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) 25 th International Conference on Low Temperature Physics, Amsterdam (NL), August 11, 2008

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

3 Conventional vs frustrated (Ising) models Consider classical Ising spins, σ i = ±1 with exchange interaction: H = J ij σ iσ j J < 0: all the spins align ferromagnetically

4 Conventional vs frustrated (Ising) models Consider classical Ising spins, σ i = ±1 with exchange interaction: H = J ij σ iσ j J < 0: all the spins align ferromagnetically J > 0: antiferromagnetic order is frustrated ( 4 ) 2 H = J σ i + const. 2 i=1 for a single tetrahedron: i σ i = 0 N gs = ( 4 2 ) = 6 ground states

5 Conventional vs frustrated (Ising) models Consider classical Ising spins, σ i = ±1 with exchange interaction: H = J ij σ iσ j J < 0: all the spins align ferromagnetically J > 0: antiferromagnetic order is frustrated ( 4 ) 2 H = J σ i + const. 2 i=1 Degeneracy is the hallmark of frustration

6 Zero-point entropy on the pyrochlore lattice Pyrochlore lattice = corner-sharing tetrahedra ( ) 2 H pyro = J σ i 2 tet i tet Pauling estimate of ground state entropy S 0 = ln N gs : N gs = 2 N ( 6 16 ) N/2 S 0 = N 2 ln 3 2 microstates vs. constraints; N spins, N/2 tetrahedra

7 Mapping from ice to spin ice In ice, water molecules retain their identity Hydrogen near oxygen spin pointing in /takagi/matuhirasan/SpinIce.jpg

8 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 by Ramirez as predicted

9 The real (dipolar) Hamiltonian of spin ice Siddharthan+Shastry 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. details Wrong model right answer... WHY???

10 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

11 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,

12 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

13 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!

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

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

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

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

18 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?

19 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

20 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

21 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?

22 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

23 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

24 Emergent particles and new order in spin ice Spin ice is an interesting model system (and material!) frustrated magnet with ground-state entropy (emergent gauge structure; dimensional reduction in a field) Magnetic monopoles as excitations fractionalisation / deconfinement in 3d material magnetic Coulomb law (felt by external test particle) would show up in monopole search

25 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]

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

27 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

28 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

29 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

30 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

31 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)