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

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1 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

2 The monopole physics of spin ice P.C.W. Holdsworth Ecole Normale Supérieure de Lyon Vojtech Kaiser, Steven Bramwell, Roderich Moessner, Ludovic Jaubert, Adam Harman-Clarke, Marion Brooks-Bartlett, Simon Banks, Michael Faulkner, Laura Bovo, Jonathon Bloxom

3 Ice rules in spin ice A good Hamiltonian for spin ice materials H = J ij S i. S j +D J and D are O(1K) S i. S j 3 r 3( S i. r ij )( S j. r ij ) 5 ij r ij 0-10K => Ising like spins along <111> axes Long range interactions are almost but not quite screened den Hertog and Gingras, PRL.84, 3430 (2000), Isakov, Moessner and Sondhi, PRL 95, , 2005 ij T. Fennell et. al.,science, 326, 415, 2009.

4 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

5 Castelnovo, Moessner, Sondhi, Nature, 451, 42, 2008 Topological defects carry magnetic charge magnetic monopoles ΔM = 2m, 4m U(r) = µ 0 4π Q i Q j r ; Q i = ± 2m a, ± 4m a A grand canonical Coulomb gas of quasi particles. H U(r ij ) µ 1 ˆN1 i> j µ 2 ˆN2 µ(j, m, a) In which case one should expect «electrolyte» physics + constraints - magnetolyte (Castelnovo)

6 Spin Ice phase diagram Dipolar spin ice phase diagram R. G. Melko and M. J. P. Gingras Journal of Physics-Condensed Matter, 16 (43) R1277 R1319 (2004). All in all out AIAO Dipolar scale Antiferro exchange scale D > 0 J < 0 Reducing µ 2J µ n = n D µ 2 = 4µ 1 CMS, Nature, 451, 42, 2008, Brooks-Bartlett et al (Phys. Rev. X, 4, (2014))

7 Electrolyte and Magnetolyte Coulomb gases 1. Electrolyte E Magnetolyte B 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)

8 Electrolyte and Magnetolyte Coulomb gases 1. Electrolyte n(t ) = n + + n = n = 2exp(βµ DH ) 1+ 2exp(βµ DH ) S(0) = 0, S( ) 1.1 Debye Huckle theory, CMS, Phys. Rev. B 84, , Magnetolyte n(t ) = n N + n S = n = 3 exp(βµ DH ) exp(βµ DH ) N S S(0) log 3 2, S( ) log 7 2, ΔS 0.84 Debye Huckle theory, V. Kaiser, J. Bloxom, Ph.D (Ryzhkin JETP, 101, 481, 2005)

9 Heat capacity numerical simulation (Single charge electrolyte 14-vertex magnetolyte DTO) magnetolyte electrolyte c [J K 1 mol 1 ] T [K]

10 Heat capacity simulation and Debye-Huckle theory (Single charge electrolyte 14-vertex magnetolyte DTO) c [J K 1 mol 1 ] magnetolyte DH theory electrolyte DH theory magnetolyte electrolyte T [K]

11 Double charge vertices become important for T> 2-3K Comparison with new specific heat data (Bovo et al unpublished) HTO magnetolyte DH theory 14-v. magnetolyte DH theory 16-v. experiment magnetolyte simulations 14-v. magnetolyte simulations 16-v. c [J K 1 mol 1 ] High density Coulomb fluid at high T T [K]

12 Double charge vertices become important for T> 2-3K Comparison with new specific heat data (Bovo et al unpublished) DTO magnetolyte DH theory 14-v. magnetolyte DH theory 16-v. experiment magnetolyte simulations 14-v. magnetolyte simulations 16-v. c [J K 1 mol 1 ] High density Coulomb fluid at high T T [K]

13 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: AIAO = Crystal of Q i = ± 4m a charges in ZnS (Zinc Blend) structure if U = U C µn 0 < 0 U C = N 0 α u(a) 2 α = Madalung constant for diamond lattice Breaking of a Z 2 symmetry µ * = µ u(a) < α 2 = µ * = 1.4 For DTO

14 Spin Ice phase diagram and Monopole Crystalization Dipolar spin ice phase diagram R. G. Melko and M. J. P. Gingras Journal of Physics-Condensed Matter, 16 (43) R1277 R1319 (2004). All in all out AIAO Dipolar scale D nn > 0 Antiferro exchange J nn < 0 scale J nn D nn = µ 2 = 4 2J D

15 Spin Ice phase diagram and Monopole Crystalization Dipolar spin ice phase diagram R. G. Melko and M. J. P. Gingras Journal of Physics-Condensed Matter, 16 (43) R1277 R1319 (2004). All in all out AIAO Dipolar scale D nn > 0 Antiferro exchange J nn < 0 scale J nn D nn = Double monopole crystalization to AIAO phase at: J nn = 4 D nn α 2 = Excellent agreement With DSI Brooks-Bartlett et al (Phys. Rev. X, 4, (2014))

16 Magnetic moments, charge and lattice gauge field Coulomb fluid satisfies magnetic Gauss law:. B = 0 = µ 0.( M + H ). M M.d S i = Q i. H =. M = ρ ( ) Three in one out defect i

17 Magnetic moments, charge and lattice gauge field Coulomb fluid satisfies magnetic Gauss law:. B = 0 = µ 0.( M + H ). M M.d S i = Q i. H =. M = ρ ( ) Lattice field M ij = M.d S = M ji = ± m a Three in one out vertex i j M ij = m a j M ij [ 1, 1, 1,1 ] = Q i Brooks-Bartlett et al (Phys. Rev. X, 4, (2014)) Maggs & Rossetto, PRL, 88, , Bramwell, Phil. Trans. R. Soc. A 370, 5738 (2012). Q i = 2m a

18 Moment fragmentation (Lattice Helmholtz decomposition) M = ψ + A = M m + M d Two fluids M ij = [ 0 ]+ m a 2 in-2 out M d only [ 1, 1,1,1 ] M ij = m a ( 1, 1, 1,1) = m a 1 2, 1 2, 1 2, m a 1 2, 1 2, 1 2, in 1 out/3 out 1 in M d and M m M ij = m a [ 1, 1, 1, 1]+ [ 0] All in/all out M m only

19 Excluding double charges gives a «simple» monopole crystal Brooks-Bartlett et al, Phys. Rev. X, 4, (2014) M = ψ + A = M m + M d Crystal of 3 in 1 out vertices 3 out 1 in ( 1, 1, 1,1) = 1 2, 1 2, 1 2, , 1 2, 1 2, 3 2 The red spins map onto dimers on a diamond lattice G.-W. Chern et al, PRL 106, (2011). Moller and Moessner, PRB 80, (2009). Divergence free part maps onto the emergent gauge field from the dimers Huse et. al. PRL 91, , 2005 Field E=1 along dimer, 7-1/(z-1) between dimers

20 Excluding double charges gives a «simple» monopole crystal Brooks-Bartlett et al, Phys. Rev. X, 4, (2014) M = ψ + A = M m + M d Crystal of 3 in 1 out vertices 3 out 1 in ( 1, 1, 1,1) = 1 2, 1 2, 1 2, , 1 2, 1 2, 3 2 Could occur here in Spin ice materials Coexisiting AIAO order and Coulomb phase

21 Deconfined monopole charge via Bramwell et al, Nature, 461, 956, 2009 The Wien effect Charges either free or bound in pairs. n = n b + n f Conduction due to free charges. Applying field changes concentration of free charges: δn f (E) n f (0) = b 2 = Eq 3 16πε 0 k B 2 T 2 BQ3 µ 0 16πk B 2 T 2 δσ(e) σ = b 2 L. Onsager, Deviations from Ohm s law in " weak electrolytes. " J. Chem. Phys. 2, 599,615 (1934)"

22 Deconfined monopole charge via Bramwell et al, Nature, 461, 956, 2009 The Wien effect Charges either free or bound in pairs. n = n b + n f Conduction due to free charges. Applying field changes concentration of free charges: 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

23 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 Monopole physics but transient currents only Ryzhkin JETP, 101, 481, Jaubert and Holdsworth, Nature Physics, 5, 258, 2009 Φ

24 The 2 nd 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 = n f + n b

25 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 V eff (r) = q 1 q 2 4πε 0 r exp( r / l D )

26 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 ) for l D >> l E,l T b = l T q3 E l E T 2 Linear in E For small field this is a non-equilibrium effect

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

28 Simulation results for the lattice electrolyte: Kaiser, Bramwell, PCWH, Moessner, Nature Materials, 12, , (2013) 4 Onsager s theory Simulations nf(b)/nf(0) 3 2 n f (0) E T*=0.14 => 0.43K for DTO,

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

30 Linear term is renormalized away by Debye screening: 4 Onsager s theory Simulations nf(b)/nf(0) E Crossover to quadratic behaviour and negative intercept Crossover when E = D See poster V. Kaiser

31 Wien effect in the magnetolyte: Switching on field at t=0 Electrolyte Magnetolyte Time scale: 1 MCS = 1 ms for DTO Jaubert and Holdsworth, Nature Phyiscs, 5, 258, 2009

32 Square AC field 8.2 secs

33 Square AC field 0.13 secs n(t) [] m(t) [] K t [1000 MC steps ' s] B(t) [mt]

34 Average monopole concentration n(t) [] m(t) [] K t [1000 MC steps ' s] B(t) [mt]

35 Magnetolyte DTO 0.5 K

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

37 An experimental signal? In equilibrium χ(b,t ) = χ 0 (T ) + χ 1 (T )B Wien contribution χ(b,t ) = χ 0 (T ) + χ 1 (T ) B +... Sinusoidal ac field χ B (ω 0 ) χ 0 (ω 0 ) n (< f B >) n f (0)

38 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

39 Relative conductivity falls below prediction (E )/ (0) σ = q 2 ωn f Onsager s theory Onsager s theory + lattice Simulations E Theory relies on mobility, ω being field independent

40 Field dependent mobility: Blowing away of Debye screening cloud (1 st Wien effect) Velocity max for Metropolis

41 Magnetic monopole crystalization breaks this symmetry Brooks-Bartlett et al, Phys. Rev. X, 4, (2014) M = ψ + A = M m + M d Crystal of 3 in 1 out vertices 3 out 1 in ( 1, 1, 1,1) = 1 2, 1 2, 1 2, , 1 2, 1 2, 3 2 The red spins map onto dimers on a diamond lattice Divergence free part maps onto the emergent gauge field from the dimers Huse et. al. PRL 91, , 2005 Field E=1 along dimer, 7-1/(z-1) between dimers

42 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)

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44 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

45 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

46 Internal field distributions in dipolar spin ice Needles: B large Dipoles B = 0 B 0 Flux confined to the needle => zero field sites Field spills out Sala et al, Phys. Rev. Lett. 108, , 2012 From 150 mt to 4.5 T

47 Examples: Ion-hole conduction Cation-anion conduction