B. D. Gaulin- Frustrated and Satisfied Ground States in Pyrochlore Magnets
Geometry, topology and frustration: the physics of spin icemoessner/Talks/Magd0305.pdf · Spin ice...
Transcript of Geometry, topology and frustration: the physics of spin icemoessner/Talks/Magd0305.pdf · Spin ice...
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