Spin Liquid and Solid in Pyrochlore Antiferromagnets
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1 Spin Liquid and Solid in Pyrochlore Antiferromagnets Doron BergmanUCSB PhysicsGreg FieteKITPRyuichi ShindouSimon TrebstQ Station“Quantum Fluids”, Nordita 2007
2 Outline Quantum spin liquids and dimer models Realization in quantum pyrochlore magnetsEinstein spin-lattice modelConstrained phase transitions and exotic criticality
3 Spin Liquids Empirical definition: Quantitatively: A magnet in which spins are strongly correlated but do not orderQuantitatively:High-T susceptibility:Frustration factor:Quantum spin liquid:f=1 (Tc=0)Not so many models can be shown to have such phases
5 D=3 Quantum Dimer Models Generically can support a stable spin liquid statevc is lattice dependent. May be positive or negativeorderedordered1Spin liquid state
6 CW = -390K,-70K,-32K for A=Zn,Cd,Hg Chromium SpinelsH. TakagiS.H. LeeACr2O4 (A=Zn,Cd,Hg)cubic Fd3mspin s=3/2no orbital degeneracyisotropicSpins form pyrochlore latticeAntiferromagnetic interactionsCW = -390K,-70K,-32K for A=Zn,Cd,Hg
7 Pyrochlore Antiferromagnets HeisenbergMany degenerate classical configurationsZero field experiments (neutron scattering)Different ordered states in ZnCr2O4, CdCr2O4, HgCr2O4Evidently small differences in interactions determine ordering
8 Magnetization Process H. Ueda et al, 2005Magnetically isotropicLow field ordered state complicated, material dependentPlateau at half saturation magnetization
9 HgCr2O4 neutronsNeutron scattering can be performed on plateau because of relatively low fields in this material.M. Matsuda et al, Nature Physics 2007Powder data on plateau indicates quadrupled (simple cubic) unit cell with P4332 space groupX-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 orderSpin-phonon coupling?- classical Einstein modellarge magnetostrictionPenc et alH. Ueda et al- effective biquadratic exchange favors collinear statesBut 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 latticeDimer on diamond link =down pointing spinEffective 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 exchangeHow 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 simplexeskagome, checkerboardpyrochlore…- Magnetization plateaus: k down spins per simplex of q sitesGauge-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 statesApply Degenerate Perturbation Theory (DPT)Ising expansionSpin wave theoryCan work directly at any sIncludes quantum tunneling(Usually) completely resolves degeneracyOnly has U(1) symmetry:- Best for larger MLarge sno tunneling (K=0)gauge-like symmetry leaves degeneracyspin-rotationally invariantOur 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 DPTAgrees exactly with large-s calculation (Hizi+Henley) in overlapping limit and resolves degeneracy at O(1/s)D Bergman et al cond-mat/S1/213/225/23Off-diagonal coefficientc1.50.880.250.0560.010.002dominantcomparablenegligible
17 Comparison to large sTruncating Heff to O(s) reproduces exactly spin wave result of XXZ model (from Henley technique)- O(s) ground states are degenerate “zero flux” configurationsCan 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>5/2 has 7-fold enlargement of unit cell and trigonal symmetryJust minority sites shown in one magnetic unit cell
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)U(1) spin liquid“frozen” state(staggered state)1S=5/2S=2S · 3/2Numerical simulations in progress: O. Sikora et al, (P. Fulde group)
19 R stateUnique state saturating upper bound on density of resonatable hexagonsQuadrupled (simple cubic) unit cellStill cubic: P43328-fold degenerateQuantum 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| » 0.25JActual order observed at T & Tplateau/2We should reconsider classical degeneracy breaking byFurther neighbor couplingsSpin-lattice interactionsC.f. “spin Jahn-Teller”: Yamashita+K.Ueda;Tchernyshyov et alConsidered identical distortions of each tetrahedral “molecule”We would prefer a model that predicts the periodicity of the distortion
21 Lowest energy state maximizes u*: Einstein Modelvector from i to jSite phononOptimal 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 plateauNot clear: which mechanism is more important?
23 Zero field Does Einstein model work at B=0? Yes! Reduced set of degenerate states“bending” states preferredConsistent with:CdCr2O4 (up to small DM-induced deformation)HgCr2O4Matsuda et al, (Nat. Phys. 07)J. H. Chung et al PRL 05Chern, 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 differentZnCr2O4 appears to have weakest spin-lattice couplingB=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:TMagnetization plateau developsT» CWR stateClassical (thermal) phase transitionClassical spin liquid“frozen” state1U(1) spin liquidLocal constraint changes the nature of the “paramagnetic”=“classical spin liquid” state- Youngblood+Axe (81): dipolar correlations in “ice-like” modelsLandau-theory assumes paramagnetic state is disordered- Local constraint in many models implies non-Landau classical criticalityBergman et al, PRB 2006
26 Dimer model = gauge theory Can consistently assign direction to dimers pointing from A ! B on any bipartite latticeBADimer constraint ) Gauss’ LawSpin 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 modelHamiltonianModel 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) Specific heatT/V“Crossings”
29 ConclusionsWe derived a general theory of quantum fluctuations around Ising states in corner-sharing simplex latticesSpin-lattice coupling probably is dominant in HgCr2O4, and a simple Einstein model predicts a unique and definite state (R state), consistent with experimentProbably 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?