Magnetic Monopoles in Spin Ice

Size: px

Start display at page:

Download "Magnetic Monopoles in Spin Ice"

Herbert Booker
5 years ago
Views:

1 Magnetic Monopoles in Spin Ice Claudio Castelnovo University of Oxford Roderich Moessner Max Planck Institut Shivaji Sondhi Princeton University Nature 451, 42 (2008) ISIS Seminars, Rutherford Appleton Laboratory, Didcot (UK), September 30, 2008

2 Outline General context Introduction, from frustrated magnetism to spin ice Dipolar interactions and the ice rule Excitations: monopoles = fractionalised dipoles Experimental evidence: I: Searching for monopoles II: The magnetic Coulomb liquid The emergent gauge structure of spin ice Conclusions

3 Outline General context Introduction, from frustrated magnetism to spin ice Dipolar interactions and the ice rule Excitations: monopoles = fractionalised dipoles Experimental evidence: I: Searching for monopoles II: The magnetic Coulomb liquid The emergent gauge structure of spin ice Conclusions

4 Collective phenomena and complexity Complementary fundamental 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

5 Magnetic monopole search Monopole passes through ring magnetic flux through ring changes e.m.f. induced in the ring countercurrent Q D is set up

6 Magnetic monopole search Monopole passes through ring magnetic flux through ring changes e.m.f. induced in the ring countercurrent Q D is set up

7 Magnetic monopole search Monopole passes through ring magnetic flux through ring changes e.m.f. induced in the ring countercurrent Q D is set up Cabrera 1982

8 Outline General context Introduction, from frustrated magnetism to spin ice Dipolar interactions and the ice rule Excitations: monopoles = fractionalised dipoles Experimental evidence: I: Searching for monopoles II: The magnetic Coulomb liquid The emergent gauge structure of spin ice Conclusions

9 Conventional vs frustrated Ising models Consider classical Ising spins, pointing either up or down: σ i = ±1 Simple exchange (strength J): H = J ij σ i σ j J < 0: ferromagnetic spins align J > 0: antiferromagnetic spins antialign... but only where possible: frustration = What happens instead?

10 Frustration leads to (classical) degeneracy Not all terms in H = J ij σ iσ j can simultaneously be minimised But we can rewrite H: ( 4 H = J 2 i=1 σ i ) 2 + const. which can be minimised for a single tetrahedron: i σ i = 0 N gs = ( 4 2 ) = 6 ground states Degeneracy is the hallmark of frustration

11 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

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

13 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

14 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???

15 Outline General context Introduction, from frustrated magnetism to spin ice Dipolar interactions and the ice rule Excitations: monopoles = fractionalised dipoles Experimental evidence: I: Searching for monopoles II: The magnetic Coulomb liquid The emergent gauge structure of spin ice Conclusions

16 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

17 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,

18 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

19 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!

20 Outline General context Introduction, from frustrated magnetism to spin ice Dipolar interactions and the ice rule Excitations: monopoles = fractionalised dipoles Experimental evidence: I: Searching for monopoles II: The magnetic Coulomb liquid The emergent gauge structure of spin ice Conclusions

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

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

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

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

25 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?

26 Deconfined magnetic monopoles The dumbbell Hamiltonian gives E(r) = µ 0 qm 2 4π r

27 Deconfined magnetic monopoles The dumbbell Hamiltonian gives E(r) = µ 0 qm 2 4π r magnetic Coulomb interaction

28 Deconfined magnetic monopoles The dumbbell Hamiltonian gives E(r) = µ 0 qm 2 4π r magnetic Coulomb interaction deconfined monopoles

29 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

Intuitive picture for monopoles Simplest picture does not work: disconnect monopoles Next best thing: no string tension between monopoles: Two

30 Intuitive picture for monopoles Simplest picture does not work: disconnect monopoles Next best thing: no string tension between monopoles: 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

31 Outline General context Introduction, from frustrated magnetism to spin ice Dipolar interactions and the ice rule Excitations: monopoles = fractionalised dipoles Experimental evidence: I: Searching for monopoles II: The magnetic Coulomb liquid The emergent gauge structure of spin ice Conclusions

32 Outline General context Introduction, from frustrated magnetism to spin ice Dipolar interactions and the ice rule Excitations: monopoles = fractionalised dipoles Experimental evidence: I: Searching for monopoles II: The magnetic Coulomb liquid The emergent gauge structure of spin ice Conclusions

33 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

34 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?

35 Outline General context Introduction, from frustrated magnetism to spin ice Dipolar interactions and the ice rule Excitations: monopoles = fractionalised dipoles Experimental evidence: I: Searching for monopoles II: The magnetic Coulomb liquid The emergent gauge structure of spin ice Conclusions

36 Experiment II: interacting Coulomb liquid Monopoles form a two-component Coulomb 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

37 Experiment II: interacting Coulomb liquid Monopoles form a two-component Coulomb 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 details B

38 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 Sakakibara+Maeno confirmed numerically

39 Outline General context Introduction, from frustrated magnetism to spin ice Dipolar interactions and the ice rule Excitations: monopoles = fractionalised dipoles Experimental evidence: I: Searching for monopoles II: The magnetic Coulomb liquid The emergent gauge structure of spin ice Conclusions

40 Conventional order and disorder Gas-crystal (e.g. rock salt): Paramagnet-ferromagnet (e.g. fridge magnet) In between: critical points Anything else???

41 Is spin ice ordered or not? details No order as in ferromagnet deconfined monopoles (in 3d)

42 Is spin ice ordered or not? details No order as in ferromagnet deconfined monopoles (in 3d) Not disordered like a paramagnet ice rules

43 Is spin ice ordered or not? details No order as in ferromagnet deconfined monopoles (in 3d) 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

44 Is spin ice ordered or not? details No order as in ferromagnet deconfined monopoles (in 3d) 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

45 Bow-ties in neutron scattering proton correlations in water ice I h Li et al. spin correlations in kagome ice Fennell+Bramwell

46 Outline General context Introduction, from frustrated magnetism to spin ice Dipolar interactions and the ice rule Excitations: monopoles = fractionalised dipoles Experimental evidence: I: Searching for monopoles II: The magnetic Coulomb liquid The emergent gauge structure of spin ice Conclusions

47 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)

48 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

Picture credits Iceberg: /images/noaa iceberg jpg image.html 3 January 2008 www.nature.

49 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]

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

51 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

of spins Other sublattices form kagome lattice Kagome

52 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

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

53 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

54 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

55 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)

Magnetic Monopoles in Spin Ice

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