The AC Wien effect: non-linear non-equilibrium susceptibility of spin ice P.C.W. Holdsworth Ecole Normale Supérieure de Lyon 1. The Wien effect 2. The dumbbell model of spin ice. 3. The Wien effect in a magnetic Coulomb gas Vojtech Kaiser, Steven Bramwell, Roderich Moessner,
L. Onsager, “Deviations from Ohm’s law in The Wien effect: weak electrolytes”. J. Chem. Phys. 2, 599,615 (1934) � Non-Ohmic conduction in low density charged fluids E n = n f + n b
Ion-hole conduction
Length scales Three length scales appear naturally: q 2 The Bjerrum length : l T = 8 πε 0 k B T Particles separated by r < l T are bound l E = k B T Field drift length: qE ⎛ ⎞ 1/ 2 l D = 2 πε 0 k B Ta 3 Debye screening length ⎜ ⎟ ⎜ ⎟ q 2 n f ⎝ ⎠
Lattice Coulomb gas: n f n u + + n b − n b = n b Three species - bound particles, + + n f − n f = n f - free particles - unoccupied sites n u
L. Onsager, “Deviations from Ohm’s law in The Wien effect weak electrolytes”. J. Chem. Phys. 2, 599,615 (1934) � + + n b + + n f − − n b = n b n f = n f n u + n b + n f = 1 n u + , n b − ] ⇔ [ n f + ] + [ n f − ] [ n u ] ⇔ [ n b dn f 2 = 0 dt = k ⇒ n b − k ⇐ n f K = k ⇒ 2 k ⇐ = n f n b
L. Onsager, “Deviations from Ohm’s law in The Wien effect weak electrolytes”. J. Chem. Phys. 2, 599,615 (1934) � + + n b + + n f − − n b = n b n f = n f n u + n b + n f = 1 n u + , n b − ] ⇔ [ n f + ] + [ n f − ] [ n u ] ⇔ [ n b ⇒ K 0 = k 0 ⇐ = n b k 0 n u K = k ⇒ 2 k ⇐ = n f n b
L. Onsager, “Deviations from Ohm’s law in The Wien effect weak electrolytes”. J. Chem. Phys. 2, 599,615 (1934) � ⇒ K = k ⇒ 2 k ⇐ = n f K 0 ( E ) = k 0 ⇐ = n b ≈ K 0 (0) ≈ K (0) + O ( E ) k 0 n u n b K ( E ) K (0) = I 2 (2 b ) = 1 + b + O ( b 2 ) l D >> l E , l T for 2 b ∝ q 3 E b = l T T 2 l E Linear in For small field E
L. Onsager, “Deviations from Ohm’s law in The Wien effect weak electrolytes”. J. Chem. Phys. 2, 599,615 (1934) � n = n f + n b n f ( E ) I 2 (2 b ) = 1 + b b ∝ q 3 E n f (0) ≈ 2 + O ( b 2 ) T 2 2 b E Linear in For small field – this is a non-equilibrium effect E
The linear field dependence => A non-equilibrium effect => Compare with Blume-Capel paramagnet. ( ) ∑ ∑ 2 , S i = 0, ± 1 Η = − H S i + Δ S i i 2exp( − β Δ ) n (0) = n ↑ + n ↓ = 1 + 2exp( − β Δ ) n ( H ) = n ↑ ( H ) + n ↓ ( H ) = n (0) 2 (exp( β H ) + exp( − β H )) = n (0) + O ( H 2 ) This scalar quantity changes quadratically with applied field
Lattice Electrolyte - Coulomb gase 1. Electrolyte + + E Hopping on a diamond lattice A grand canonical Coulomb gas. ∑ H ≈ ij ) − µ ˆ U ( r N i > j n = N << k B T Weak electrolyte limit: µ > k B T N 0
Results: Kaiser, Bramwell, PCWH, Moessner, Nature Materials, 12 , 1033-1037, (2013) Onsager’s theory 4 Simulations 3 ∆ n f ( B ) / n f (0) 2 1 0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 E ∗ Lattice Electrolyte Linear in to lowest order E
Linear term is renormalized away by Debye screening: Onsager’s theory 4 Simulations 3 ∆ n f ( B ) / n f (0) 2 1 T * 0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 E ∗ Negative offset Δ n f = − (1 − γ ) E * n f E > D Crossover
Relative conductivity falls below prediction σ = q 2 κ n f κ Theory relies on mobility, being field independent
Field dependent mobility: Blowing away of Debye screening cloud (1 st Wien effect) Velocity max for Metropolis Fuoss-Onsager theory + Metropolis
Spin ice- a magnetic Coulomb gas Spin Ice – a dipolar magnet S i . 3 − 3 ( ij )( S i . S j . ⎡ ⎤ S i . S j r r ij ) ∑ ∑ ⎢ ⎥ H = J S j +D ⎢ ⎥ 5 r r ⎣ ⎦ ij ij ij ij 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 Six equivalent configs for each tetrehedron
Magnetic ice rules => Pauling entropy. 1 2 ln 3 S P = Nk B 2 Magnetic « Giauque and Stout » experiment: Ramirez et al, Nature Glassy behaviour: 399,333, (1999) Schiffer et al, Castelnovo Moessner Sondhi, Cugliandolo et al, Davis et al,
Extension of the point dipoles into magnetic needles/dumbbelles Möller and Moessner PRL. 96, 237202, 2006, Castelnovo, Moessner, Sondhi, Nature, 451, 42, 2008 Needles DSI Gingras et al Configurations of ⇒ magnetic charge at tetrehedron centres a Magnetic ice rules two-in two-out N S m An extensive degeneracy of states satisfy these rules – Monopole vacuum
S. V. Isakov, K. Gregor, R. Moessner, � Ice rules, topological constraints and S. L. Sondhi PRL 93, 167204, 2004 ∇ . M = 0 M= divergence free field M = ∇ ∧ A = M d Emergent gauge field Monopole vacuum has divergence free configurations - « Coulomb phase » Physics . Pinch Points: T. Fennell et. al ., Magnetic Coulomb Phase in the Spin Ice Ho 7 O 2 Ti 2 Science , 326, 415, 2009. Topological sector fluctuations: Jaubert et. al. Phys. Rev. X, 3, 011014, (2013)
Topological excitations back to paramagnetic phase space Castelnovo, Moessner, Sondhi, Nature, 451, 42, 2008 -CMS, Ryzhkin JETP, 101 , 481, 2005. Extensive phase space of topologically constrained states = Vacuum for quasi-particle excitations
Topological constraints Excitations back to paramagnet….
Topological constraints Spin flip creates two defects 3 out- 1 in 3 in 1 out Δ E ≈ 4 J eff
Topological constraints Spin flip creates two defects 3 out- 1 in 3 in 1 out Δ E ≈ 4 J eff Δ E ≈ 0
Castelnovo, Moessner, Sondhi, Nature, 451, 42, 2008 (Ryzhkin JETP, 101 , 481, 2005) Topological defects carry magnetic charge – magnetic monopoles ⇒ U ( r ) = µ 0 ⇒ Q i Q j Δ M = 2 m ; Q i = ± 2 m 4 π r a A grand canonical Coulomb gas of quasi particles. ∑ H ≈ ij ) − µ ˆ U ( r N µ ( J , m , a ) i > j In which case one should expect « electrolyte » physics + constraints - magnetolyte (Castelnovo)
Electrolyte and Magnetolyte Coulomb gases 1. Electrolyte + + E 2. Magnetolyte N N H Chemical potential /particle for Dy 2 Ti 2 O 7 µ 1 = − 4.35 K µ 1 = − 5.7 K /particle for Ho 2 Ti 2 O 7 CMS, Phys. Rev. B 84, 144435, 2011, Melko, Gingras JPCM, 16 (43) R1277–R1319 (2004)
Monopole dynamics polarizes the medium Coulomb gas physics with transient currents Ryzhkin JETP, 101 , 481, 2005. Jaubert and Holdsworth, Nature Physics, 5, 258, 2009 ⎛ ⎞ j = d M = 1 M τ m H − Time ⎜ ⎟ τ m χ T ⎝ ⎠ dt
Wien effect in the magnetolyte: Kaiser et al, to appear in Phys Rev Lett. Magnetolyte Electrolyte Switching on field at t=0 Time scale: 1 MCS = 1 ms for DTO Jaubert and Holdsworth, Nature Phyiscs, 5, 258, 2009
Square AC field – 8.2 secs
Monopole concentration with time 0.0020 0.0015 n ( t ) [ ] 0.0010 0.0005 DTO @ 0.5 K 0.0000 60 0.02 40 B ( t ) [mT] m ( t ) [ ] 20 0.00 0 � 20 � 0.02 � 40 � 60 0.0 0.2 0.4 0.6 0.8 1.0 t [1000 MC steps ' s] 1 τ L 1 τ m
Magnetolyte DTO 0.5 K Electrolyte DTO 0.43 K Onsager’s theory 4 Simulations 3 ∆ n f ( B ) / n f (0) 2 1 0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 E ∗
An experimental signal ? H = H 0 sin( ω t ) χ ( H , ω ) = χ 0 ( ω ) + χ 1 ( ω ) H 2 + ....... In equilibrium χ ( H , ω ) = χ 0 ( ω ) + χ 1 ( ω ) H + ....... Wien contribution χ B ( ω 0 ) χ 0 ( ω 0 ) ≈ n f ( < B > ) n f (0)
Analytic approach –two coupled equations τ 0 ⎛ ⎞ j = d M 1 M τ m ∝ H − ⎜ ⎟ τ m ⎝ χ T ⎠ n f ( H ) dt ⎛ ⎞ d Δ n f ∝ h − m ( t ) − Δ n f 0 1 dn f dt = k ⇒ n b − 1 ⇒ 2 ⎜ ⎟ 2 k ⇐ n f 0 0 ⎝ ⎠ n f dt n f
Deconfined monopole charge via The Wien effect Bramwell et al, Nature, 461, 956, 2009 Muon relaxation = BQ 3 µ 0 δσ ( E ) ⇒ δν ( B ) σ ν 16 π k B 2 T 2 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 ( ) ∇ . H = − ∇ . M = ρ ∇ ∧ M = ∇ ψ + A = M m + M d When ρ = 0, M = M d Monopolar and dipolar parts (largely) decoupled and dynamics is from monopole movement Perfect Coulomb gas within frequency window 1 < ω < 1 τ L τ m
Conclusions 1. The Wien effect is a model non- equilibrium process. 2. The Wien effect emerges from the magnetic Coulomb gas. 3. Spin ice proves to be a perfect, symmetric Coulombic system. 0.0020 0.0015 n ( t ) [ ] 0.0010 0.0005 DTO @ 0.5 K 0.0000 60 0.02 40 B ( t ) [mT] m ( t ) [ ] 20 0.00 0 � 20 � 0.02 � 40 � 60 0.0 0.2 0.4 0.6 0.8 1.0 Franco-Japanese seminar, Kyoto, August 2015 t [1000 MC steps ' s]
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