Geometry, topology and frustration: the physics of spin ice

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1 Geometry, topology and frustration: the physics of spin ice Roderich Moessner CNRS and LPT-ENS 9 March 25, Magdeburg

2 Overview Spin ice: experimental discovery and basic model Spin ice in a field dimensional reduction to kagome ice : magnetisation plateau: entropy, correlations tilted field: Kasteleyn transition termination of plateau: entropy peak Conservation law and gauge theory: algebraic dipolar correlations exact N = solution Why spin ice obeys the ice rules Conclusions and outlook Collaborators: K. Gregor, S. V. Isakov, K. Raman, S. L. Sondhi

3 Spin ice Experimental discovery of a frustrated pyrochlore Ising ferromagnet by Harris et al.: Ho 2 Ti 2 O 7, followed by Dy 2 Ti 2 O 7. Missing entropy (Ramirez et al.) at low T is precisely the Pauling entropy for cubic ice I c, S Pauling = (1/2) ln(3/2):

4 Crystal structure of spin ice Spins reside on pyrochlore lattice Pyrochlore lattice consists of corner-sharing tetrahedra ll sites and bonds are symmetry-equivalent not a Bravais lattice

5 Spin ice, a frustrated ferromagnet (Θ W > ) Ising spins are forced to point along local [111] axes there are four sublattices, κ anisotropy generates Ising (a) (b) pseudospins: S i = σ iˆd κ(i) J changes sign exchange favours two-in two-out states (Bernal- Fowler ice rules): i σ i = H = J S i S j E ) 2 (ˆd κ(i) S i = (J/3) ij i <ij> σ i σ j

6 The Pauling entropy n anisotropic pyrochlore ferromagnet leads to a pseudospin antiferromagnet Pyrochlore antiferromagnets are highly frustrated. Frustration leads to a large ground-state degeneracy. Degeneracy of single tetrahedron: ground-state configurations: n g = ( 4 2) = 6 (ice rules) fraction of ground-state configurations: f = n g /4 2 = 3/8 Pauling estimate of entropy for full lattice: total number of ground states: 2 n s f n t = 2 n s f n s/2 ground state entropy per spin S = 1 2 ln 3 2 Connection between ice and pyrochlores due to nderson

7 Spin ice in a field Ising spins are forced to point along local [111] axes there are four sublattices, κ anisotropy generates Ising pseudospins: S i = σ iˆd κ(i) J changes sign B becomes staggered H = J ij S i S j E i (a) (b) (c) [1] [11] [111] (d) <1> <111> <11> ) 2 (ˆd κ(i) S i gµb J B S i i = (J/3) <ij> σ i σ j gµ B J i B ˆd κ(i) σ i

8 Magnetisation plateau in a [111] field exchange favours two-in two-out states [111] field has projection of 1 on one sublattice, 1/3 on other three favours three-in one-out state low field we get a magnetisation plateau with 2/3 of maximal magnetisation (predicted Bramwell et al., measured Hiroi et al.) Low-field plateau has a non-vanishing entropy of.96k B /Dy or.78k B /Dy. M (µ B /Dy) (b) (a) H // [111].48 K.99 K 1.65 K H (koe) Dy 2 Ti 2 O 7

9 From pyrochlore to kagome Pyrochlore lattice in [111] direction: One sublattice forms triangular planes Three other sublattices form kagome planes Kagome and triangular planes alternate

10 The plateau regime as a kagome magnet One sublattice pinned by field; triangular planes fully polarised Kagome [111] planes now decoupled (other three sublattices): each triangle has two up and one down spin. y This is equivalent to kagome Ising in a field = hexagonal dimer model =triangular height model. x Plateau is exactly soluble, for entropy S =.8k B /Dy, and for correlations: structure factor with logarithmic peaks. Dimensional reduction in bulk!

11 Tilting the field out of the plateau regime Equivalence between kagome sites is lost Logarithmic peaks in structure factor wander inwards until commensurate incommensurate transition with anisotropic scaling. This is Kasteleyn transition, mixed first/second order character. Plot: 1/ξ x and 1/ξ y and m vs. tilt of field

12 Termination of pleateau: entropy spike.3 S t high field, monomers (tetrahedra in 3-in 1-out state) proliferate. This behaviour is captured by monomer-dimer model, with entropy peak at transition larger than zero-field entropy. In experiment, first order transition takes over at low T oki et al. S per pyrochlore atom B B c T fugacity

13 Back to zero field: correlations Longer-range correlations are present in pyrochlores: Pyrochlore structure factor in [hhk] plane has sharp features ( bowties ). Zinkin et al. These are (apparently) unimportant for thermodynamics. They are due to the ground state constraint, L =, for each tetrahedron. This constraint gives rise to a local conservation law, which can be resolved by a gauge theory. Youngblood et al., Henley, Huse et al., Hermele et al.

14 Gauge theory for the pyrochlore ground states e e 4 2 z x e 1 e 3 same as for six-vertex in d = 2: y Turn the Ising spins, S i, into oriented link vectors, B i = S i ê i, on (bipartite!) diamond lattice. Ground-state constraint becomes conservation law gauge theory. L i = B = = B = For continuous spins (XY, Heisenberg models), define a separate gauge field for each spin components.

15 Long-wavelength analysis: coarse-graining Vector potential takes care of constraint Flippable loops have zero average flux: low average flux many microstates nsatz: upon coarse-graining, obtain energy functional of entropic origin: Z = D exp[s cl ]; S cl = K ( ) 2 2 Resulting correlators are transverse and algebraic: e.g. B z (q)b z ( q) q 2 /q 2 (3 cos 2 θ 1)/r 3.

16 Bow-ties in nature inelastic neutrons on YSc 2 Ballou et al. neutrons on water ice I h Li et al.

17 Large-N treatment compared to finite N Strategy: Gauge ansatz works for any number of spin components. Therefore, consider classical O(N) model, N =. O( ) model is soluble Canals+Garanin no free parameter. S [hhk] = 32( cos( q x ) 4 cos(q z )) 2 4 sin( q x 4 )2 5 cos(q x ) 4 cos( q x ) 2 cos(q z ) 2 Ising, T = Heisenberg Zinkin et al. N = Canals+Garanin

18 Pyrochlore real space correlations Multiplied by r 3 : lines large-n; dots Monte Carlo different directions, sublattices, and system sizes Ising Heisenberg, T = J/2

19 Dipolar spin ice: the real Hamiltonian Real interactions, D, are mainly dipolar: long-ranged! D ij = S i S j 3(S i ˆr ij )(S i ˆr ij ) r 3 ij Phenomenology well understood Gingras et al., Siddharthan et al. Question: Why is Pauling entropy (from nearest-neighbour ice model) measured in dipolar spin ice??? nswer: T = correlations of n.n. model are also dipolar: G ij lim T (H/T + λ1) 1 ij D ij for r ij Dipolar D and n.n. H are (approximately) diagonalised by the same unitary transformation. Spectrum (in mean-field theory) very similar

20 Spin ice vs. dipolar spin ice: D = G + nn + fn 2Π 2Π 2Π 2Π Π 2Π 2Π 1 2Π 2Π 2Π spectra of H G + nn G H 1 T= G + nn 2Π 2Π + fn Π 2Π 2Π 2Π

21 Why spin ice obeys the ice rules t zero temperature, only the zero-energy modes play a role. These form a flat band for the nearest-neighbour model. The same flat band is found for dipolar interactions. t zero (but not at finite T ), the physics is the same projective equivalence Dipolar spins are ice because ice is dipolar

22 Summary Spin ice, a ferromagnet, is highly frustrated (geometry) Dimensional reduction in applied field extended string defects restore higher dimensionality in weak field (topology) Kagome ice with non-trivial thermodynamics and correlations Topological Kasteleyn transition in tilted field Plateau termination: entropy peak (monomer-dimer model) Large-N approach applicable to dipolar interactions dipolar spins are ice because ice is dipolar Gauge structure can be generalised to quantum problems: artificial light

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