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**Spin Liquid and Solid in Pyrochlore Antiferromagnets**

Doron Bergman UCSB Physics Greg Fiete KITP Ryuichi Shindou Simon Trebst Q Station “Quantum Fluids”, Nordita 2007

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**Outline Quantum spin liquids and dimer models**

Realization in quantum pyrochlore magnets Einstein spin-lattice model Constrained phase transitions and exotic criticality

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**Spin Liquids Empirical definition: Quantitatively:**

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

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**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

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**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

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**CW = -390K,-70K,-32K for A=Zn,Cd,Hg**

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

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**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

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**Magnetization Process**

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

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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?

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**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

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**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?

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**Effective Hamiltonian**

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

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**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

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**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

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**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/ 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

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Diagonal term

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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)!) Ground state for s>5/2 has 7-fold enlargement of unit cell and trigonal symmetry Just minority sites shown in one magnetic unit cell

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**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) 1 S=5/2 S=2 S · 3/2 Numerical simulations in progress: O. Sikora et al, (P. Fulde group)

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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.

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**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

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**Lowest energy state maximizes u*:**

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

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**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?

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**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)

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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?

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**Constrained Phase Transitions**

Schematic phase diagram: T Magnetization plateau develops T» CW R state Classical (thermal) phase transition Classical spin liquid “frozen” state 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

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**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

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**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.

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**Numerics (courtesy S. Trebst)**

Specific heat T/V “Crossings”

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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?

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