1. Degeneracy Breaking in Some Frustrated Magnets Doron BergmanUCSB Physics Greg FieteKITP Ryuichi ShindouUCSB Physics Simon TrebstQ Station HFM Osaka, August 2006 cond-mat: 0510202 (prl) 0511176 (prb) 0605467 0607210 0608131

2. Outline Chromium spinels and magnetization plateau Ising expansion for effective models of quantum fluctuations Einstein spin-lattice model Constrained phase transitions and exotic criticality

3. Chromium Spinels ACr 2 O 4 (A=Zn,Cd,Hg) spin s=3/2 no orbital degeneracy isotropic Spins form pyrochlore lattice cubic Fd3m Antiferromagnetic interactions  CW = -390K,-70K,-32K for A=Zn,Cd,Hg Takagi S.H. Lee

4. Pyrochlore Antiferromagnets Many Many degenerate classical configurations Heisenberg Zero field experiments (neutron scattering) -Different ordered states in ZnCr 2 O 4, CdCr 2 O 4 -HgCr 2 O 4 ? c.f.  CW = -390K,-70K,-32K for A=Zn,Cd,Hg Evidently small differences in interactions determine ordering

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

6. HgCr 2 O 4 neutrons M. Matsuda et al, unpublished Powder data on plateau indicates quadrupled (simple cubic) unit cell with P4 3 32 space group Neutron scattering can be performed on plateau because of relatively low fields in this material. S.H. Lee talk: ordering stabilized by lattice distortion - Why this order?

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

8. 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 Effective dimer model: What splits the degeneracy? -Classical: -further neighbor interactions? -Lattice coupling beyond Penc et al? -Quantum fluctuations?

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

10. Ising Expansion XXZ model: Ising model (J  =0) has collinear ground states Apply Degenerate Perturbation Theory (DPT) Ising expansionSpin 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 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

11. Form of effective Hamiltonian The leading diagonal term assigns energy E a (s) to plaquette “type” a: the same for any such lattice at any applicable M kagome, pyrochlore checkerboard The leading off-diagonal term also depends only on plaquette size and s. It becomes very high order for large s. Energies are a little complicated e.g. hexagonal plaquettes:

12. Some results Checkerboard lattice at M=1/2: - “columnar” state for all s. Kagome lattice at M=1/3: - state for s>1

13. Pyrochlore plateau case State Diagonal term: ++ Extrapolated V  -2.3K Dominant? Checks: -Two independent techniques to sum 6 th order DPT -Agrees exactly with large-s calculation (Hizi+Henley) in overlapping limit and resolves degeneracy at O(1/s) for s=3/2

14. Resolution of spin wave degeneracy Truncating H eff 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)!) Ground state for s>3/2 has 7-fold enlargement of unit cell and trigonal symmetry Just minority sites shown in one magnetic unit cell

15. Quantum Dimer Model ++ Expected T=0 phase diagram (various arguments) 0 1 Rokhsar-Kivelson Point U(1) spin liquid Maximally “resonatable” R state “frozen” state -2.3 on diamond lattice Interesting phase transition between R state and spin liquid! Will return to this. Quantum dimer model is expected to yield the R state structure S=1

16. R state Unique state saturating upper bound on density of resonatable hexagons Quadrupled (simple cubic) unit cell Still cubic: P4 3 32 8-fold degenerate Quantum dimer model predicts this state uniquely.

17. Is this the physics of HgCr 2 O 4 ? Probably not: –Quantum ordering scale » |V| » 0.02J –Actual order observed at T & T plateau /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

18. Einstein Model Site phonon vector from i to j Optimal distortion: Lowest energy state maximizes u*: “bending rule”

19. Bending Rule States At 1/2 magnetization, only the R state satisfies the bending rule globally - Einstein model predicts R state! Zero field classical spin-lattice ground states? collinear states with bending rule satisfied for both polarizations ground state remains degenerate  Consistent with different zero field ground states for A=Zn,Cd,Hg  Simplest “bending rule” state (weakly perturbed by DM) appears to be consistent with CdCr 2 O 4 Chern et al, cond-mat/0606039 SH Lee talk

20. Constrained Phase Transitions Schematic phase diagram: 0 1 U(1) spin liquid R state T “frozen” state Classical spin liquid Classical (thermal) phase transition Magnetization plateau develops T   CW 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

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

22. 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.

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

24. Conclusions We derived a general theory of quantum fluctuations around Ising states in corner-sharing simplex lattices Spin-lattice coupling probably is dominant in HgCr 2 O 4, 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?