

• [111] field has projection of 1on one sublattice, −1/3 onother three → favoursthree-in one-out state

• low field we get amagnetisation plateau with2/3 of maximalmagnetisation (predicted Bramwell

et al., measured Hiroi et al.)

• Low-field plateau has anon-vanishing entropy of0.096kB/Dy or 0.078kB/Dy.

(a) (b) (c)10

1

2

3

4

5

0 5 10 15 20 25 30

0.48 K

0.99 K

1.65 K

M (

µ B/D

y)

H (kOe)

// [111]H

D y2Ti2O7

Geometry, topology and frustration:the physics of spin ice

From pyrochlore to kagome

Pyrochlore lattice in [111] direction:• One sublattice forms triangular

planes• Three other sublattices form

kagome planes• Kagome and triangular planes

alternate

Geometry, topology and frustration:the physics of spin ice

The plateau regime as a kagome magnet

• One sublattice pinned by field;triangular planes fully polarised.

• Kagome [111] planes nowdecoupled (other three sublattices):each triangle has two ’up’ and one’down’ spin.

• This is equivalent to kagome Isingin a field = hexagonal dimer model=triangular height model.

• Plateau is exactly soluble, forentropy S = 0.08kB/Dy, and forcorrelations: structure factor withlogarithmic peaks.

• Dimensional reduction in bulk!

x

y

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Geometry, topology and frustration:the physics of spin ice

Tilting the field out of the plateau regime

• Equivalence between kagomesites is lost

• Logarithmic peaks in structurefactor wander inwards untilcommensurate incommensuratetransition with anisotropicscaling.

• This is Kasteleyn transition, mi-xed first/second order character.Plot: 1/ξx and 1/ξy and m vs. tiltof field.

-0.5 -0.25 0.25 0.5 0.75 1

-0.25

0.25

0.5

0.75

1

1.25

1.5

Geometry, topology and frustration:the physics of spin ice

Termination of pleateau: entropy spike

• At high field, monomers(tetrahedra in 3-in 1-outstate) proliferate.

• This behaviour is capturedby monomer-dimer model,with entropy peak attransition – larger thanzero-field entropy.

• In experiment, first ordertransition takes over at low TAoki et al.

-10 -5 0 5 10fugacity

0.05

0.1

0.15

0.2

0.25

0.3

Spe

rpy

roch

lore

atom

B−Bc

T

S

Geometry, topology and frustration:the physics of spin ice

Back to zero field: correlations

• Longer-range correlations are present in pyrochlores:

Pyrochlore structure factor in[hhk] plane has sharp features(‘bowties’). Zinkin et al.

• These are (apparently) unimportant for thermodynamics.• They are due to the ground state constraint, L = 0, for each

tetrahedron.• This constraint gives rise to a local conservation law, which

can be resolved by a gauge theory. Youngblood et al., Henley, Huse et

al., Hermele et al.

Geometry, topology and frustration:the physics of spin ice

Gauge theory for the pyrochlore ground states

e`2

e`3

e`1

e`4

xy

z

same as for six-vertex in d = 2:

• Turn the Ising spins, Si, intooriented link vectors, Bi = Siei, on(bipartite!) diamond lattice.

• Ground-state constraint becomesconservation law ⇒ gauge theory.

Li = 0 ⇔ ∇ · B = 0 =⇒ B = ∇× A

• For continuous spins (XY, Heisen-berg models), define a separategauge field for each spin com-ponents.

Geometry, topology and frustration:the physics of spin ice

Long-wavelength analysis: coarse-graining

• Vector potential takes care of constraint• ‘Flippable’ loops have zero average flux:

low average flux ⇔ many microstates

• Ansatz: upon coarse-graining, obtain energyfunctional of entropic origin:

Z =

∫

D ~A exp[Scl];Scl = −K

2

∫

(∇× ~A)2

• Resulting correlators are transverse andalgebraic: e.g.

〈Bz(q)Bz(−q)〉 ∝ q2⊥/q2 ↔ (3 cos2 θ − 1)/r3.

Geometry, topology and frustration:the physics of spin ice

Bow-ties in nature

inelastic neutrons on YSc2 Ballou et al.neutrons on water

ice Ih Li et al.

Geometry, topology and frustration:the physics of spin ice

Large- N treatment compared to finite N

Strategy: Gauge ansatz works for any number of spincomponents. Therefore, consider classical O(N) model, N = ∞.O(∞) model is soluble Canals+Garanin – no free parameter.

S[hhk] =32

(

cos( qx

4) − cos( qz

4))2

sin( qx

4)2

5 − cos(qx) − 4 cos( qx

2) cos( qz

2)

Ising, T = 0 HeisenbergZinkin et al.

N = ∞Canals+Garanin

Geometry, topology and frustration:the physics of spin ice

Pyrochlore real space correlations

• Multiplied by r3: lines ↔ large-N ; dots ↔ Monte Carlo• different directions, sublattices, and system sizes

2 4 6 8 10 12 14

-0.05

0

0.05

0.1

0.15

0.2

2 4 6 8 10 12 14-0.3-0.25-0.2-0.15-0.1-0.05

00.05

Ising

2 4 6 8

-0.05

0

0.05

0.1

0.15

0.2

2 4 6 8-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

Heisenberg, T = J/200

Geometry, topology and frustration:the physics of spin ice

Dipolar spin ice: the real Hamiltonian

• Real interactions, D, are mainly dipolar: long-ranged!

Dij =Si · Sj − 3(Si · rij)(Si · rij)

r3ij

• Phenomenology well understood Gingras et al., Siddharthan et al.

• Question: Why is Pauling entropy (from nearest-neighbourice model) measured in dipolar spin ice???

• Answer: T = 0 correlations of n.n. model are also dipolar:

Gij ≡ limT→0

(H/T + λ1)−1ij ∝ −Dij for rij → ∞

⇒ Dipolar D and n.n. H are (approximately) diagonalised bythe same unitary transformation.

⇒ Spectrum (in mean-field theory) very similar

Geometry, topology and frustration:the physics of spin ice

Spin ice vs. dipolar spin ice: D = −G + ∆nn + ∆fn

-2 Π0

2 Π

-2 Π

0

2 Π

-4

-3

-2

-1

0

-2 Π0

-2 Π

0

-2 Π0

2 Π

-2 Π

0

2 Π

0

1

2

3

-2 Π0

-2 Π

0

-2 Π0

2 Π

-2 Π

0

2 Π

-5

0

-2 Π0

-2 Π

0

-2 Π0

2 Π

-2 Π

0

2 Π

-5

0

-2 Π0

-2 Π

0

spectra of

H G ∼ H−1T=0

G + ∆nn G + ∆nn

+∆fn

Geometry, topology and frustration:the physics of spin ice

Why spin ice obeys the ice rules

• At zero temperature, only the zero-energy modes play a role.• These form a flat band for the nearest-neighbour model.• The same flat band is found for dipolar interactions.• At zero (but not at finite T ), the physics is the same⇒ “projective equivalence”

• Dipolar spins are ice because ice is dipolar

Geometry, topology and frustration:the physics of spin ice

Summary

• Spin ice, a ferromagnet, is highly frustrated (geometry)• Dimensional reduction in applied field – extended string

defects restore higher dimensionality in weak field (topology)• Kagome ice with non-trivial thermodynamics and

correlations• Topological Kasteleyn transition in tilted field• Plateau termination: entropy peak (monomer-dimer model)• Large-N approach applicable to dipolar interactions

– dipolar spins are ice because ice is dipolar

• Gauge structure can be generalised to quantum problems:‘artificial light’

Geometry, topology and frustration:the physics of spin ice