Magnetic Monopoles in Spin Ice

Magnetic Monopoles in Spin Ice
Claudio Castelnovo
University of Oxford
Roderich Moessner
Max Planck Institut
Shivaji Sondhi
Princeton University
Nature 451, 42 (2008)
25th International Conference on Low Temperature Physics,
Amsterdam (NL), August 11, 2008
Outline
I
the spin-ice model
I
low temperature behaviour: from spins to monopoles
I
experimental evidence of deconfined monopolar excitations
I
conclusions
Conventional vs frustrated (Ising) models
Consider classical Ising spins, σi = ±1
P
with exchange interaction: H = J hiji σi σj
I
J < 0: all the spins align ferromagnetically
Conventional vs frustrated (Ising) models
Consider classical Ising spins, σi = ±1
P
with exchange interaction: H = J hiji σi σj
I
J < 0: all the spins align ferromagnetically
I
J > 0: antiferromagnetic order is frustrated
H =
J
2
4
X
⇒ Ngs =
σi
+ const.
i=1
for a single tetrahedron:
( 42 )
!2
P
i
σi = 0
= 6 ground states
Conventional vs frustrated (Ising) models
Consider classical Ising spins, σi = ±1
P
with exchange interaction: H = J hiji σi σj
I
J < 0: all the spins align ferromagnetically
I
J > 0: antiferromagnetic order is frustrated
H =
J
2
4
X
!2
σi
+ const.
i=1
Degeneracy is the hallmark of frustration
Zero-point entropy on the pyrochlore lattice
I
Pyrochlore lattice = corner-sharing tetrahedra
!2
JX X
Hpyro =
σi
2 tet
i∈tet
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Pauling estimate of ground state
entropy S0 = ln Ngs :
Ngs = 2
I
N
6
16
N/2
N 3
⇒ S0 = ln
2 2
microstates vs. constraints;
N spins, N/2 tetrahedra
Mapping from ice to spin ice
I
In ice, water molecules retain their identity
I
Hydrogen near oxygen ↔ spin pointing in
150.69.54.33/takagi/matuhirasan/SpinIce.jpg
Pauling entropy in spin ice
Anderson 1956; Harris+Bramwell 1997
Ho2 Ti2 O7 (and Dy2 Ti2 O7 ) are pyrochlore Ising magnets
Pauling entropy measured by Ramirez as predicted
The real (dipolar) Hamiltonian of spin ice
I
Siddharthan+Shastry
The nearest-neighbour model Hnn for spin ice is not correct
details
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Leading term is dipolar energy (µ0 µ2 /4πa3 > J):
H = Hnn +
~i · µ
~ j − 3(~
µi · r̂ij )(~
µi · r̂ij )
µ0 X µ
3
4π
rij
ij
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Both give same entropy (!!!)
Gingras et al.
Wrong model → right answer . . .
WHY???
The ‘dumbbell’ model (1)
Dipole ≈ pair of opposite charges (µ = qa):
I
Sum over dipoles ≈ sum over charges:
2Ndip.
H=
X
i,j=1
2Ndip.
v (rij ) =
X µ0 qi qj
4π rij
i,j=1
The ‘dumbbell’ model (2)
Choose a = ad , separation between centres of tetrahedra
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v ∝ q 2 /r is the usual Coulomb interaction (regularised):
 µ0 qi qj
rij 6= 0
 4π rij
q v (rij ) =
 ±vo ( µa )2 = ± J3 + 4 D3 (1 + 23 )
rij = 0,
Origin of the ice rules
Resum tetrahedral charges Qα =
P
i∈α qi :
(
H≈
X
ij
v (rij ) −→
X
αβ
V (rαβ ) =
µ0 Qα Qβ
4π rαβ
1
2
2 vo Qα
α 6= β
α=β
Origin of the ice rules
Resum tetrahedral charges Qα =
P
i∈α qi :
(
H≈
X
ij
I
v (rij ) −→
X
αβ
V (rαβ ) =
µ0 Qα Qβ
4π rαβ
1
2
2 vo Qα
α 6= β
α=β
Ice configurations (Qα ≡ 0) degenerate ⇒ Pauling entropy!
Excitations: dipoles or charges?
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Ground-state
I
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no net charge
Excited states:
I
I
flipped spin ↔ dipole excitation
same as two charges?
Excitations: dipoles or charges?
I
Ground-state
I
I
no net charge
Excited states:
I
I
flipped spin ↔ dipole excitation
same as two charges?
Excitations: dipoles or charges?
I
Ground-state
I
I
no net charge
Excited states:
I
I
flipped spin ↔ dipole excitation
same as two charges?
Excitations: dipoles or charges?
I
Ground-state
I
I
no net charge
Excited states:
I
I
flipped spin ↔ dipole excitation
same as two charges?
Fractionalisation in d = 1
Excitations in spin ice: dipolar or charged?
Single spin-flip (dipole µ)
≡
two charged tetrahedra
(charges qm = 2µ/ad )
Are charges independent?
⇒ Fractionalisation in d = 3?
Deconfined magnetic monopoles
The dumbbell Hamiltonian gives
E (r ) = −
2
µ0 qm
4π r
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magnetic Coulomb interaction
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deconfined monopoles
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monopoles in H, not B
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charge qm = 2µ/ad =
(2µ/µB )(αλC /2πad )qD
≈ qD /8000
Experiment I: Stanford monopole search
Monopole passes through ring
⇒ magnetic flux through ring changes
⇒ e.m.f. induced in the ring ⇒ countercurrent ∝ qm is set up
Experiment I: Stanford monopole search
Monopole passes through ring
⇒ magnetic flux through ring changes
⇒ e.m.f. induced in the ring ⇒ countercurrent ∝ qm is set up
I
I
‘Works’ for both fundamental cosmic and spin ice monopoles
signal-noise ratio a problem
How do we know if a particle is elementary?
Experiment II: interacting Coulomb liquid
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Monopoles form a two-component Coulomb liquid
I
[111] magnetic field acts as staggered chemical potential
~
B
⇑
=⇒ we can tune ρmonopole and T separately
details
Liquid-gas transition in spin ice in a [111] field
I
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Hnn predicts crossover to maximally polarised state
dipolar H: first-order transition with critical endpoint
Fisher et al.
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observed
experimentally
Hiroi+Maeno groups
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confirmed
numerically
Emergent particles and new order in spin ice
Spin ice is an interesting model system (and material!)
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frustrated magnet with ‘ground-state entropy’
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(emergent gauge structure; dimensional reduction in a field)
Magnetic monopoles as excitations
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fractionalisation / deconfinement in 3d material
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magnetic Coulomb law (felt by external test particle)
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would show up in monopole search
Picture credits
Iceberg:
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Field lines:
no.7174
mcatpearls.com/master/img911.png
POLES
APART
A magnetic north–south
divide in spin ice
3.1 cover UK 1
NaCl:
greenfacts.org/images/glossary/crystallattice.jpg
01
9 770028 083095
18/12/07 4:31:29 pm
[artwork by Alessandro
Canossa]
Kagome ice: dimensional reduction in a field
Ising axes are not collinear
I
back
[111] field pins one sublattice of
spins
~
B
⇑
Kagome ice: dimensional reduction in a field
Ising axes are not collinear
back
I
[111] field pins one sublattice of
spins
I
Other sublattices form kagome
lattice
~
B
⇑
Kagome ice: dimensional reduction in a field
Ising axes are not collinear
back
I
[111] field pins one sublattice of
spins
I
Other sublattices form kagome
lattice
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Kagome lattice: two-dimensional
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How many dimensions are there?
~
B
⇑
Emergent gauge structure
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Ground states differ by reversing
spins around closed loops, for
which the average h~
µi = 0
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Upon coarse-graining: low
average h~
µi preferred
~ 2 ⇒ artificial
⇒ E ∼ (∇ × A)
magnetostatics
back
Ansatz: upon coarse-graining, obtain energy functional of entropic
origin:
Z
Z
K
~
~ 2
Scl = −
Z = DA exp[Scl ],
(∇ × A)
2
The resulting correlators are transverse and algebraic:
2
3 cos2 θ − 1
q⊥
∝− 2
↔
q
r3
Energy scale hierarchy in spin ice materials
(Dy, Ho magnetic moment ∼ 10µB )
Energy scales:
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crystal field in the local
[111] direction ∼ 200 K
back
Energy scale hierarchy in spin ice materials
(Dy, Ho magnetic moment ∼ 10µB )
Energy scales:
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crystal field in the local
[111] direction ∼ 200 K
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exchange interaction
∼1−2 K
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dipolar interaction
∼ 2.5 K (at nn distance)
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