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Magnetic Monopoles in Spin Ice
Claudio CastelnovoUniversity of Oxford
Roderich MoessnerMax Planck Institut
Shivaji SondhiPrinceton University
ISIS Seminars, Rutherford Appleton Laboratory, Didcot (UK),September 30, 2008
Nature 451, 42 (2008)
Outline
General context
Introduction, from frustrated magnetism to spin ice
Dipolar interactions and the ice rule
Excitations: monopoles = fractionalised dipoles
Experimental evidence:I: Searching for monopolesII: The magnetic Coulomb liquid
The emergent gauge structure of spin ice
Conclusions
Outline
General context
Introduction, from frustrated magnetism to spin ice
Dipolar interactions and the ice rule
Excitations: monopoles = fractionalised dipoles
Experimental evidence:I: Searching for monopolesII: The magnetic Coulomb liquid
The emergent gauge structure of spin ice
Conclusions
Collective phenomena and complexity
Complementary fundamental questions:
I What are the fundamental building blocksof matter, and how do they interact?
⇒ high energy + particle physics
I Given building blocks and interactions: what is the resultingcollective behaviour?
⇒ many-body physics and complexity
Magnetic monopole search
Monopole passes through ring
⇒ magnetic flux through ring changes
⇒ e.m.f. induced in the ring ⇒ countercurrent ∝ QD is set up
Magnetic monopole search
Monopole passes through ring
⇒ magnetic flux through ring changes
⇒ e.m.f. induced in the ring ⇒ countercurrent ∝ QD is set up
Magnetic monopole search
Monopole passes through ring
⇒ magnetic flux through ring changes
⇒ e.m.f. induced in the ring ⇒ countercurrent ∝ QD is set up
Cabrera 1982
Outline
General context
Introduction, from frustrated magnetism to spin ice
Dipolar interactions and the ice rule
Excitations: monopoles = fractionalised dipoles
Experimental evidence:I: Searching for monopolesII: The magnetic Coulomb liquid
The emergent gauge structure of spin ice
Conclusions
Conventional vs frustrated Ising models
I Consider classical Ising spins, pointing eitherup or down: σi = ±1
I Simple exchange (strength J):
H = J∑〈ij〉
σiσj
I J < 0: ferromagnetic – spins align
I J > 0: antiferromagnetic – spins antialign
I . . . but only where possible: ‘frustration’
=⇒ What happens instead?
Frustration leads to (classical) degeneracy
Not all terms in H = J∑
〈ij〉 σiσj can simultaneously be minimised
I But we can rewrite H:
H =J
2
(4∑
i=1
σi
)2
+ const.
which can be minimised
I for a single tetrahedron:∑
i σi = 0
⇒ Ngs = (42) = 6 ground states
Degeneracy is the hallmark of frustration
Zero-point entropy on the pyrochlore lattice
I Pyrochlore lattice = corner-sharingtetrahedra
Hpyro =J
2
∑tet
(∑i∈tet
σi
)2
I Pauling estimate of ground stateentropy S0 = ln Ngs:
Ngs = 2N
(6
16
)N/2
⇒ S0 =N
2ln
3
2
I 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
Ho2Ti2O7 (and Dy2Ti2O7) are pyrochlore Ising magnets
Pauling entropy measured by Ramirez as predicted
The real (dipolar) Hamiltonian of spin ice Siddharthan+Shastry
I The nearest-neighbour model Hnn for spin ice is not correctdetails
I Leading term is dipolar energy (µ0µ2/4πa3 > J):
H = Hnn +µ0
4π
∑ij
~µi · ~µj − 3(~µi · r̂ij)(~µi · r̂ij)r3ij
I Both give same entropy (!!!) Gingras et al.
Wrong model → right answer . . .
WHY???
Outline
General context
Introduction, from frustrated magnetism to spin ice
Dipolar interactions and the ice rule
Excitations: monopoles = fractionalised dipoles
Experimental evidence:I: Searching for monopolesII: The magnetic Coulomb liquid
The emergent gauge structure of spin ice
Conclusions
The ‘dumbbell’ model (1)
Dipole ≈ pair of opposite charges (µ = qa):
I Sum over dipoles ≈ sum over charges:
H =
2Ndip.∑i ,j=1
v(rij) =
2Ndip.∑i ,j=1
µ0
4π
qiqj
rij
The ‘dumbbell’ model (2)
Choose a = ad , separation between centres of tetrahedra
I v ∝ q2/r is the usual Coulomb interaction (regularised):
v(rij) =
µ04π
qiqj
rijrij 6= 0
±vo(µa )2 = ±
[J3 + 4D
3 (1 +√
23)
]rij = 0,
Origin of the ice rules
Resum tetrahedral charges Qα =∑
i∈α qi :
H ≈∑ij
v(rij) −→∑αβ
V (rαβ) =
{µ04π
QαQβ
rαβα 6= β
12voQ2
α α = β
I Ice configurations (Qα ≡ 0) degenerate ⇒ Pauling entropy!
Origin of the ice rules
Resum tetrahedral charges Qα =∑
i∈α qi :
H ≈∑ij
v(rij) −→∑αβ
V (rαβ) =
{µ04π
QαQβ
rαβα 6= β
12voQ2
α α = β
I Ice configurations (Qα ≡ 0) degenerate ⇒ Pauling entropy!
Outline
General context
Introduction, from frustrated magnetism to spin ice
Dipolar interactions and the ice rule
Excitations: monopoles = fractionalised dipoles
Experimental evidence:I: Searching for monopolesII: The magnetic Coulomb liquid
The emergent gauge structure of spin ice
Conclusions
Excitations: dipoles or charges?
I Ground-state
I no net charge
I Excited states:
I flipped spin ↔ dipole excitation
I same as two charges?
Fractionalisation in d = 1
Excitations: dipoles or charges?
I Ground-state
I no net charge
I Excited states:
I flipped spin ↔ dipole excitation
I same as two charges?
Fractionalisation in d = 1
Excitations: dipoles or charges?
I Ground-state
I no net charge
I Excited states:
I flipped spin ↔ dipole excitation
I same as two charges?
Fractionalisation in d = 1
Excitations: dipoles or charges?
I Ground-state
I no net charge
I Excited states:
I flipped spin ↔ dipole excitation
I 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) = −µ0
4π
q2m
r
I magnetic Coulomb interaction
I deconfined monopoles
I monopoles in H, not B
I charge qm = 2µ/ad =(2µ/µB)(αλC/2πad)qD
≈ qD/8000
Deconfined magnetic monopoles
The dumbbell Hamiltonian gives
E (r) = −µ0
4π
q2m
r
I magnetic Coulomb interaction
I deconfined monopoles
I monopoles in H, not B
I charge qm = 2µ/ad =(2µ/µB)(αλC/2πad)qD
≈ qD/8000
Deconfined magnetic monopoles
The dumbbell Hamiltonian gives
E (r) = −µ0
4π
q2m
r
I magnetic Coulomb interaction
I deconfined monopoles
I monopoles in H, not B
I charge qm = 2µ/ad =(2µ/µB)(αλC/2πad)qD
≈ qD/8000
Deconfined magnetic monopoles
The dumbbell Hamiltonian gives
E (r) = −µ0
4π
q2m
r
I magnetic Coulomb interaction
I deconfined monopoles
I monopoles in H, not B
I charge qm = 2µ/ad =(2µ/µB)(αλC/2πad)qD
≈ qD/8000
Intuitive picture for monopoles
Simplest picture does not work: disconnect monopoles
Next best thing: no string tension between monopoles:
Two monopoles form a dipole:
I connected by tensionless ‘Dirac string’
I Dirac string is observable
⇒ qm ≈ qD/8000 not in conflict with quantisation of e
Outline
General context
Introduction, from frustrated magnetism to spin ice
Dipolar interactions and the ice rule
Excitations: monopoles = fractionalised dipoles
Experimental evidence:I: Searching for monopolesII: The magnetic Coulomb liquid
The emergent gauge structure of spin ice
Conclusions
Outline
General context
Introduction, from frustrated magnetism to spin ice
Dipolar interactions and the ice rule
Excitations: monopoles = fractionalised dipoles
Experimental evidence:I: Searching for monopolesII: The magnetic Coulomb liquid
The emergent gauge structure of spin ice
Conclusions
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 ‘Works’ for both fundamental cosmic and spin ice monopoles
I signal-noise ratio a problem
How do we know if a particle is elementary?
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 ‘Works’ for both fundamental cosmic and spin ice monopoles
I signal-noise ratio a problem
How do we know if a particle is elementary?
Outline
General context
Introduction, from frustrated magnetism to spin ice
Dipolar interactions and the ice rule
Excitations: monopoles = fractionalised dipoles
Experimental evidence:I: Searching for monopolesII: The magnetic Coulomb liquid
The emergent gauge structure of spin ice
Conclusions
Experiment II: interacting Coulomb liquid
Monopoles form a two-component Coulomb liquid
I any characteristic collective behaviour?
I interaction strength Γ ∝ (q2m/〈r〉)/T ∼ exp[−cv0/T ]/T
vanishes at both high and low T
I solution: [111] magnetic field acts as chemical potential
⇒ can tune 〈r〉 and T separately details
~B⇑
Experiment II: interacting Coulomb liquid
Monopoles form a two-component Coulomb liquid
I any characteristic collective behaviour?
I interaction strength Γ ∝ (q2m/〈r〉)/T ∼ exp[−cv0/T ]/T
vanishes at both high and low T
I solution: [111] magnetic field acts as chemical potential
⇒ can tune 〈r〉 and T separately details
~B⇑
Liquid-gas transition in spin ice in a [111] field
I Hnn predicts crossover to maximally polarised state
I dipolar H: first-order transition with critical endpointFisher et al.
I observedexperimentallySakakibara+Maeno
I confirmednumerically
Outline
General context
Introduction, from frustrated magnetism to spin ice
Dipolar interactions and the ice rule
Excitations: monopoles = fractionalised dipoles
Experimental evidence:I: Searching for monopolesII: The magnetic Coulomb liquid
The emergent gauge structure of spin ice
Conclusions
Conventional order and disorder
Gas-crystal (e.g. rock salt):
Paramagnet-ferromagnet (e.g. fridge magnet)
In between: critical points
Anything else???
Is spin ice ordered or not? details
No order as in ferromagnet
I deconfined monopoles (in 3d)
Not disordered like a paramagnet
I ice rules
⇒ ‘conservation law’
Consider magnetic moments ~µi
as (lattice) ‘flux’ vector field
I Ice rules ⇔ ∇ · ~µ = 0 ⇒ ~µ = ∇× ~A
I Local constraint⇒ ‘emergent gauge structure’
I Bow-tie motif in neutron scattering
I Algebraic (but not critical!) correlations
Is spin ice ordered or not? details
No order as in ferromagnet
I deconfined monopoles (in 3d)
Not disordered like a paramagnet
I ice rules
⇒ ‘conservation law’
Consider magnetic moments ~µi
as (lattice) ‘flux’ vector field
I Ice rules ⇔ ∇ · ~µ = 0 ⇒ ~µ = ∇× ~A
I Local constraint⇒ ‘emergent gauge structure’
I Bow-tie motif in neutron scattering
I Algebraic (but not critical!) correlations
Is spin ice ordered or not? details
No order as in ferromagnet
I deconfined monopoles (in 3d)
Not disordered like a paramagnet
I ice rules ⇒ ‘conservation law’
Consider magnetic moments ~µi
as (lattice) ‘flux’ vector field
I Ice rules ⇔ ∇ · ~µ = 0 ⇒ ~µ = ∇× ~A
I Local constraint⇒ ‘emergent gauge structure’
I Bow-tie motif in neutron scattering
I Algebraic (but not critical!) correlations
Is spin ice ordered or not? details
No order as in ferromagnet
I deconfined monopoles (in 3d)
Not disordered like a paramagnet
I ice rules ⇒ ‘conservation law’
Consider magnetic moments ~µi
as (lattice) ‘flux’ vector field
I Ice rules ⇔ ∇ · ~µ = 0 ⇒ ~µ = ∇× ~A
I Local constraint⇒ ‘emergent gauge structure’
I Bow-tie motif in neutron scattering
I Algebraic (but not critical!) correlations
Bow-ties in neutron scattering
proton correlations in waterice Ih Li et al.
spin correlations in kagome iceFennell+Bramwell
Outline
General context
Introduction, from frustrated magnetism to spin ice
Dipolar interactions and the ice rule
Excitations: monopoles = fractionalised dipoles
Experimental evidence:I: Searching for monopolesII: The magnetic Coulomb liquid
The emergent gauge structure of spin ice
Conclusions
Emergent particles and new order in spin ice
Spin ice is an interesting model system (and material!)
I frustrated magnet with ‘ground-state entropy’
I emergent gauge structure; (dimensional reduction in a field)
Magnetic monopoles as excitations
I fractionalisation / deconfinement in 3d material
I magnetic Coulomb law (felt by external test particle)
I would show up in monopole search
Emergent particles and new order in spin ice
Spin ice is an interesting model system (and material!)
I frustrated magnet with ‘ground-state entropy’
I emergent gauge structure; (dimensional reduction in a field)
Magnetic monopoles as excitations
I fractionalisation / deconfinement in 3d material
I magnetic Coulomb law (felt by external test particle)
I would show up in monopole search
Picture credits
Iceberg:windows.ucar.edu/tour/link=/earth/polar
/images/NOAA iceberg jpg image.html
Levitation:math.ucr.edu/home/baez/physics/General
/Levitation/levitation.html
Field lines:mcatpearls.com/master/img911.png
NaCl:greenfacts.org/images/glossary/crystal-
lattice.jpg
NATUREJOBSNew Year’s resolutions
A magnetic north–south divide in spin ice
POLES APARTPOLES APART
GEOPOLITICS Turf wars on
the ocean bed
ARCTIC CLIMATEWarming with
altitude
CANCER SUPPRESSION
The Down’s syndrome link
GEOPOLITICS Turf wars on
the ocean bed
ARCTIC CLIMATEWarming with
altitude
CANCER SUPPRESSION
The Down’s syndrome link
3 January 2008 | www.nature.com/nature | £10 THE INTERNATIONAL WEEKLY JOURNAL OF SCIENCE
451, 1–10
6 3 January 200
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no.7174
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[artwork by AlessandroCanossa]
Kagome ice: dimensional reduction in a field
Ising axes are not collinear back
I [111] field pins one sublattice ofspins
I Other sublattices form kagomelattice
I Kagome lattice: two-dimensional
I How many dimensions are there?
~B
⇑
Kagome ice: dimensional reduction in a field
Ising axes are not collinear back
I [111] field pins one sublattice ofspins
I Other sublattices form kagomelattice
I Kagome lattice: two-dimensional
I How many dimensions are there?
~B
⇑
Kagome ice: dimensional reduction in a field
Ising axes are not collinear back
I [111] field pins one sublattice ofspins
I Other sublattices form kagomelattice
I Kagome lattice: two-dimensional
I How many dimensions are there?
~B
⇑
Emergent gauge structure back
I Ground states differ by reversingspins around closed loops, forwhich the average 〈~µ〉 = 0
I Upon coarse-graining: lowaverage 〈~µ〉 preferred⇒ E ∼ (∇× ~A)2 ⇒ artificialmagnetostatics
Ansatz: upon coarse-graining, obtain energy functional of entropicorigin:
Z =
∫D~A exp[Scl], Scl = −K
2
∫(∇× ~A)2
The resulting correlators are transverse and algebraic:
∝ −q2⊥
q2↔
(3 cos2 θ − 1
)r3
Energy scale hierarchy in spin ice materials
(Dy, Ho magnetic moment ∼ 10µB) back
Energy scales:
I crystal field in the local[111] direction ∼ 200 K
I exchange interaction∼ 1− 2 K
I dipolar interaction∼ 2.5 K (at nn distance)