Degeneracy Breaking in Some Frustrated Magnets

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Degeneracy Breaking in Some Frustrated Magnets
cond-mat 0510202 (prl) 0511176 (prb) 0605467
0607210 0608131
HFM Osaka, August 2006
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Outline

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

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Chromium Spinels
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

c.f. ?CW -390K,-70K,-32K for AZn,Cd,Hg
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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M. Matsuda et al, unpublished

• Powder data on plateau indicates quadrupled
  (simple cubic) unit cell with P4332 space group

• S.H. Lee talk 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

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

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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
• 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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Form of effective Hamiltonian

• The leading diagonal term assigns energy Ea(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

• The leading off-diagonal term also depends only
  on plaquette size and s. It becomes very high
  order for large s.

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Some results

• Checkerboard lattice at M1/2
• - columnar state for all s.

• Kagome lattice at M1/3
• -

state for sgt1
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Pyrochlore plateau case
Diagonal term
State
Dominant?

• 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)

Extrapolated V ¼ -2.3K
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Resolution of spin wave degeneracy

• 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 sgt3/2 has 7-fold enlargement of
unit cell and trigonal symmetry
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Quantum Dimer Model
on diamond lattice

• Expected T0 phase diagram (various arguments)

U(1) spin liquid
Maximally resonatable R state
frozen state
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S1
-2.3
0
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
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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?

• Probably not
• Quantum ordering scale V 0.02J
• 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

• bending rule

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Bending Rule States

• At 1/2 magnetization, only the R state satisfies
  the bending rule globally
• - Einstein model predicts R state!

SH Lee talk

• 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 AZn,Cd,Hg
• Simplest bending rule state (weakly perturbed
  by DM) appears to be consistent with CdCr2O4

Chern et al, cond-mat/0606039
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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?