Classical and Quantum Frustrated Magnets Yong Baek Kim University of Toronto University of Virginia, November 1, 2007 Collaborators: J. Hopkinson (T oronto), S. Isakov (Toronto), M. Lawler (Toronto), H. Y. Kee (Toronto) F. Wang (Berkeley), A. Vishwanath (Berkeley), S. Sachdev (Harvard), L. Friz (Harvard)
Outline Introduction to Frustrated Magnets Spin-1/2 Quantum Magnets; Real Materials Review of the Heisenberg model on the Kagome lattice Distorted Kagome (Volborthites) Lattices Hyper-Kagome (Na4Ir3O8) Lattice Zn-Paratacamite Lattice
Introduction to Frustrated Magnets Geometric Frustration: the arrangement of spins on a lattice precludes (fully) satisfying all interactions at the same time Modern: large degeneracy of the (classical) ground state manifold ∼ e α N Consequence: No energy scale of its own; any perturbation is strong (reminiscent of the lowest Landau level physics) Mother of the conventional and exotic phases
Introduction to Frustrated Magnets Susceptibility ‘fingerprint’: Θ CW Curie-Weiss temperature mean-field ordering temp. interaction energy scale T F / Θ CW ≪ 1 strong frustration T < T F T F < T < Θ CW Cooperative paramagnet: Magnetically ordered ? correlations remain weak Spin liquid ? Glassy ? more universal not universal
Origin of Classical Ground State Degeneracy Classical nearest-neighbor antiferromagnetic Heisenberg model on lattices with corner-sharing simplexes (simplex = triangle, tetrahedron) 2 S i · S j = J � � � S i H = J 2 � i,j � simplex i ǫ simplex is a vector with a fixed length S i Classical ground state should satisfy � S i = 0 i ǫ simplex These constraints are not independent; counting is subtle Nonetheless there exists macroscopic degeneracy
Kagome Pyrochlore Hyper-Kagome
Order by Disorder Order by Disorder via Thermal Fluctuations: Different entropic weighting to each ground state Softer the fluctuations around a particular ground state, more likely this ground state will be entropically favored. Order by Disorder via Quantum Fluctuations: Quantum zero point energy and anharmonic contributions may select an ordered ground state. Sufficiently strong quantum fluctuations (S=1/2 for example), however, may destabilize any ordered phase; possible quantum spin liquid - Disorder by Disorder
S=1/2 Frustrated Magnets; Real Materials Ideal Kagome lattice Herbertsmithite t ZnCu 3 (OH) 6 Cl 2 . J ≈ 197 K (fit to series expansion) Θ CW = − 300 ± 20 K J. S. Helton et al, PRL 98, 107204 (2007)
S=1/2 Frustrated Magnets; Real Materials Volborthite ets. In this context, the r Distorted Kagome lattice te Cu 3 V 2 O 7 (OH) 2 � 2H 2 O ( b ) 1000 H = 7 T J 1 800 1/ � ( mol Cu/cm 3 ) J 2 Fig.2a, Hiroi et al. J 1 600 Curie-Weiss fitting p eff = 1.95 µ B (g = 2.26) 400 � = -115 K C u 1 200 0 C u 2 0 100 200 300 400 T (K) Θ CW = − 115 K Z. Hiroi et al, JPSJ 70, 3377 (2001); F. Bert et al, PRL 95, 087203 (2005)
S=1/2 Frustrated Magnets; Real Materials 2000 Hyper-Kagome (a) Na 4 Ir 3 O 8 e, Na 4 Ir 3 O 8 , � 1 (mol Ir/emu ) 1.6 1500 -3 emu /mol Ir) 0.01 T 1.4 0.1 T 1 T % 1.2 1000 '&* 5 T 1.0 0.8 (10 500 !" 0.6 0.1 1 10 100 T (K) 0 3 2 mol Ir) 100 C m � T (b) 2 mol Ir) 60 C m / T (mJ/K 2 10 C m � T C m / T (mJ/K 0 T 8 T 40 12 T 4 T 1 Θ CW = − 650 K 1 10 100 T (K) 20 0 S m (J/Kmol Ir) (c) 6 Y. Okamoto, M. Nohara, H. Agura-Katrori, 4 and H. Takagi, arXiv:0705.2821 (2007) 2 0 0 50 100 150 200 250 300 T (K)
Classical Heisenberg Model on Kagome Lattice Consider first co-planar states � S i = 0 A i ǫ △ 3-state Potts spins = B C √ √ q = 0 3 × 3 A A B A A C B A B C B C B C C B B A A A A C B B C B A C B A C B C C A A B A
Degeneracy of the co-planar states is extensive; e 0 . 379 N (from 3-state Potts antiferromagnet) D. A. Huse, A. Rutenberg, PRB 45, 7536 (1992) Non-planar states can be generated by continuous distortions of a planar state Weathervane loop All ground states can be generated by repeated introduction of ‘defects’ into the different parent planar states
Planar ground states have more soft modes (introduction of ‘defect’ removes certain soft modes) Planar ground states are favored at small temperatures Expect growing correlation of nematic order (broken spin-rotation symmetry); Nematic long range order as T → 0 2 g ( r a − r b ) = 3 2 � ( n a · n b ) 2 � − 1 n a = √ 3( S 1 × S 2 + S 2 × S 3 + S 3 × S 1 ) , 2 , 3 J. T. Chalker, P . C. W.Holdsworth, E. F. Shender, PRL 68, 855 (1992) D. A. Huse, A. Rutenberg, PRB 45, 7536 (1992) g ( r ) → 1 for coplanar ground states N --- nearest-neighbor triangles NN --- next-nearest-neighbor triangles
Small distortions from an arbitrary coplanar ground state � H = H 0 + H n ( ǫ n ) n quadratic potential; identical for all coplanar states H 2 : zero mode (for all q) H 3 + H 4 : Quartic potential not the same for different coplanar states Boltzman weight not the same; √ √ 3 favored as T → 0 3 × D. A. Huse, A. Rutenberg, PRB 45, 7536 (1992) A. Chubukov, PRL 69, 832 (1992)
Quantum Heisenberg Model on Kagome Lattice What about spin-1/2 quantum model ? Exact Diagonalization Singlet Ground State ? Effective Field Theory Spin Liquid ? Valence Bond Solid ? Series Expansion Nature of the ground state not understood ...
Distorted Kagome (Volborthite) Lattice consider mainly th J AB = J CA � = J BC . � H = ( S A · S B + α S B · S C + S C · S A ) triangles = α [(1 / α ) S A + S B + S C ] 2 − constant � L 2 triangles CA � B J’ d J BC = α . t J AB = J CA = 1 J J A Volborthite; t α > 1 B C M � K Constraint on classical ground states [(1 / α ) S A + S B + S C = 0
Classical Heisenberg Model angle between A-site s s θ 0 = arccos( − 1 / 2 α ) ( � 0 Single triangle (3-state Potts does not apply) o α ≤ 1 / 2. ‘cluster spin’ cannot be zero collinear ground state; no degeneracy; ferrimagnet � < 1/2 r α > 1 / 2 c coplanar ground states � 0 chirality variables are more useful η = ± 1
Classical Ground State Degeneracy Constraints on the chirality variables A Isotropic Kagome � 1 0 to � 6 i =1 η i = ± 6 or 0. � 2 � 6 B C ingle hexagon out of 2 Volborthite Kagome D mensurate to 2 π and e: � 6 i =1 η i = ± 6 or, � � 5 nd the constraint is more restric- 3 e last equation is the r, � 6 E F i =1 η i = 0 and η 1 + η 4 = 0. � 4 Degeneracy of coplanar ground states omagnet (or 3- Isotropic Kagome e, exp(0 . 379 N ), Volborthite Kagome t exp(2 . 2 L ). Sub-extensive ! Direct enumeration Transfer Matrix Method
Consequence of Sub-extensive Degeneracy No local weather-vane mode not equivalent cannot be in the same direction Non-local weather-vane modes exist - the number do not scale as the area of the system Classical ground state manifold of the Volborthite Kagome is much less connected than the isotropic case requires moving an infinite number of spins; large kinetic barriers; may expect freezing at low temperatures
Application to Volborthtite Low temperature NMR experiments on Volborthite; spin freezing below 1.5 K (J/60) (isotropic Herbertsmithite; no freezing observed) 51 V NMR; V atoms at the hexagon centers 1 /T 1 rises rapidly through the glass transition temperature 51 V two distinct local environments for the lower static field - 80% higher static field - 20% F. Bert et al, Phys. Rev. Lett. 95, 087203 (2005) Assume that the glassy state locally resembles certain classical ground state Volume average of a local quantity in the glassy state = ensemble average over classical ground states
α ≈ 1 case: three different field values are possible re H Cu : the field from a single spin , three di ff erent eld values ≈ √ , H ≈ 3 H cu , , H ≈ 3 H Cu d H ≈ 0, √ √ ( 3 × 3) ≈ In the experiment, assume , H ≈ 3 H cu , for high static field copper moment per site = 0 . 4 µ B Constraint on the classical ground states (via transfer matrix method) leads to , three di ff erent eld values √ - 25% - 75% , H ≈ 3 H Cu d H ≈ 0, √ copper moment per site = 3 × 0 . 4 µ B = 0 . 7 µ B
Summary Distorted Kagome Lattice Classical Heisenberg model: sub-extensive degeneracy of the classical ground states much less connected than the isotropic case; glassy behavior ? thermal fluctuations favor Chirality Stripe state L J’ J J Quantum spin-1/2 Heisenberg model A singlet ground state with a spin gap; B C spin liquid ? F. Wang, A. Vishwanath, Y. B. Kim, Phys. Rev. B 76, 094421(2007)
Three-dimensional S=1/2 Frustrated Magnet has a Hyper-Kagome sublattice of Ir ions e, Na 4 Ir 3 O 8 , Ir Na 3/4 Ir, 1/4 Na Pyrochlore Hyper-Kagome All Ir-Ir bonds are equivalent 5 4+ carries S=1/2 moment (low spin state) Ir ( 5d ) Y. Okamoto, M. Nohara, H. Agura-Katrori, and H. Takagi, arXiv:0705.2821 (2007)
Hyper-Kagome Lattice
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