Magnetic monopoles in spin ice Claudio Castelnovo Oxford University Roderich Moessner MPI-PKS Dresden Shivaji Sondhi Princeton University Nature 451, 42 (2008)
The fundamental question astroparticle physics 18 15 12 9 6 3 0 3 6 9 scale (metres) cosmology astrophysics our world nuclear physics? string theory
The fundamental question scale (metres) astroparticle physics 18 15 12 9 6 3 0 3 6 9 cosmology astrophysics our world nuclear physics emergent phenomena many body physics complexity? string theory
Collective phenomena and complexity Complementary questions: What are the fundamental building blocks of matter, and how do they interact? high energy+particle physics Given building blocks and interactions: what is the resulting collective behaviour? many-body physics and complexity
Outline frustrated Ising models and the (spin) ice model the spin ice compounds zero-point entropy long-range (dipolar) interactions survival of the ground-state degeneracy excitations: magnetic monopoles and their properties Is spin ice ordered?
Conventional vs frustrated Ising models Consider classical Ising spins, pointing either up or down: σ i = ±1 Simple exchange (strength J): H = Jσ i σ j J < 0: ferromagnetic spins align J > 0: antiferromagnetic spins antialign... but only where possible: frustration = What happens instead??
Frustration leads to (classical) degeneracy Not all terms in H = ij σ iσ j can simultaneously be minimised But we can rewrite H:? H = J 2 ( q i=1 σ i ) 2 + const which can be minimised for tetrahedron: i σ i = 0 N gs = ( 4 ) = 6 ground states 2 Degeneracy is hallmark of frustration
Zero-point entropy on the pyrochlore lattice Pyrochlore lattice = corner-sharing tetrahedra H pyro = J 2 ( σ i ) 2 tet i tet Pauling estimate of ground state entropy S 0 = ln N gs : ( ) N/2 6 N gs = 2 N S 0 = 1 16 2 ln 3 2 microstates vs. constraints; N spins, N/2 tetrahedra
Pauling entropy in spin ice Anderson 1956; Harris+Bramwell 1997 Ho 2 Ti 2 O 7 (and Dy 2 Ti 2 O 7 ) are pyrochlore Ising magnets Pauling entropy measured Ramirez as predicted
Mapping from ice to spin ice In ice, water molecules retain their identity Hydrogen near oxygen spin pointing in 150.69.54.33/takagi/matuhirasan/SpinIce.jpg axes non-collinear everything seems to hang together
The real (dipolar) Hamiltonian of spin ice Siddharthan+Shastr The nearest-neighbour model H nn for spin ice is not correct Leading term is dipolar energy (µ 0 µ 2 /4πa 3 > J): H = H nn + µ 0 4π ij µ i µ j 3( µ i ˆr ij )( µ i ˆr ij ) r 3 ij Both give same entropy (!!!) Gingras et al. Wrong model right answer... WHY???
The dumbell model Dipole pair of opposite charges (µ = qa): Sum over dipoles sum over charges: +q H ij = 2 v(r mn ij ) µ = q a m,n=1 v q 2 /r is the usual Coulomb interaction (regularised): v(r mn ij ) = { µ0 q m i qn j /(4πrmn ij ) i j v o ( µ a )2 = J 3 + 4D 3 (1 + 2 3 ) i = j,
Origin of the ice rules Choose a = a d, separation between centres of tetrahedra Resum tetrahedral charges Q α = r m i α qm i : H mn ij v(r ij,mn ) αβ V (r αβ ) = { µ0 Q α Q β r αβ α β 4π 1 v 2 oq 2 α α = β Ice configurations (Q α 0) degenerate Pauling entropy!
Excitations: dipoles or charges? Ground-state no net charge Excited states: flipped spin dipole excitation same as two charges? one dipole Q=0 two charges
Excitations: dipoles or charges? Ground-state no net charge Excited states: flipped spin dipole excitation same as two charges? Q=0 one dipole two charges Fractionalisation in d = 1
Excitations in spin ice: dipolar or charged? Single spin-flip (dipole µ) two charged tetrahedra (charges q m = 2µ/a d ) Are charges independent? Fractionalisation in d = 3?
Deconfined magnetic monopoles Dumbell Hamiltonian gives E(r) = µ 0 4π q 2 m r magnetic Coulomb interaction
Deconfined magnetic monopoles Dumbell Hamiltonian gives E(r) = µ 0 4π q 2 m r magnetic Coulomb interaction deconfined monopoles
Deconfined magnetic monopoles Dumbell Hamiltonian gives E(r) = µ 0 4π q 2 m r magnetic Coulomb interaction deconfined monopoles charge q m = 2µ/a = (2µ/µ b )(αλ C /2πa d )q D q D /8000 monopoles in H, not B
Intuitive picture for monopoles Simplest picture does not work: disconnect monopoles N S N Next best thing: no string tension between monopoles: S N S N S Two monopoles form a dipole: connected by tensionless Dirac string Dirac string is observable q m q D /8000 not in conflict with quantisation of e
Experiment I: Stanford monopole search Monopole passes through superconducting ring magnetic flux through ring changes e.m.f. induced in the ring countercurrent q m is set up Works for both fundamental cosmic and spin ice monopoles signal-noise ratio a problem
Experiment I: Stanford monopole search Monopole passes through superconducting ring magnetic flux through ring changes e.m.f. induced in the ring countercurrent q m is set up Works for both fundamental cosmic and spin ice monopoles signal-noise ratio a problem How do we know if a particle is elementary?
Experiment II: interacting Coulomb liquid Monopoles form a two-component liquid any characteristic collective behaviour? interaction strength Γ (q 2 m / r )/T exp[ cv 0/T]/T vanishes at both high and low T
Experiment II: interacting Coulomb liquid Monopoles form a two-component liquid any characteristic collective behaviour? interaction strength Γ (q 2 m / r )/T exp[ cv 0/T]/T vanishes at both high and low T solution: [111] magnetic field acts as chemical potential can tune r and T separately B
Liquid-gas transition in spin ice in a [111] field H nn predicts crossover to maximally polarised state dipolar H: first-order transition with critical endpoint Fisher et al. observed experimentally Hiroi+Maeno groups confirmed numerically
Kagome ice: dimensional reduction in a field Ising axes are not collinear [111] field pins one sublattice of spins B
Kagome ice: dimensional reduction in a field Ising axes are not collinear [111] field pins one sublattice of spins B Other sublattices form kagome lattice
Kagome ice: dimensional reduction in a field Ising axes are not collinear [111] field pins one sublattice of spins Other sublattices form kagome lattice Kagome lattice: two-dimensional How many dimensions are there? B
Conventional order and disorder Gas-crystal (e.g. rock salt): Paramagnet-ferromagnet (e.g. fridge magnet) In between: critical points Anything else???
Is spin ice ordered or not? No order as in ferromagnet deconfined monopoles
Is spin ice ordered or not? No order as in ferromagnet deconfined monopoles Not disordered like a paramagnet ice rules
Is spin ice ordered or not? No order as in ferromagnet deconfined monopoles Not disordered like a paramagnet ice rules conservation law Consider magnetic moments µ i as (lattice) flux vector field Ice rules µ = 0 = µ = A
Is spin ice ordered or not? No order as in ferromagnet deconfined monopoles Not disordered like a paramagnet ice rules conservation law Consider magnetic moments µ i as (lattice) flux vector field Ice rules µ = 0 = µ = A Local constraint emergent gauge structure Bow-tie motif in neutron scattering Algebraic (but not critical!) correlations
Bow-ties in neutron scattering proton correlations in water ice I h Li et al. spin correlations in kagome ice Fennell+Bramwell
Emergent particles and new order in spin ice Spin ice is an interesting model system (and material!) frustrated magnet with ground-state entropy dimensional reduction in a field; emergent gauge structure
Emergent particles and new order in spin ice Spin ice is an interesting model system (and material!) frustrated magnet with ground-state entropy dimensional reduction in a field; emergent gauge structure Magnetic monopoles as excitations magnetic Coulomb law (felt by external test particle) fractionalisation/deconfinement in 3d material would show up in monopole search: q m q D /8000
Thanks Claudio Castelnovo Oxford John Chalker Oxford Karol Gregor Caltech Peter Holdsworth ENS Lyon Sergei Isakov ETH Zürich Ludovic Jaubert ENS Lyon Kumar Raman UC Riverside Shivaji Sondhi Princeton Alessandro Canossa
Picture credits Iceberg: www.windows.ucar.edu/tour/link=/earth/polar/images/noaa_iceberg_jpg_image.html Levitation: math.ucr.edu/home/baez/physics/general/levitation/levitation.html Field lines: www.mcatpearls.com/master/img911.png NaCl: http://www.greenfacts.org/images/glossary/crystal-lattice.jpg