1. Spin Liquid and Solid in Pyrochlore Antiferromagnets
Doron Bergman
UCSB Physics
Greg Fiete
KITP
Ryuichi Shindou
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: Quantitatively:
A magnet in which spins are strongly correlated but do not order
Quantitatively:
High-T susceptibility:
Frustration factor:
Quantum spin liquid:
f=1 (Tc=0)
Not so many models can be shown to have such phases

4. 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 + 1
O. Syljuansen 2005

5. D=3 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. CW = -390K,-70K,-32K for A=Zn,Cd,Hg
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
CW = -390K,-70K,-32K for A=Zn,Cd,Hg

7. 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

9. 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?

10. 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

11. 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?

12. Effective Hamiltonian
Due to 3:1 constraint and locality, must be a QDM +
Ring exchange
How to derive it?

13. 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): p=+1

14. 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 (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

15. 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/ 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
dominant
comparable
negligible

16. Diagonal term

17. 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

18. Quantum Dimer Model, s · 5/2
In this range, can approximate diagonal term: +
Expected phase diagram
(D Bergman et al PRL 05, PRB 06)
Maximally “resonatable” R state (columnar state)
U(1) spin liquid
“frozen” state
(staggered state) 1
S=5/2
S=2
S · 3/2
Numerical simulations in progress: O. Sikora et al, (P. Fulde group)

19. 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.

20. 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

21. Lowest energy state maximizes u*:
Einstein Model
vector from i to j
Site phonon
Optimal distortion:
Lowest energy state maximizes u*:

22. 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?

23. 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)
HgCr2O4
Matsuda et al, (Nat. Phys. 07)
J. H. Chung et al PRL 05
Chern, Fennie, Tchernyshyov (PRB 06)

24. 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?

25. Constrained Phase Transitions
Schematic phase diagram: T
Magnetization plateau develops
T» CW
R state
Classical (thermal) phase transition
Classical spin liquid
“frozen” state 1
U(1) spin liquid
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

26. 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

27. 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.

28. Numerics (courtesy S. Trebst)
Specific heat
T/V
“Crossings”

29. 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?