the monopole physics of spin ice - tcm groupthe monopole physics of spin ice p.c.w. holdsworth ecole...
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The monopole physics of spin ice
P.C.W. Holdsworth Ecole Normale Supérieure de Lyon
1. The dumbbell model of spin ice. 2. Specific heat for lattice Coulomb gas and spin ice 3. Spin ice phase diagram and moment fragmentation 4. The Wien effect in a lattice electrolyte and magnetolyte
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Vojtech Kaiser, Steven Bramwell, Roderich Moessner,
Ludovic Jaubert, Adam Harman-Clarke, Marion Brooks-Bartlett, Simon Banks, Michael Faulkner, Laura Bovo, Jonathon Bloxom
The monopole physics of spin ice
P.C.W. Holdsworth Ecole Normale Supérieure de Lyon
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Ice rules in spin ice
0 - 10K => Ising like spins along <111> axes
H = JSi.Sj +D
Si.Sjrij3 −
3(Si.rij )(Sj.rij )
rij5
⎡
⎣⎢⎢
⎤
⎦⎥⎥ij
∑ij∑
A good Hamiltonian for spin ice materials
|J| and D are O(1K)
Long range interactions are almost but not quite screened den Hertog and Gingras, PRL.84, 3430 (2000), Isakov, Moessner and Sondhi, PRL 95, 217201, 2005
T. Fennell et. al.,Science, 326, 415, 2009.
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Extension of the point dipoles into magnetic needles/dumbbelles Möller and Moessner PRL. 96, 237202, 2006, Castelnovo, Moessner, Sondhi, Nature, 451, 42, 2008
Configurations of magnetic charge at tetrehedron centres
⇒
NS
m
Magnetic ice rules two-in two-out
a
An extensive degeneracy of states satisfy these rules – Monopole vacuum
DSI Needles
Gingras et al
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H ≈ U(rij )− µ1 N̂1i> j∑ − µ2 N̂2
Castelnovo, Moessner, Sondhi, Nature, 451, 42, 2008
A grand canonical Coulomb gas of quasi particles.
⇒
ΔM = 2m, 4m ⇒
µ(J,m,a)
Topological defects carry magnetic charge – magnetic monopoles
In which case one should expect « electrolyte » physics + constraints - magnetolyte (Castelnovo)
U(r) = µ04π
Qi Qj
r; Qi = ± 2m
a,± 4m
a
⇒
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Spin Ice phase diagram
R. G. Melko and M. J. P. Gingras Journal of Physics-Condensed Matter, 16 (43) R1277–R1319 (2004).
µReducing
Dipolar spin ice phase diagram
µn = −n2 2J3
+ 831+ 2
3⎛⎝⎜
⎞⎠⎟D
⎡
⎣⎢⎢
⎤
⎦⎥⎥
D > 0Dipolar scale
J < 0Antiferro exchange scale
µ2 = 4µ1
CMS, Nature, 451, 42, 2008, Brooks-Bartlett et al (Phys. Rev. X, 4, 011007 (2014))
All in all out AIAO
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N N
+ +
Electrolyte and Magnetolyte Coulomb gases
1. Electrolyte
2. Magnetolyte
Chemical potential /particle for Dy2Ti2O7 /particle for Ho2Ti2O7
µ1 = −4.35K
E
B
CMS, Phys. Rev. B 84, 144435, 2011, Melko, Gingras JPCM, 16 (43) R1277–R1319 (2004)
µ1 = −5.7K
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Electrolyte and Magnetolyte Coulomb gases
1. Electrolyte
2. Magnetolyte
+ − n = 2exp(βµDH )1+ 2exp(βµDH )
N S
n =
43exp(βµDH )
1+ 43exp(βµDH )
n(T →∞) = n+ + n− = 23
n(T →∞) = nN + nS = 47
Debye Huckle theory, CMS, Phys. Rev. B 84, 144435, 2011
Debye Huckle theory, V. Kaiser, J. Bloxom, Ph.D. 2014
S(0) = 0, S(∞) ≈1.1
S(0) ≈ log 32
⎛⎝⎜
⎞⎠⎟ , S(∞) ≈ log
72
⎛⎝⎜
⎞⎠⎟ , ΔS ≈ 0.84
(Ryzhkin JETP, 101, 481, 2005)
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100 101
T [K]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5c⌫[J
K�1mol�
1]
magnetolyte
electrolyte
Heat capacity – numerical simulation (Single charge electrolyte 14-vertex magnetolyte DTO)
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100 101
T [K]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
c⌫[J
K�1mol�
1]
magnetolyte DH theory
electrolyte DH theory
magnetolyte
electrolyte
Heat capacity – simulation and Debye-Huckle theory (Single charge electrolyte 14-vertex magnetolyte DTO)
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100 101
T [K]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
c⌫[J
K�1mol�
1]
magnetolyte DH theory 14-v.
magnetolyte DH theory 16-v.
experiment
magnetolyte simulations 14-v.
magnetolyte simulations 16-v.
Double charge vertices become important for T> 2-3K Comparison with new specific heat data (Bovo et al unpublished)
HTO
High density Coulomb fluid at high T
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100 101
T [K]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
c⌫[J
K�1mol�
1]
magnetolyte DH theory 14-v.
magnetolyte DH theory 16-v.
experiment
magnetolyte simulations 14-v.
magnetolyte simulations 16-v.
Double charge vertices become important for T> 2-3K Comparison with new specific heat data (Bovo et al unpublished)
DTO
High density Coulomb fluid at high T
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Spin Ice phase diagram and Monopole Crystalization
For spin ice materials the ground state is a monopole vacuum. This could change by changing the chemical potential:
U =UC − µN0 < 0
AIAO = Crystal of charges in ZnS (Zinc Blend) structure if
UC = N0αu(a)2
⎛⎝⎜
⎞⎠⎟
α =1.638Madalung constant for diamond lattice
µ* = µu(a)
< α2= 0.819
Breaking of a Z2 symmetry µ* =1.4
For DTO
Qi = ± 4ma
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Spin Ice phase diagram and Monopole Crystalization
R. G. Melko and M. J. P. Gingras Journal of Physics-Condensed Matter, 16 (43) R1277–R1319 (2004).
Dipolar spin ice phase diagram
Dnn > 0Dipolar scale
Jnn < 0Antiferro exchange scale
All in all out AIAO
JnnDnn
= −0.905
µ2 = −4 2J3
+ 831+ 2
3⎛⎝⎜
⎞⎠⎟D
⎡
⎣⎢⎢
⎤
⎦⎥⎥
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Spin Ice phase diagram and Monopole Crystalization
R. G. Melko and M. J. P. Gingras Journal of Physics-Condensed Matter, 16 (43) R1277–R1319 (2004).
Dipolar spin ice phase diagram
Dnn > 0Dipolar scale
Jnn < 0Antiferro exchange scale
Brooks-Bartlett et al (Phys. Rev. X, 4, 011007 (2014))
All in all out AIAO
Double monopole crystalization to AIAO phase at:
JnnDnn
= − 451+ 2
31− α
2⎛⎝⎜
⎞⎠⎟
⎡
⎣⎢
⎤
⎦⎥ = −0.918
JnnDnn
= −0.905
Excellent agreement With DSI
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∇.M ⇒
M.dSi = −Qi∫
Magnetic moments, charge and lattice gauge field
Coulomb fluid satisfies magnetic Gauss’ law:
∇.H = −
∇.M = ρ( )
iThree in one out defect
�
∇ . B = 0 = µ0
∇ .(
M + H )
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∇.M ⇒
M.dSi = −Qi∫
Mij =M.dS = −M ji = ± m
aLattice field
Magnetic moments, charge and lattice gauge field
Mij⎡⎣ ⎤⎦ =ma
−1,−1,−1,1[ ]
Qi =2ma
Coulomb fluid satisfies magnetic Gauss’ law:
∇.H = −
∇.M = ρ( )
⇒ Mijj∑ = −Qi
ij
Three in one out vertex
�
∇ . B = 0 = µ0
∇ .(
M + H )
Brooks-Bartlett et al (Phys. Rev. X, 4, 011007 (2014)) Maggs & Rossetto, PRL, 88, 196402, 2002. Bramwell, Phil. Trans. R. Soc. A 370, 5738 (2012).
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Mij⎡⎣ ⎤⎦ =ma(−1,−1,−1,1) = m
a− 12,− 12,− 12,− 12
⎛⎝⎜
⎞⎠⎟ +
ma
− 12,− 12,− 12, 32
⎛⎝⎜
⎞⎠⎟
Moment fragmentation (Lattice Helmholtz decomposition) M = ∇ψ +
∇∧A =
Mm +
Md
3 in – 1 out/3 out – 1 in Md and Mm
Mij⎡⎣ ⎤⎦ = 0[ ]+ ma
−1,−1,1,1[ ]
Mij⎡⎣ ⎤⎦ =ma
−1,−1,−1,−1[ ]+ 0[ ]
2 in-2 out Md only
All in/all out Mm only
Two fluids
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(−1,−1,−1,1) = − 12,− 12,− 12,− 12
⎛⎝⎜
⎞⎠⎟ + − 1
2,− 12,− 12, 32
⎛⎝⎜
⎞⎠⎟
Excluding double charges gives a « simple » monopole crystal Brooks-Bartlett et al, Phys. Rev. X, 4, 011007 (2014)
7
The red spins map onto dimers on a diamond lattice
Divergence free part maps ontothe emergent gauge field from thedimers
Huse et. al. PRL 91, 167004, 2005 Field E=1 along dimer, -1/(z-1) between dimers
M = ∇ψ +
∇∧A =
Mm +
Md
Crystal of 3 in – 1 out vertices 3 out – 1 in
G.-W. Chern et al , PRL 106, 207202 (2011). Moller and Moessner, PRB 80, 140409(2009).
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(−1,−1,−1,1) = − 12,− 12,− 12,− 12
⎛⎝⎜
⎞⎠⎟ + − 1
2,− 12,− 12, 32
⎛⎝⎜
⎞⎠⎟
Excluding double charges gives a « simple » monopole crystal Brooks-Bartlett et al, Phys. Rev. X, 4, 011007 (2014)
M = ∇ψ +
∇∧A =
Mm +
Md
Crystal of 3 in – 1 out vertices 3 out – 1 in
Coexisiting AIAO order and Coulomb phase
Could occur here in Spin ice materials
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Deconfined monopole charge via Bramwell et al, Nature, 461, 956, 2009
Charges either free or bound in pairs. Conduction due to free charges. Applying field changes concentration of free charges:
�
δσ(E)σ
= b2
L. Onsager, “Deviations from Ohm’s law in "weak electrolytes”. "J. Chem. Phys. 2, 599,615 (1934)"
�
n = nb + n f
�
δn f (E)n f (0)
= b2
=Eq3
16πε0kB2T 2
⇒BQ3µ0
16πkB2T 2
The Wien effect
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Deconfined monopole charge via Bramwell et al, Nature, 461, 956, 2009
Charges either free or bound in pairs. Conduction due to free charges. Applying field changes concentration of free charges:
�
n = nb + n f
The Wien effect
�
δσ(E)σ
⇒ δν(B)ν
=BQ3µ0
16πkB2T 2
Muon relaxation
Highly controversial !
Dunsiger et al, Phys Rev. Lett, 107, 207207, 2011 Sala et al, Phys. Rev. Lett. 108, 217203, 2012 Blundell, Phys. Rev. Lett. 108, 147601, 2012
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Large internal fields even in the absence of charges
M = ∇ψ +
∇∧A =
Mm +
Md
∇.H = −
∇.M = ρ( )
When ρ = 0,M =
Md
Monopole physics but transient currents only Ryzhkin JETP, 101, 481, 2005. Jaubert and Holdsworth, Nature Physics, 5, 258, 2009
Φ
Monopolar and dipolar parts (largely) decoupled and dynamics is from monopole movement
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The 2nd Wien effect: L. Onsager, “Deviations from Ohm’s law in weak electrolytes”. J. Chem. Phys. 2, 599,615 (1934)"
Non-Ohmic conduction in low density charged fluids
n = nf + nb
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Length scales
Three length scales appear naturally:
The Bjerrum length :
Field drift length: lE =kBTqE
Particles separated by r < lT are bound
lT =q2
8πε0kBT
Debye screening length lD ⇒Veff (r) =q1q24πε0r
exp(−r / lD )
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The Wien effect
b = lTlE
∝ q3ET 2
L. Onsager, “Deviations from Ohm’s law in weak electrolytes”. J. Chem. Phys. 2, 599,615 (1934)"
for
lD >> lE, lT
Linear in E For small field – this is a non-equilibrium effect
nf (E)nf (0)
≈ I2 (2 b )2b
=1+ b2+O(b2 )
n = nf + nb
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The linear field dependence => A non-equilibrium effect => Compare with Blume-Capel paramagnet.
H = −B Si + Δ Si( )2 , Si = 0,±1∑i∑
n(0) = n↑ + n↓ =2exp(−βΔ)1+ 2exp(−βΔ)
n(B) = n↑(B)+ n↓(B) =n(0)2(exp(βB)+ exp(−βB))
= n(0)+O(B2 )
This scalar quantity changes quadratically with applied field
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Simulation results for the lattice electrolyte: Kaiser, Bramwell, PCWH, Moessner, Nature Materials, 12, 1033-1037, (2013)
0.00 0.02 0.04 0.06 0.08 0.10 0.12E ⇤
0
1
2
3
4�n f(B
)/n f(0)Onsager’s theory
Simulations
T*=0.14 => 0.43K for DTO,
nf (0) ≈10−4
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Results: Kaiser, Bramwell, PCWH, Moessner, Nature Materials, 12, 1033-1037, (2013)
Linear in to lowest order E
0.00 0.02 0.04 0.06 0.08 0.10 0.12E ⇤
0
1
2
3
4
�n f(B
)/n f(0)
Onsager’s theory
Simulations
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Linear term is renormalized away by Debye screening:
Crossover to quadratic behaviour and negative intercept
E = DCrossover when
0.00 0.02 0.04 0.06 0.08 0.10 0.12E ⇤
0
1
2
3
4
�n f(B
)/n f(0)
Onsager’s theory
Simulations
See poster V. Kaiser
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Switching on field at t=0
Electrolyte Magnetolyte
Time scale: 1 MCS = 1 ms for DTO Jaubert and Holdsworth, Nature Phyiscs, 5, 258, 2009
Wien effect in the magnetolyte:
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Square AC field – 8.2 secs
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0.0000
0.0005
0.0010
0.0015
0.0020
n(t)
[]
DTO @ 0.5 K
0.0 0.2 0.4 0.6 0.8 1.0t [1000 MC steps ' s]
�0.020.00
0.02
m(t)[]
�60�40�200204060
B(t)[m
T]
Square AC field – 0.13 secs
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0.0000
0.0005
0.0010
0.0015
0.0020
n(t)
[]
DTO @ 0.5 K
0.0 0.2 0.4 0.6 0.8 1.0t [1000 MC steps ' s]
�0.020.00
0.02
m(t)[]
�60�40�200204060
B(t)[m
T]
Average monopole concentration
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Magnetolyte DTO 0.5 K
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0.00 0.02 0.04 0.06 0.08 0.10 0.12E ⇤
0
1
2
3
4
�n f(B
)/n f(0)
Onsager’s theory
Simulations
Electrolyte DTO 0.43 K Magnetolyte DTO 0.5 K
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An experimental signal ?
χ(B,T ) = χ0 (T )+ χ1 (T )B2 + .......In equilibrium
Wien contribution χ(B,T ) = χ0 (T )+ χ1 (T )B + .......
Sinusoidal ac field
χB (ω0 )χ0 (ω0 )
≈nf (<
B >)
nf (0)
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Conclusions
1. The dumbbell (needle) model reproduces
many characteristics of spin ice.
2. Include specific heat, phase boundary and monopole crystalization.
3. The Wien effect can emerge from the
magnetolyte.
HFM2014, Cambridge, 7-11 July 2014
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Relative conductivity falls below prediction
Theory relies on mobility, being field independent
σ = q2ωnf
ω
0.02 0.04 0.06 0.08 0.10 0.12E ⇤
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5��(E⇤ )/�
(0)
Onsager’s theoryOnsager’s theory + latticeSimulations
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Field dependent mobility: Blowing away of Debye screening cloud (1st Wien effect)
Velocity max for Metropolis
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(−1,−1,−1,1) = − 12,− 12,− 12,− 12
⎛⎝⎜
⎞⎠⎟ + − 1
2,− 12,− 12, 32
⎛⎝⎜
⎞⎠⎟
Magnetic monopole crystalization breaks this symmetry Brooks-Bartlett et al, Phys. Rev. X, 4, 011007 (2014)
7
The red spins map onto dimers on a diamond lattice
Divergence free part maps ontothe emergent gauge field from thedimers
Huse et. al. PRL 91, 167004, 2005 Field E=1 along dimer, -1/(z-1) between dimers
M = ∇ψ +
∇∧A =
Mm +
Md
Crystal of 3 in – 1 out vertices 3 out – 1 in
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�
∇ .
M = 0 M= divergence free field
Ice rules, topological constraints S. V. Isakov, K. Gregor, R. Moessner, "and S. L. Sondhi PRL 93, 167204, 2004
M =
∇∧A =
MdEmergent gauge field
Monopole vacuum has divergence free configurations - « Coulomb phase » Physics.
Pinch Points:
T. Fennell et. al., Magnetic Coulomb Phase in the Spin Ice Ho7O2Ti2 Science, 326, 415, 2009.
Topological sector fluctuations: Jaubert et. al. Phys. Rev. X, 3, 011014, (2013)
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Three species - bound particles, - free particles - unoccupied sites
Lattice Coulomb gas:
nb = nb+ + nb
−
nf = nf+ + nf
−
nu
nu
nf
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nu + nb + nf = 1
[nu ]⇔ [nb+ ,nb
− ]⇔ [nf+ ]+ [nf
− ]
�
K =k⇒
k⇐ =n f2
nb
The Wien effect
nb = nb+ + nb
− nf = nf+ + nf
− nu
L. Onsager, “Deviations from Ohm’s law in weak electrolytes”. J. Chem. Phys. 2, 599,615 (1934)"
dnfdt
= k⇒nb − k⇐nf
2 = 0
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Sala et al, Phys. Rev. Lett. 108, 217203, 2012
Needles:
Flux confined to the needle => zero field sites
B = 0
B large Dipoles
Field spills out
B ≠ 0
From 150 mT to 4.5 T
Internal field distributions in dipolar spin ice
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Ion-hole conduction Cation-anion conduction
Examples: