Download Degeneracy Breaking in Some Frustrated Magnets

Spin Liquid and Solid in
Pyrochlore Antiferromagnets
Doron Bergman
Greg Fiete
UCSB Physics
KITP
Ryuichi Shindou
Simon Trebst
UCSB Physics
Q Station
“Quantum Fluids”, Nordita 2007
Outline
• Quantum spin liquids and dimer models
• Realization in quantum pyrochlore
magnets
• Einstein spin-lattice model
• Constrained phase transitions and exotic
criticality
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:
– f=∞ (Tc=0)
– Not so many models can be shown to have such
phases
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
+
O. Syljuansen 2005
1
D=3 Quantum Dimer Models
• Generically can support a stable spin
liquid state
ordered
ordered
1
Spin liquid state
• vc is lattice dependent. May be positive or
negative
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
ΘCW = -390K,-70K,-32K
for A=Zn,Cd,Hg
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
Magnetization Process
H. Ueda et al, 2005
• Magnetically isotropic
• Low field ordered state complicated, material dependent
• Plateau at half saturation magnetization
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?
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
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?
Effective Hamiltonian
• Due to 3:1 constraint and locality, must be
a QDM
+
Ring exchange
• How to derive it?
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): τp=+1
Ising Expansion
• XXZ model:
• Ising model (J⊥ =0) has collinear ground states
• Apply Degenerate Perturbation Theory (DPT)
Ising expansion
• Can work directly at any s
• Includes quantum tunneling
• (Usually) completely resolves
degeneracy
• Only has U(1) symmetry:
- Best for larger M
Spin wave theory
• 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
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/0607210
Off-diagonal
coefficient
S
1/2
1
3/2
2
5/2
3
c
1.5
0.88
0.25
0.056
0.01
0.002
dominant
comparable
negligible
Diagonal term
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
Quantum Dimer Model, s · 5/2
• In this range, can approximate diagonal term:
+
+
• Expected phase diagram
Maximally “resonatable” R
state (columnar state)
(D Bergman et al PRL 05, PRB 06)
0
S=5/2 S=2
“frozen” state
(staggered state)
U(1) spin
liquid
1
S · 3/2
Numerical simulations in progress: O. Sikora et al, (P. Fulde group)
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?
• 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
Einstein Model
vector from i to j
• Site
phonon
• Optimal distortion:
• Lowest energy state maximizes u*:
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?
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)
J. H. Chung et al PRL 05
Chern, Fennie, Tchernyshyov (PRB 06)
HgCr2O4
Matsuda et al, (Nat. Phys. 07)
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?
Constrained Phase Transitions
• Schematic phase diagram:
T
Classical
(thermal) phase
transition
R state
Magnetization plateau develops
T≈ ΘCW
Classical spin liquid
0
U(1) spin liquid
“frozen” state
1
• 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
B
A
• Can consistently assign
direction to dimers
pointing 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.
Numerics (courtesy S. Trebst)
C
Specific heat
T/V
“Crossings”
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?