GEOMETRICLLY FRUSTRTED MGNETS John Chalker Physics Department, Oxford University
Outline How are geometrically frustrated magnets special? What they are not Evading long range order Degeneracy and fluctuations - models and experiment Statistical physics of underconstrained systems Emergent degrees of freedom and classical fractionalization
Condensed matter at low temperature Crystalline solids Symmetry breaking as the norm
Broken symmetry in Bose liquids 4 He phase diagram Quantum fluctuations suppress crystalisation
Ordering in ferromagnets High temperature Susceptibility χ T T c Ground state
Unfrustrated antiferromagnetic order Néel order Neutron diffraction Inverse susceptibility χ 1 T θ CW T N Shull and Smart (1949)
lternative to symmetry breaking # 1... a unique ground state In the Fermi gas n(p) low T high T p Momentum distribution
Spin system with unique ground state J Weakly coupled singlet pairs χ vs T SrCu 2 (BO 3 ) 2
lternative to symmetry breaking # 2... strong fluctuations Frustration and degeneracy? nderson 1956, Villain 1979
ntiferromagnetic spin clusters - frustration and degeneracy Ising triangle Heisenberg tetrahedron 4 3? φ 1 2 a Ground states: cluster spin L i S i minimised H = J pairss i S j J 2 L 2 + c
Examples of frustrated lattices Building block: corner-sharing frustrated units 2D: kagome lattice 3D: pyrochlore lattice
Frustrated lattices beyond physics...
Characteristics of frustrated magnets SrGa 12 x Cr x O 19 (SCGO) as an example Paramagnetic even for T Θ CW Strong short-range correlations χ 1 vs T Martinez et al, PRB 46, 10786 (1992) Elastic neutron scattering S.H. Lee et al, Europhys Lett 35, 127 (1996)
Selected examples of frustrated magnets Layered materials Pyrochlore antiferromagnets SCGO pyrochlore slabs Cr 3+ S = 3/2 Θ CW 500K T F 4K hydromium iron jarosite kagome layers Fe 3+ S = 5/2 Θ CW 700K T F 14K Herbertsmithite kagome layers Cu 2+ S = 1/2 Θ CW 300K Y 2 Mo 2 O 7 Mo 4+ S = 1 Θ CW 200K T F 22K Cs Ni Cr F 6 Ni 2+ S = 1 Cr 3+ S = 3/2 Θ CW 70K T F 2.3K Spin ice materials Dy 2 Ti 2 O 7 and Ho 2 Ti 2 O 7 ferromagnets with single-ion anisotropy hence frustration J eff 1K 2K
Ground state degeneracy in classical Heisenberg models Maxwellian constraint-counting
Ground state degeneracy in Heisenberg pyrochlore FM H = J S i S j J 2 bonds Total number of degrees of freedom: Constraints satisfied in ground state: L α 2 + c units F = 2 (number of spins) K = 3 (number of units) Ground state dimension: D=F-K Geometric Frustration Macroscopic D
Schematics of behaviour at low temperature Classical cooperative paramagnet: JS k B T JS 2 000 111 00000000000 11111111111 00000000000000 11111111111111 000000000000000 111111111111111 0000000000 1111111111 00000 11111 0000000 1111111 0000 1111 0000000 1111111 0000 1111 0000 1111 0000000 1111111 000 111 0000000 1111111 000000 111111 000 111 0000000000 1111111111 000000 111111 000 111 00000000000 11111111111 00000 11111 0000 1111 000000000000 111111111111 00000 11111 000 111 0000000 1111111 00000 11111 0000 1111 00000 11111 000000 111111 0000 1111 00000 11111 0000000 1111111 0000000000 1111111111 000 111 000000000000000000 111111111111111111 0000000 1111111 000 111 00000000000000000000 11111111111111111111 00000 11111 0000 1111 00000000 11111111 0000 1111 0000 1111 0000 1111 000 111 0000 1111 000 111 0000 1111 Ground state 000 111 00000 11111 0000 1111 00000 11111 00000 11111 0000000 1111111 manifold 00000000000000000 11111111111111111 000000000 111111111 0000000000000000000 1111111111111111111 00000000 11111111 0000000000000 1111111111111 ccessible states at low T Phase space
Ground state selection by fluctuations? Order by disorder Villain (1980), Shender (1982) Some states have soft modes Others don t δθ δθ δθ δθ E = J 2 L 2 (δθ) 4 E = J 2 L 2 (δθ) 2
Ground state selection? Thermal fluctuations Probability distribution on ground states y x dy e ωy2 /k B T P(x) l kb T ( ) kb T ω l (x) ω Thermal fluctuations kagome coplanar pyrochlore disordered
Dynamics of Heisenberg systems How does system explore ground state manifold? Equation of motion: ds i dt = S i H i = JS i j S j Harmonic approximation ρ(ω) zero modes spin waves ω Normal mode frequencies ω Langevin: nharmonic interactions finite spinwave lifetime Brownian motion between groundstates ds/dt(0) = S h(t) random fluctuations in H i S(0) S(t) exp( ck B Tt/ )
Quasielastic neutron scattering in Y 2 Ru 2 O 7 Θ CW = 1100K T N = 77K Scattering vs Q & ω Linewidth vs temperature Γ = Ck B T C = 1.17 van Dujin et al (2008)
Frustration and residual entropy Spin ice Water ice nisotropy + ferromagnetic exchange Ground states: two-in, two-out Pauling 1935
One tetrahedron Pauling s entropy estimate Total number of states: 16 Fraction that are ground states: 6 16 Pyrochlore lattice Estimate for number of ground states: ( (total #states) 6 (# tetrahedra) 16) ( ) = 3 (# spins/2) 2
Pauling entropy in experiment Dy 2 Ti 2 O 7, Ramirez et al, Nature 399, 333 (1999).
Correlations induced by ground state constraints Local constraints tet S i = 0 Long range correlations Sharp structure in S q S q
Gauge theory of ground state correlations Youngblood et al (1980), Huse et al (2003), Henley (2004) Map spin configurations...... to vector fields B(r) two-in two out groundstates...... map to divergenceless B(r)
Ground states as flux loops Entropic distribution: P[B(r)] exp( κ B 2 (r)d 3 r) Power-law correlations: B i (r)b j (0) r 3
Low T correlations from neutron diffraction Bramwell and Harris, unpublished Ho 2 Ti 2 O 7
Classical fractionalised excitations Fractionalisation in one dimension Ground state n excited state... two separated excitations
Fractionalisation in spin ice Monopole excitations Excited states Ground state + Castelnovo, Moessner and Sondhi (2008) +
Candidate quantum Spin Liquids κ-(et) 2 Cu 2 (CN) 3 Interaction scale J 250 K No order to T = 30 mk K J Herbertsmithite ZnCu 3 (OH) 6 Cl 2 Interaction scale J 200 K No order to T = 50 mk
Metallic characteristics in an insulator κ-(et) 2 Cu 2 (CN) 3 Interaction scale J 250 K No order to T = 30 mk Finite low-t susceptibility Heat capacity at + bt 3
Summary Geometric frustration macroscopic classical degeneracies long-range order avoided Frustrated magnets at low T soft modes and slow dynamics emergent degrees of freedom exotic excitations Collaborators M. J. Bhaseen R. Coldea P. Conlon J. F. G. Eastham P.C.W. Holdsworth L. D. C. Jaubert R. Moessner T. S. Pickles T. E. Saunders E. F. Shender S. E. Palmer S. Powell M. Y. Veillette