Degeneracy Breaking in Some Frustrated Magnets
Doron Bergman UCSB PhysicsGreg Fiete KITPRyuichi Shindou UCSB PhysicsSimon Trebst Q Station
HFM Osaka, August 2006
cond-mat:0510202 (prl)0511176 (prb)060546706072100608131
Outline
• Chromium spinels and magnetization plateau
• Ising expansion for effective models of quantum fluctuations
• Einstein spin-lattice model• Constrained phase transitions and exotic
criticality
Chromium Spinels
ACr2O4(A=Zn,Cd,Hg)
• spin s=3/2• no orbital degeneracy• isotropic
• Spins form pyrochlore lattice
cubic Fd3m
• Antiferromagnetic interactions
ΘCW = -390K,-70K,-32K for A=Zn,Cd,Hg
TakagiS.H. Lee
Pyrochlore Antiferromagnets
• ManyMany degenerate classical configurations
• Heisenberg
• Zero field experiments (neutron scattering)-Different ordered states in ZnCr2O4, CdCr2O4-HgCr2O4?
c.f. ΘCW = -390K,-70K,-32K for A=Zn,Cd,Hg
• Evidently small differences in interactions determine ordering
Magnetization Process
• Magnetically isotropic• Low field ordered state complicated, material dependent
• Plateau at half saturation magnetization
H. Ueda et al, 2005
HgCr2O4 neutrons
M. Matsuda et al, unpublished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?
• Neutron scattering can be performed on plateau because of relatively low fields in this material.
Collinear Spins• Half-polarization = 3 up, 1 down spin?
- Presence of plateau indicates no transverse order
Penc et al
- effective biquadratic exchange favors collinear states
• Spin-phonon coupling?- classical Einstein model
large magnetostriction
But no definite order
H. Ueda et al
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
• Effective dimer model: What splits the degeneracy?-Classical:
-further neighbor interactions?-Lattice coupling beyond Penc et al?
-Quantum fluctuations?
Spin Wave Expansion
• Henley and co.: lattices of corner-sharing simplexeskagome, checkerboardpyrochlore…
• Quantum zero point energy of magnons:- O(s) correction to energy: - favors collinear states:
- 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): τp=+1
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
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
• The leading off-diagonal term also depends only on plaquettesize and s. It becomes very high order for large s.
e.g. hexagonal plaquettes:
Some results
• Checkerboard lattice at M=1/2:- “columnar” state for all s.
• Kagome lattice at M=1/3:- state for s>1
Pyrochlore plateau case
State
Diagonal term:
+ +
Extrapolated V … -2.3K
Dominant?
• 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)
for s=3/2
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)!)
Ground state for s>3/2 has 7-fold enlargement of unit cell and trigonal symmetry
Just minority sites shown in one magnetic unit cell
Quantum Dimer Model+ +
• Expected T=0 phase diagram (various arguments)
0 1Rokhsar-Kivelson Point
U(1) spin liquid
Maximally “resonatable” R state
“frozen” state
-2.3
on diamond lattice
• Interesting phase transition between R state and spin liquid! Will return to this.
Quantum dimer model is expected to yield the R state structure
S=1
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.
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”: 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
Einstein Model
• Site phonon
vector from i to j
• Optimal distortion:
• Lowest energy state maximizes u*:
• “bending rule”
Bending Rule States• At 1/2 magnetization, only the R state satisfies the bending rule globally
- Einstein model predicts R state!
• 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 A=Zn,Cd,HgSimplest “bending rule” state (weakly perturbed by DM) appears to be
consistent with CdCr2O4 Chern et al, cond-mat/0606039
SH Lee talk
Constrained Phase Transitions• Schematic phase diagram:
0 1U(1) spin liquid
R state
T
“frozen” stateClassical spin liquid
Classical (thermal) phase transition
Magnetization plateau developsT≈ ΘCW
• 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
Dimer model = gauge theory
A
B • Can consistently assign direction to dimerspointing from A → B on any bipartite lattice
• Dimer constraint ) Gauss’ Law
• Spin fluctuations, like polarization fluctuations in a dielectric, have power-law dipolar form reflecting charge conservation
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.
ConclusionsConclusions• 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?