Spin Liquid and Solid in Pyrochlore Antiferromagnets

1. Spin Liquid and Solid in Pyrochlore Antiferromagnets “Quantum Fluids”, Nordita 2007

2. Outline • Quantum spin liquids and dimer models • Realization in quantum pyrochlore magnets • Einstein spin-lattice model • Constrained phase transitions and exotic criticality

3. Spin Liquids • Empirical definition: • A magnet in which spins are strongly correlated but do not order • Quantitatively: • High-T susceptibility: • Frustration factor: • Quantum spin liquid: • f=1 (Tc=0) • Not so many models can be shown to have such phases

4. + + Quantum Dimer Models • “RVB” Hamiltonian • Hilbert space of dimer coverings • D=2: lattice highly lattice dependent • E.g. square lattice phase diagram (T=0) Ordered except exactly at v=1 + 1 O. Syljuansen 2005

5. D=3 Quantum Dimer Models • Generically can support a stable spin liquid state • vc is lattice dependent. May be positive or negative ordered ordered 1 Spin liquid state

6. cubic Fd3m Chromium Spinels H. Takagi S.H. Lee ACr2O4 (A=Zn,Cd,Hg) • spin s=3/2 • no orbital degeneracy • isotropic • Spins form pyrochlore lattice • Antiferromagnetic interactions CW = -390K,-70K,-32K for A=Zn,Cd,Hg

7. Pyrochlore Antiferromagnets • Heisenberg • Many degenerate classical configurations • Zero field experiments (neutron scattering) • Different ordered states in ZnCr2O4, CdCr2O4, HgCr2O4 • Evidently small differences in interactions determine ordering

8. Magnetization Process H. Ueda et al, 2005 • Magnetically isotropic • Low field ordered state complicated, material dependent • Plateau at half saturation magnetization

9. HgCr2O4 neutrons • Neutron scattering can be performed on plateau because of relatively low fields in this material. M. Matsuda et al, Nature Physics 2007 • Powder data on plateau indicates quadrupled (simple cubic) unit cell with P4332 space group • X-ray experiments: ordering stabilized by lattice distortion • - Why this order?

10. Collinear Spins • Half-polarization = 3 up, 1 down spin? • - Presence of plateau indicates no transverse order • Spin-phonon coupling? • - classical Einstein model large magnetostriction Penc et al H. Ueda et al - effective biquadratic exchange favors collinear states But no definite order

11. 3:1 States • Set of 3:1 states has thermodynamic entropy • - Less degenerate than zero field but still degenerate • - Maps to dimer coverings of diamond lattice Dimer on diamond link = down pointing spin • Effective dimer model: What splits the degeneracy? • Classical: • further neighbor interactions? • Lattice coupling beyond Penc et al? • Quantum fluctuations?

12. + Effective Hamiltonian • Due to 3:1 constraint and locality, must be a QDM Ring exchange • How to derive it?

13. Spin Wave Expansion • Quantum zero point energy of magnons: • O(s) correction to energy: • favors collinear states: • Henley and co.: lattices of corner-sharing simplexes kagome, checkerboard pyrochlore… - Magnetization plateaus:k down spins per simplex of q sites • Gauge-like symmetry: O(s) energy depends only upon “Z2 flux” through plaquettes - Pyrochlore plateau (k=2,q=4): p=+1

14. Ising Expansion • XXZ model: • Ising model (J =0) has collinear ground states • Apply Degenerate Perturbation Theory (DPT) Ising expansion Spin wave theory • Can work directly at any s • Includes quantum tunneling • (Usually) completely resolves degeneracy • Only has U(1) symmetry: • - Best for larger M • Large s • no tunneling (K=0) • gauge-like symmetry leaves degeneracy • spin-rotationally invariant • Our group has recently developed techniques to carry out DPT for any lattice of corner sharing simplexes

15. + Effective Hamiltonian derivation • DPT: • Off-diagonal term is 9th order! [(6S)th order] • Diagonal term is 6th order (weakly S-dependent)! • Checks: • Two independent techniques to sum 6th order DPT • Agrees exactly with large-s calculation (Hizi+Henley) in overlapping limit and resolves degeneracy at O(1/s) D Bergman et al cond-mat/0607210 S 1/2 1 3/2 2 5/2 3 Off-diagonal coefficient c 1.5 0.88 0.25 0.056 0.01 0.002 dominant comparable negligible

16. Diagonal term

17. Comparison to large s • Truncating Heff to O(s) reproduces exactly spin wave result of XXZ model (from Henley technique) • - O(s) ground states are degenerate “zero flux” configurations • Can break this degeneracy by systematically including terms of higher order in 1/s: • - Unique state determined at O(1/s) (not O(1)!) Just minority sites shown in one magnetic unit cell Ground state for s>5/2 has 7-fold enlargement of unit cell and trigonal symmetry

18. + + Quantum Dimer Model, s · 5/2 • In this range, can approximate diagonal term: • Expected phase diagram (D Bergman et al PRL 05, PRB 06) Maximally “resonatable” R state (columnar state) “frozen” state (staggered state) U(1) spin liquid 0 1 S=5/2 S=2 S · 3/2 Numerical simulations in progress: O. Sikora et al, (P. Fulde group)

19. R state • Unique state saturating upper bound on density of resonatable hexagons • Quadrupled (simple cubic) unit cell • Still cubic: P4332 • 8-fold degenerate • Quantum dimer model predicts this state uniquely.

20. Is this the physics of HgCr2O4? • Not crazy but the effect seems a little weak: • Quantum ordering scale » |K| » 0.25J • Actual order observed at T & Tplateau/2 • We should reconsider classical degeneracy breaking by • Further neighbor couplings • Spin-lattice interactions • C.f. “spin Jahn-Teller”: Yamashita+K.Ueda;Tchernyshyov et al Considered identical distortions of each tetrahedral “molecule” We would prefer a model that predicts the periodicity of the distortion

21. Einstein Model vector from i to j • Site phonon • Optimal distortion: • Lowest energy state maximizes u*:

22. Einstein model on the plateau • Only the R-state satisfies the bending rule! • Both quantum fluctuations and spin-lattice coupling favor the same state! • Suggestion: all 3 materials show same ordered state on the plateau • Not clear: which mechanism is more important?

23. Zero field • Does Einstein model work at B=0? • Yes! Reduced set of degenerate states “bending” states preferred • Consistent with: CdCr2O4 (up to small DM-induced deformation) HgCr2O4 Matsuda et al, (Nat. Phys. 07) J. H. Chung et al PRL 05 Chern, Fennie, Tchernyshyov (PRB 06)

24. Conclusions (I) • Both effects favor the same ordered plateau state (though quantum fluctuations could stabilize a spin liquid) • Suggestion: the plateau state in CdCr2O4 may be the same as in HgCr2O4, though the zero field state is different • ZnCr2O4 appears to have weakest spin-lattice coupling • B=0 order is highly non-collinear (S.H. Lee, private communication) • Largest frustration (relieved by spin-lattice coupling) • Spin liquid state possible here?

25. Constrained Phase Transitions • Schematic phase diagram: T Magnetization plateau develops T»CW R state Classical (thermal) phase transition Classical spin liquid “frozen” state 1 0 U(1) spin liquid • Local constraint changes the nature of the “paramagnetic”=“classical spin liquid” state • - Youngblood+Axe (81): dipolar correlations in “ice-like” models • Landau-theory assumes paramagnetic state is disordered • - Local constraint in many models implies non-Landau classical criticality Bergman et al, PRB 2006

26. Dimer model = gauge theory • Can consistently assign direction to dimers pointing from A ! B on any bipartite lattice B A • Dimer constraint ) Gauss’ Law • Spin fluctuations, like polarization fluctuations in a dielectric, have power-law dipolar form reflecting charge conservation

27. A simple constrained classical critical point • Classical cubic dimer model • Hamiltonian • Model has unique ground state – no symmetry breaking. • Nevertheless there is a continuous phase transition! • - Analogous to SC-N transition at which magnetic fluctuations are quenched (Meissner effect) • - Without constraint there is only a crossover.

28. Numerics (courtesy S. Trebst) C Specific heat T/V “Crossings”

29. Conclusions • We derived a general theory of quantum fluctuations around Ising states in corner-sharing simplex lattices • Spin-lattice coupling probably is dominant in HgCr2O4, and a simple Einstein model predicts a unique and definite state (R state), consistent with experiment • Probably spin-lattice coupling plays a key role in numerous other chromium spinels of current interest (possible multiferroics). • Local constraints can lead to exotic critical behavior even at classical thermal phase transitions. • Experimental realization needed! Ordering in spin ice?