the monopole physics of spin ice - tcm groupthe monopole physics of spin ice p.c.w. holdsworth ecole...

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

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

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.

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

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

⇒

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

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

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)

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)

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)

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.


experiment

magnetolyte simulations 14-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

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]



experiment



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

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




Dnn > 0Dipolar scale

Jnn < 0Antiferro exchange scale

All in all out AIAO

JnnDnn

= −0.905

µ2 = −4 2J3

+ 831+ 2

3⎛⎝⎜

⎞⎠⎟D

⎡

⎣⎢⎢

⎤

⎦⎥⎥




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

∇.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 )

∇.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).

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

(−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).

(−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


Coexisiting AIAO order and Coulomb phase

Could occur here in Spin ice materials

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

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

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

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

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 )

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

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

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

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

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

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:

Square AC field – 8.2 secs

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

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

Magnetolyte DTO 0.5 K

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

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)

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

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

Field dependent mobility: Blowing away of Debye screening cloud (1st Wien effect)

Velocity max for Metropolis

(−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


�

∇ .

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)

Three species - bound particles, - free particles - unoccupied sites

Lattice Coulomb gas:

nb = nb+ + nb

−

nf = nf+ + nf

−

nu

nu

nf

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

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

Ion-hole conduction Cation-anion conduction

Examples:

the monopole physics of spin ice - tcm groupthe monopole physics of spin ice p.c.w. holdsworth ecole...

Documents