Spin Liquid and Solid in Pyrochlore Antiferromagnets

1
Spin Liquid and Solid in Pyrochlore
Antiferromagnets
Doron Bergman UCSB Physics
Greg Fiete KITP
Ryuichi Shindou UCSB Physics
Simon Trebst Q Station
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
• f1 (Tc0)
• Not so many models can be shown to have such
  phases

4
Quantum Dimer Models

• RVB Hamiltonian
• Hilbert space of
• dimer coverings
• D2 lattice highly lattice dependent
• E.g. square lattice phase diagram (T0)

Ordered except exactly at v1

1
O. Syljuansen 2005
5
D3 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
Chromium Spinels
H. Takagi S.H. Lee
ACr2O4 (AZn,Cd,Hg)

• spin s3/2
• no orbital degeneracy
• isotropic

• Spins form pyrochlore lattice

• Antiferromagnetic interactions

?CW -390K,-70K,-32K for AZn,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

8
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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31 States

• Set of 31 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 31 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 (k2,q4) ?p1
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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 (K0)
• 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
  (HiziHenley) 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
comparable
dominant
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)!)

Just minority sites shown in one magnetic unit
cell
Ground state for sgt5/2 has 7-fold enlargement of
unit cell and trigonal symmetry
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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)
U(1) spin liquid
Maximally resonatable R state (columnar state)
frozen state (staggered state)
1
0
S2
S5/2
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 YamashitaK.UedaTchernys
  hyov 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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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 B0?
• 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
• B0 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
0
U(1) spin liquid

• Local constraint changes the nature of the
  paramagneticclassical spin liquid state
• - YoungbloodAxe (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)
C
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?