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Degeneracy Breaking in Some Frustrated Magnets

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Ising expansion for effective models of quantum fluctuations. Einstein spin ... Ground state for s 3/2 has 7-fold enlargement of unit cell and trigonal symmetry ... – PowerPoint PPT presentation

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Title: Degeneracy Breaking in Some Frustrated Magnets


1
Degeneracy Breaking in Some Frustrated Magnets
cond-mat 0510202 (prl) 0511176 (prb) 0605467
0607210 0608131
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
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
4
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
5
Magnetization Process
H. Ueda et al, 2005
  • Magnetically isotropic
  • Low field ordered state complicated, material
    dependent
  • Plateau at half saturation magnetization

6
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M. Matsuda et al, unpublished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?

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
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?

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 Z2 flux through plaquettes

- Pyrochlore plateau (k2,q4) ?p1
10
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

11
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.

12
Some results
  • Checkerboard lattice at M1/2
  • - columnar state for all s.
  • Kagome lattice at M1/3
  • -

state for sgt1
13
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
14
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
15
Quantum Dimer Model
on diamond lattice
  • Expected T0 phase diagram (various arguments)

U(1) spin liquid
Maximally resonatable R state
frozen state
1
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
16
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.

17
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
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!

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
20
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
21
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

22
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.

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
    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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