Magnetic Monopoles in Spin Ice

Transcription

1 Magnetic Monopoles in Spin Ice Claudio Castelnovo University of Oxford Roderich Moessner Max Planck Institut Shivaji Sondhi Princeton University Nature 451, 42 (2008) International Conference on Highly Frustrated Magnetism, Braunschweig (DE), September 8, 2008

2 Outline the spin-ice model low temperature behaviour: from spins to monopoles experimental evidence of deconfined monopolar excitations conclusions

3 Spin Ice in a nutshell Ising spins: σ i = ±1 H = J ij σ iσ j (J > 0) ( ) 6 N/2 N gs = 2 N S 0 = N 16 2 ln but the interactions are long range (dipolar)!

4 The dumbbell model (1) Dipole pair of opposite charges (µ = qa): Sum over dipoles sum over charges: H = 2N dip. i,j=1 v(r ij ) = 2N dip. i,j=1 µ 0 4π q i q j r ij

5 The dumbbell model (2) Choose a = a d, separation between centres of tetrahedra v q 2 /r is the usual Coulomb interaction (regularised): v(r ij ) = µ 0 q i q j 4π r ij r ij 0 [ ] ±v o ( µ a )2 J = ± D 3 ( ) r ij = 0,

6 Origin of the ice rules Resum tetrahedral charges Q α = i α q i: H ij v(r ij ) αβ V (r αβ ) = { µ0 4π Q αq β r αβ α β 1 2 v oqα 2 α = β

7 Origin of the ice rules Resum tetrahedral charges Q α = i α q i: H ij v(r ij ) αβ V (r αβ ) = { µ0 4π Q αq β r αβ α β 1 2 v oqα 2 α = β Ice configurations (Q α 0) degenerate Pauling entropy!

8 Excitations: dipoles or charges? Ground-state no net charge Excited states: flipped spin dipole excitation same as two charges?

9 Excitations: dipoles or charges? Ground-state no net charge Excited states: flipped spin dipole excitation same as two charges?

10 Excitations: dipoles or charges? Ground-state no net charge Excited states: flipped spin dipole excitation same as two charges?

11 Excitations: dipoles or charges? Ground-state no net charge Excited states: flipped spin dipole excitation same as two charges? Fractionalisation in d = 1

12 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?

13 Deconfined magnetic monopoles The dumbbell Hamiltonian gives E(r) = µ 0 qm 2 4π r magnetic Coulomb interaction deconfined monopoles monopoles in H, not B charge q m = 2µ/a d = (2µ/µ B )(αλ C /2πa d )q D q D /8000

14 Experiment I: Stanford monopole search Monopole passes through ring magnetic flux through ring changes e.m.f. induced in the ring countercurrent q m is set up

15 Experiment I: Stanford monopole search Monopole passes through 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?

16 Experiment II: interacting Coulomb liquid Monopoles form a two-component Coulomb liquid [111] magnetic field acts as staggered chemical potential B = we can tune ρ monopole and T separately details

17 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

18 Emergent particles and new order in spin ice Spin ice is an interesting model system (and material!) I frustrated magnet with ground-state entropy I (emergent gauge structure; dimensional reduction in a field) 3 January I magnetic Coulomb law (felt by external test particle) I would show up in monopole search NATUREJOBS New Year s resolutions no.7174 fractionalisation / deconfinement in 3d material GEOPOLITICS Turf wars on the ocean bed ARCTIC CLIMATE Warming with altitude CANCER SUPPRESSION The Down s syndrome link I 451, January 2008 Magnetic monopoles as excitations THE INTERNATIONAL WEEKLY JOURNAL OF SCIENCE POLES APART A magnetic north south divide in spin ice 3.1 cover UK /12/07 4:31:29 pm [artwork by A. Canossa]

19 Picture credits Iceberg: /images/noaa iceberg jpg image.html 3 January GEOPOLITICS Turf wars on the ocean bed ARCTIC CLIMATE Warming with altitude CANCER SUPPRESSION The Down s syndrome link Levitation: math.ucr.edu/home/baez/physics/general /Levitation/levitation.html THE INTERNATIONAL WEEKLY JOURNAL OF SCIENCE 451, January 2008 windows.ucar.edu/tour/link=/earth/polar NATUREJOBS New Year s resolutions Field lines: no.7174 mcatpearls.com/master/img911.png POLES APART A magnetic north south divide in spin ice 3.1 cover UK 1 NaCl: greenfacts.org/images/glossary/crystallattice.jpg /12/07 4:31:29 pm [artwork by Alessandro Canossa]

20 Kagome ice: dimensional reduction in a field Ising axes are not collinear back [111] field pins one sublattice of spins B

21 Kagome ice: dimensional reduction in a field Ising axes are not collinear back [111] field pins one sublattice of spins Other sublattices form kagome lattice B

22 Kagome ice: dimensional reduction in a field Ising axes are not collinear back [111] field pins one sublattice of spins Other sublattices form kagome lattice Kagome lattice: two-dimensional How many dimensions are there? B

23 Emergent gauge structure back Ground states differ by reversing spins around closed loops, for which the average µ = 0 Upon coarse-graining: low average µ preferred E ( A) 2 artificial magnetostatics Ansatz: upon coarse-graining, obtain energy functional of entropic origin: Z = DA exp[s cl ], S cl = K ( A) 2 2 The resulting correlators are transverse and algebraic: ( 3 cos 2 θ 1 ) q2 q 2 r 3

24 Energy scale hierarchy in spin ice materials (Dy, Ho magnetic moment 10µ B ) back Energy scales: crystal field in the local [111] direction 200 K

25 Energy scale hierarchy in spin ice materials (Dy, Ho magnetic moment 10µ B ) back Energy scales: crystal field in the local [111] direction 200 K exchange interaction 1 2 K dipolar interaction 2.5 K (at nn distance)