Magnetic monopoles in spin ice Magnetic monopoles in spin ice

Magnetic monopoles in spin ice Magnetic monopoles in spin ice

Magnetic monopoles in spin ice
Claudio Castelnovo
Oxford University
Roderich Moessner
MPI-PKS Dresden
Shivaji Sondhi
Princeton University
Nature 451, 42 (2008)
The fundamental question
scale (metres)
1018
astroparticle physics
1015
1012
109
106
cosmology
astrophysics
103
100
our world
10−3
10−6
10−9
nuclear physics
10?
string theory
The fundamental question
scale (metres)
1018
astroparticle physics
1015
1012
109
106
cosmology
astrophysics
emergent phenomena
103
100
our world
10−3
many body physics
complexity
10−6
10−9
nuclear physics
10?
string theory
Collective phenomena and complexity
Complementary questions:
• What are the fundamental building blocks
of matter, and how do they interact?
⇒ high energy+particle physics
• Given building blocks and interactions: what is the resulting
collective behaviour?
⇒ many-body physics and complexity
Outline
• frustrated Ising models and the (spin) ice model
• the spin ice compounds
–
‘zero-point entropy’
• long-range (dipolar) interactions
–
–
survival of the ground-state degeneracy
excitations: magnetic monopoles and their properties
• Is spin ice ordered?
Conventional vs frustrated Ising models
• Consider classical Ising spins, pointing
either up or down: σi = ±1
• Simple exchange (strength J):
H = Jσi σj
–
–
–
J < 0: ferromagnetic – spins align
J > 0: antiferromagnetic – spins antialign
. . . but only where possible: ‘frustration’
=⇒ What happens instead?
?
Frustration leads to (classical) degeneracy
Not all terms in H =
P
hiji
σi σj can simultaneously be minimised
?
• But we can rewrite H:
J
H=
2
q
X
i=1
σi
!2
+ const
which can be minimised
P
• for tetrahedron:
i σi = 0
⇒ Ngs = ( 42 ) = 6 ground states
Degeneracy is hallmark of frustration
Zero-point entropy on the pyrochlore lattice
• Pyrochlore lattice =
corner-sharing tetrahedra
Hpyro
JX
=
2 tet
X
σi
i∈tet
!2
• Pauling estimate of ground state
entropy S0 = ln Ngs :
N
Ngs = 2
6
16
N/2
1 3
⇒ S0 = ln
2 2
• microstates vs. constraints;
N spins, N/2 tetrahedra
Pauling entropy in spin ice Anderson 1956; Harris+Bramwell 1997
Ho2 Ti2 O7 (and Dy2 Ti2 O7 ) are pyrochlore Ising magnets
Pauling entropy measured Ramirez as predicted
Mapping from ice to spin ice
• In ice, water molecules retain their identity
• Hydrogen near oxygen ↔ spin pointing in
150.69.54.33/takagi/matuhirasan/SpinIce.jpg
• axes non-collinear
• everything seems to hang together
The real (dipolar) Hamiltonian of spin ice Siddharthan+Shastr
• The nearest-neighbour model Hnn for spin ice is not correct
• Leading term is dipolar energy (µ0 µ2 /4πa3 > J):
µ0 X ~µi · ~µj − 3(~µi · r̂ij )(~µi · r̂ij )
H = Hnn +
4π ij
rij3
• Both give same entropy (!!!)
Gingras et al.
Wrong model → right answer . . .
WHY???
The ‘dumbell’ model
Dipole ≈ pair of opposite charges (µ = qa):
+q
• Sum over dipoles ≈ sum over charges:
Hij =
2
X
µ
=
a
−q
v(rijmn )
m,n=1
• v ∝ q 2 /r is the usual Coulomb interaction (regularised):
v(rijmn )
=
(
µ0 qim qjn /(4πrijmn )
vo ( µa )2 =
J
3
+ 4 D3 (1 +
i 6= j
q
2
)
3
i = j,
Origin of the ice rules
Choose a = ad , separation between centres of tetrahedra
Resum tetrahedral charges Qα =
H≈
mn
X
ij
v(rij,mn ) −→
X
αβ
P
rim ∈α
V (rαβ ) =
(
qim :
µ0 Qα Qβ
4π rαβ
1
2
v
Q
o
α
2
α 6= β
α=β
• Ice configurations (Qα ≡ 0) degenerate⇒ Pauling entropy!
Excitations: dipoles or charges?
• Ground-state
–
no net charge
• Excited states:
–
–
flipped spin ↔ dipole excitation
same as two charges?
Q=0
one dipole
two charges
Excitations: dipoles or charges?
• Ground-state
–
no net charge
• Excited states:
–
–
flipped spin ↔ dipole excitation
same as two charges?
Q=0
one dipole
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
Dumbell Hamiltonian gives
2
µ0 qm
E(r) = −
4π r
• magnetic Coulomb interaction
Deconfined magnetic monopoles
Dumbell Hamiltonian gives
2
µ0 qm
E(r) = −
4π r
• magnetic Coulomb interaction
• deconfined monopoles
Deconfined magnetic monopoles
Dumbell Hamiltonian gives
2
µ0 qm
E(r) = −
4π r
• magnetic Coulomb interaction
• deconfined monopoles
–
–
charge qm = 2µ/a =
(2µ/µb )(αλC /2πad )qD
≈ qD /8000
monopoles in H, not B
Intuitive picture for monopoles
Simplest picture does not work: disconnect monopoles
N
S
N
S
N
S
N
S
Next best thing: no string tension between monopoles:
Two monopoles form a dipole:
• connected by tensionless ‘Dirac string’
• Dirac string is observable
⇒ qm ≈ qD /8000 not in conflict with quantisation of e
Experiment I: Stanford monopole search
Monopole passes through superconducting ring
⇒ magnetic flux through ring changes
⇒ e.m.f. induced in the ring ⇒ countercurrent ∝ qm is set up
• ‘Works’ for both fundamental cosmic and spin ice monopoles
• signal-noise ratio a problem
Experiment I: Stanford monopole search
Monopole passes through superconducting ring
⇒ magnetic flux through ring changes
⇒ e.m.f. induced in the ring ⇒ countercurrent ∝ qm is set up
• ‘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
Monopoles form a two-component liquid
• any characteristic collective behaviour?
2
• interaction strength Γ ∝ (qm
/hri)/T ∼ exp[−cv0 /T ]/T
vanishes at both high and low T
Experiment II: interacting Coulomb liquid
Monopoles form a two-component liquid
• any characteristic collective behaviour?
2
• interaction strength Γ ∝ (qm
/hri)/T ∼ exp[−cv0 /T ]/T
vanishes at both high and low T
• solution: [111] magnetic field acts as chemical potential
⇒ can tune hri and T separately
~
B
⇑
Liquid-gas transition in spin ice in a [111] field
• Hnn predicts crossover to maximally polarised state
• dipolar H: first-order transition with critical endpoint
• observed
experimentally
Hiroi+Maeno groups
• confirmed
numerically
Fisher et al.
Kagome ice: dimensional reduction in a field
Ising axes are not collinear
• [111] field pins one sublattice of spins
~
B
⇑
Kagome ice: dimensional reduction in a field
Ising axes are not collinear
• [111] field pins one sublattice of spins
~
B
• Other sublattices form kagome lattice
⇑
Kagome ice: dimensional reduction in a field
Ising axes are not collinear
• [111] field pins one sublattice of spins
~
B
• Other sublattices form kagome lattice
• Kagome lattice: two-dimensional
• How many dimensions are there?
⇑
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?
No order as in ferromagnet
• deconfined monopoles
Is spin ice ordered or not?
No order as in ferromagnet
• deconfined monopoles
Not disordered like a paramagnet
• ice rules
Is spin ice ordered or not?
No order as in ferromagnet
• deconfined monopoles
Not disordered like a paramagnet
• ice rules ⇒ ‘conservation law’
Consider magnetic moments
(lattice) ‘flux’ vector field
~µi
~
• Ice rules ⇔ ∇ · ~
µ = 0 =⇒ ~µ = ∇ × A
as
Is spin ice ordered or not?
No order as in ferromagnet
• deconfined monopoles
Not disordered like a paramagnet
• ice rules ⇒ ‘conservation law’
Consider magnetic moments
(lattice) ‘flux’ vector field
~µi
as
~
• Ice rules ⇔ ∇ · ~
µ = 0 =⇒ ~µ = ∇ × A
• Local constraint
⇒ ‘emergent gauge structure’
• Bow-tie motif in neutron scattering
• Algebraic (but not critical!) correlations
Bow-ties in neutron scattering
proton correlations in
water ice Ih Li et al.
spin correlations in
kagome ice Fennell+Bramwell
Emergent particles and new order in spin ice
Spin ice is an interesting model system (and material!)
• frustrated magnet with ‘ground-state entropy’
• dimensional reduction in a field; emergent gauge structure
Emergent particles and new order in spin ice
Spin ice is an interesting model system (and material!)
• frustrated magnet with ‘ground-state entropy’
• dimensional reduction in a field; emergent gauge structure
Magnetic monopoles as excitations
• magnetic Coulomb law (felt by external test particle)
• fractionalisation/deconfinement in 3d material
• would show up in monopole search: qm ≈ qD /8000
Thanks
• Claudio Castelnovo
• John Chalker Oxford
• Karol Gregor
Oxford
Caltech
• Peter Holdsworth
• Sergei Isakov
ENS Lyon
ETH Zürich
• Ludovic Jaubert
ENS Lyon
• Kumar Raman UC Riverside
• Shivaji Sondhi Princeton
Alessandro Canossa
Picture credits
Iceberg:
www.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: www.mcatpearls.com/master/img911.png
NaCl: http://www.greenfacts.org/images/glossary/crystal-lattice.jpg