Degeneracy Breaking in Some Frustrated Magnets

Transcription

1 Degeneracy Breaking in Some Frustrated Magnets Doron Bergman Greg Fiete Ryuichi Shindou Simon Trebst UCSB Physics KITP UCSB Physics Q Station cond-mat: (prl) (prb) HFM Osaka, August 2006

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

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

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

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 M. Matsuda et al, unpublished Powder data on plateau indicates quadrupled (simple cubic) unit cell with P space group S.H. Lee talk: ordering stabilized by lattice distortion - Why this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

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 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 Z 2 flux through plaquettes - Pyrochlore plateau (k=2,q=4): τ p =+1

10 Ising Expansion XXZ model: Ising model (J =0) has collinear ground states Apply Degenerate Perturbation Theory (DPT) Ising expansion Can work directly at any s Includes quantum tunneling (Usually) completely resolves degeneracy Only has U(1) symmetry: - Best for larger M Spin wave theory 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 The leading off-diagonal term also depends only on plaquette size and s. It becomes very high order for large s. 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 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 Diagonal term: State Dominant? for s=3/2 Checks: + + -Two independent techniques to sum 6 th order DPT -Agrees exactly with large-s calculation (Hizi+Henley) in overlapping limit Extrapolated and resolves V -2.3K degeneracy at O(1/s)

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

16 R state Unique state saturating upper bound on density of resonatable hexagons Quadrupled (simple cubic) unit cell Still cubic: P 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 vector from i to j Site phonon 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? SH Lee talk 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/

20 Constrained Phase Transitions Schematic phase diagram: T Magnetization plateau develops T Θ CW Classical (thermal) phase transition R state Classical spin liquid frozen state 0 1 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

21 Dimer model = gauge theory B A 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 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. A simple constrained classical critical point Classical cubic dimer model Hamiltonian

23 Numerics (courtesy S. Trebst) C Specific heat 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?