The AC Wien effect: non-linear non-equilibrium susceptibility of spin ice. P.C.W. Holdsworth Ecole Normale Supérieure de Lyon

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1 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,

2 The 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 E n = n f + n b

3 Ion-hole conduction

4 Length scales Three length scales appear naturally: The Bjerrum length : q 2 l T = 8πε 0 k B T Particles separated by r < l T are bound Field drift length: l E = k B T qe Debye screening length l D = 2πε 0 k B Ta3 q 2 n f 1/ 2

5 Lattice Coulomb gas: n f n u Three species - bound particles, - free particles - unoccupied sites n b = n b + + n b n f = n f + + n f n u

6 The Wien effect L. Onsager, Deviations from Ohm s law in weak electrolytes. J. Chem. Phys. 2, 599,615 (1934)! n b = n b + + n b n f = n f + + n f n u n u + n b + n f = 1 [n u ] [n b +,n b ] [n f + ] + [n f ] dn f dt = k n b k n f 2 = 0 K = k k = n 2 f n b

7 The Wien effect L. Onsager, Deviations from Ohm s law in weak electrolytes. J. Chem. Phys. 2, 599,615 (1934)! n b = n b + + n b n f = n f + + n f n u n u + n b + n f = 1 [n u ] [n b +,n b ] [n f + ] + [n f ] K 0 = k 0 k = n b 0 n u K = k k = n 2 f n b

8 The Wien effect L. Onsager, Deviations from Ohm s law in weak electrolytes. J. Chem. Phys. 2, 599,615 (1934)! K 0 (E) = k 0 k = n b K 0 (0) 0 n u K = k k = n 2 f K(0) + O(E) n b K(E) K(0) = I 2 (2 b) 2b = 1+ b + O(b 2 ) for l D >> l E,l T b = l T q3 E l E T 2 Linear in E For small field

9 The Wien effect L. Onsager, Deviations from Ohm s law in weak electrolytes. J. Chem. Phys. 2, 599,615 (1934)! n = n f + n b n f (E) n f (0) I 2 (2 b) 2b = 1+ b 2 + O(b2 ) b q3 E T 2 E Linear in E For small field this is a non-equilibrium effect

10 The linear field dependence => A non-equilibrium effect => Compare with Blume-Capel paramagnet. Η = H S i + Δ ( S ) 2 i, S i = 0, ±1 i n(0) = n + n = 2exp( βδ) 1+ 2exp( βδ) n(h ) = n (H ) + n (H ) = n(0) 2 = n(0) + O(H 2 ) (exp(βh ) + exp( βh )) This scalar quantity changes quadratically with applied field

11 Lattice Electrolyte - Coulomb gase 1. Electrolyte E + + Hopping on a diamond lattice A grand canonical Coulomb gas. H i> j U(r ij ) µ ˆN Weak electrolyte limit: µ > k B T n = N N 0 << k B T

12 Results: Kaiser, Bramwell, PCWH, Moessner, Nature Materials, 12, , (2013) 4 Onsager s theory Simulations nf(b)/nf(0) E Lattice Electrolyte Linear in E to lowest order

Linear term is renormalized away by Debye screening: 4 Onsager s theory Simulations nf(b)/nf(0) 3

13 Linear term is renormalized away by Debye screening: 4 Onsager s theory Simulations nf(b)/nf(0) E T * Negative offset Δn f = (1 γ ) n f E * Crossover E > D

14 Relative conductivity falls below prediction σ = q 2 κ n f Theory relies on mobility, κ being field independent

15 Field dependent mobility: Blowing away of Debye screening cloud (1 st Wien effect) Velocity max for Metropolis Fuoss-Onsager theory + Metropolis

16 Spin ice- a magnetic Coulomb gas Spin Ice a dipolar magnet H = J ij S i. S j +D ij S i. S j 3 r 3( S i. r ij )( S j. r ij ) 5 ij r ij Long range interactions are almost but not quite screened den Hertog and Gingras, PRL.84, 3430 (2000), Isakov, Moessner and Sondhi, PRL 95, , 2005 Six equivalent configs for each tetrehedron

17 Magnetic ice rules => Pauling entropy. S P = Nk B 1 2 ln 3 2 Magnetic «Giauque and Stout» experiment: Ramirez et al, Nature 399,333, (1999) Glassy behaviour: Schiffer et al, Castelnovo Moessner Sondhi, Cugliandolo et al, Davis et al,

18 Extension of the point dipoles into magnetic needles/dumbbelles Möller and Moessner PRL. 96, , 2006, Castelnovo, Moessner, Sondhi, Nature, 451, 42, 2008 Gingras et al DSI Needles Configurations of magnetic charge at tetrehedron centres S a N Magnetic ice rules two-in two-out m An extensive degeneracy of states satisfy these rules Monopole vacuum

Ice rules, topological constraints M= divergence free field S. V. Isakov, K. Gregor, R. Moessner,! and S. L. Sondhi PRL 93, 167204, 2004.

19 Ice rules, topological constraints M= divergence free field S. V. Isakov, K. Gregor, R. Moessner,! and S. L. Sondhi PRL 93, , M = 0 Emergent gauge field M = A = M d 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, Topological sector fluctuations: Jaubert et. al. Phys. Rev. X, 3, , (2013)

20 Topological excitations back to paramagnetic phase space Castelnovo, Moessner, Sondhi, Nature, 451, 42, CMS, Ryzhkin JETP, 101, 481, Extensive phase space of topologically constrained states = Vacuum for quasi-particle excitations

21 Topological constraints Excitations back to paramagnet.

22 Topological constraints Spin flip creates two defects 3 out- 1 in 3 in 1 out ΔE 4J eff

23 Topological constraints Spin flip creates two defects 3 out- 1 in 3 in 1 out ΔE 4J eff ΔE 0

24 Castelnovo, Moessner, Sondhi, Nature, 451, 42, 2008 (Ryzhkin JETP, 101, 481, 2005) Topological defects carry magnetic charge magnetic monopoles ΔM = 2m U(r) = µ 0 4π Q i Q j r ; Q i = ± 2m a A grand canonical Coulomb gas of quasi particles. H i> j U(r ij ) µ ˆN µ(j, m, a) In which case one should expect «electrolyte» physics + constraints - magnetolyte (Castelnovo)

25 Electrolyte and Magnetolyte Coulomb gases 1. Electrolyte E Magnetolyte H N N Chemical potential µ 1 = 4.35K /particle for Dy 2 Ti 2 O 7 µ 1 = 5.7K /particle for Ho 2 Ti 2 O 7 CMS, Phys. Rev. B 84, , 2011, Melko, Gingras JPCM, 16 (43) R1277 R1319 (2004)

26 Monopole dynamics polarizes the medium Coulomb gas physics with transient currents Ryzhkin JETP, 101, 481, Jaubert and Holdsworth, Nature Physics, 5, 258, 2009 Time τ m j = d M dt = 1 τ m H M χ T

27 Wien effect in the magnetolyte: Kaiser et al, to appear in Phys Rev Lett. Switching on field at t=0 Magnetolyte Electrolyte Time scale: 1 MCS = 1 ms for DTO Jaubert and Holdsworth, Nature Phyiscs, 5, 258, 2009

28 Square AC field 8.2 secs

29 Monopole concentration with time n(t) [] m(t) [] K t [1000 MC steps ' s] B(t) [mt] 1 1 τ L τ m

30 Magnetolyte DTO 0.5 K Electrolyte DTO 0.43 K 4 Onsager s theory Simulations nf(b)/nf(0) E

31 An experimental signal? H = H 0 sin(ωt) In equilibrium χ(h,ω ) = χ 0 (ω ) + χ 1 (ω )H Wien contribution χ(h,ω ) = χ 0 (ω ) + χ 1 (ω ) H +... χ B (ω 0 ) χ 0 (ω 0 ) n (< f B >) n f (0)

32 Analytic approach two coupled equations j = d M dt 1 τ m H M χ T τ m τ 0 n f (H ) dn f dt = k n b 1 2 k n f 2 1 n f 0 dδn f dt h m(t) Δn 0 f 0 n f

33 Deconfined monopole charge via Bramwell et al, Nature, 461, 956, 2009 The Wien effect Muon relaxation δσ(e) σ δν(b) ν = BQ3 µ 0 16πk B 2 T 2 Highly controversial! Dunsiger et al, Phys Rev. Lett, 107, , 2011 Sala et al, Phys. Rev. Lett. 108, , 2012 Blundell, Phys. Rev. Lett. 108, , 2012

34 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 τ L < ω < 1 τ m

Conclusions 1. The Wien effect is a model nonequilibrium process. 2. The Wien effect emerges from the magnetic Coulomb gas. 3. Spin ice proves to be a perfect, symmetric Coulombic system. 0.

35 Conclusions 1. The Wien effect is a model nonequilibrium process. 2. The Wien effect emerges from the magnetic Coulomb gas. 3. Spin ice proves to be a perfect, symmetric Coulombic system n(t) [] Franco-Japanese seminar, Kyoto, August 2015 m(t) [] 0.5 K t [1000 MC steps ' s] B(t) [mt]

The monopole physics of spin ice. P.C.W. Holdsworth Ecole Normale Supérieure de Lyon

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