Kagome Lattice Antiferromagnets: Theory and Experiments.

Kagome Lattice
Antiferromagnets
RRP Singh UC DAVIS
M. Rigol Georgetown Univ.
D. Huse Princeton Univ.
PRL 2007 and cond-mat 2007

Motivation
• Frustration is a key concept in studying
complex systems
Magnetism, Glasses, Protein-folding, …..
• Competing Tendencies can be resolved in
many ways—competing phases
• New Physics emerges from this
competition
• Challenge to Computational Methods

Triangular-Kagome Lattice Magnets
Triangular-Lattice:
Edge sharing triangles
Kagome-Lattice:
Corner sharing triangles
Site-depletion makes Kagome-Lattice more frustrated

Classic example of Frustration
Ising Model
?
A Triangle: 6 out of 8 states are ground
states)
uud udu duu udd dud ddu have same
energy
uuu ddd have higher energy
Lattice Models are Exactly Soluble
TLM: Ground State Entropy (T=0 critical
point)
KLM: Ground State Entropy ( finite
correlation length at T=0)

Classical Heisenberg Models
• Ground state has 120
degree structure
• TLM: Unique Ground
State (apart from
symmetry) (Fully
Constrained)
• KLM: Finite ground
state entropy (see TLM)
(Underconstrained)
• Order by Disorder
TLM
Q=0

Quantum Heisenberg Model
Spin is a good quantum number
Pair of spins like to form rotaionally
invariant singlets –entangled state

Many Open Questions
• Is Ground state magnetically ordered? SSB
• Is the ground state a VBC?
• Is there a Quantum Spin-Liquid? RVB
• Is there a spin-gap?
• Is there algebraic spin order?
• Are there fractional-spin excitations? FQHE
• Are there massless Dirac spinons?

Magnetic Long Range Order
Many Candidates
TLM [root(3)by root(3)]
Q=0
Doubled Unit Cell along Y
Answer is NO
• Spectra from exact
diagonalization
• Series expansions
• Other numerics

Is there a VBC?: SU(N) Large N:
Many Possibilities Here Too
Large N: Max-Perfect Hexagons
Marston
Zeng
Nikolic
Senthil
36-site
unit cell
Honeycomb
Stripes

Kagome Lattice
Shastry-Sutherland Lattice
Dimer Expansion for spin-half
Empty Triangles are Key
The rest are in local ground state

Series Expansion around arbitrary
Dimer Configuration
Graphs
defined by
triangles
All graphs
to 5 th order

Degeneracy Lifts in 3 rd /4 th Order
But Not Completely
3 rd Order: Bind 3Es
into H
4 th Order:
Honeycomb over
Stripe
Leftover: Pinwheels
2^(N/36) Low
energy states

Series show excellent Convergence
Order & Honeycomb & Stripe VBC & 36-site PBC \\
0 & -0.375 & -0.375 & -0.375 \\
1 & -0.375 & -0.375 & -0.375 \\
2 & -0.421875 & -0.421875 & -0.421875 \\
3 & -0.42578125 & -0.42578125 & -0.42578125 \\
4 & -0.431559245 & -0.43101671 & -0.43400065 \\
5 & -0.432088216 & -0.43153212 & -0.43624539 \\
Ground State Energy per site
Estimated H-VBC energy: -0.433(1)
36-site PBC: Energy=-0.43837653

36-site PBC too many wraps
New graphs start contributing in 4 th order
Closed Loops of 4 triangles

Exact Diagonalization
• Lhuillier et al: (E=-0.43--0.44)
• Gap of J/20 or less, maybe 0 (No VBC?)
• Gap for upto 36-site extrapolated by 1/N
• Significant spread with size
• Very little triplet dispersion
• May be indicative of exotic state! (Why so
many singlets below lowest triplet?)
VBC provides an explanation

Spin-gap is zero or small?
Exact Diagonalization upto 36-site
Most robust message
From finite clusters
Lots of singlets below
triplet

Can we calculate the spin spectra
18 by 18 matrix
Left for
Future
work

• New material:
Herbertsmithite
ZnCu_3(OH)_6Cl_2
Cu atoms carry spinhalf
Kagome-layers of Cu
Separated by layers of
Zn
Experimental Status

Some experimental properties
• Curie-Weiss T=300K
• No LRO down to
50mK
BUT
• Susceptibility turns up
at low T
Helton et al PRL
Ofer et al cond-mat

Specifc heat sublinear at low-T

Is the upturn due to impurity?
Rigol+
RRPS
Misguich+sindzingre

But muSR tracks bulk susceptibility
suggests it is intrinsic!

Interesting Crossover (Classical
Dimer Liquid)

Dzyloshinski-Moria Interactions
Cross Product between spins
Both Dz and Dp are allowed! (Of order 10% of J
in related Fe-based spin-5/2 material)

Clusters for finite-size studies with
Periodic Boundary Conditions

Susceptibility with Dp and Dz
XY ORDER
CANTING

Entropy
Misguich and Sinzindgre
Lowering of entropy due to
DM Interactions

DM Interactions
Finite-T conclusions
• D_z: Reduces entropy, reduces isotropic
susceptibility—Leads to long-range XY order
(each sign favors one chirality)-----cant be the
answer
• D_p No change in entropy, increases
susceptibility suddenly, makes it highly
anisotropic—could be the answer
induced Ising anisotropy
• Must have D_p greater than D_z to match with
experiments

Conclusion for the Material
• DM Interactions with Dp>Dz present.
• Impurities also present but not simple
additive—must be embedded
• DM+Impurities+Dilution (stoichiometry
requires Zn/Cu to substitute each other)
• Single Crystals can measure anisotropy

Summary and Conclusions
• Kagome Lattice appears to have a VBC
ground state (Debate is not over)
• Spectra and spin-gap calculations are
essential to further understand it
• DM interactions are allowed- Will be there
only magnitudes can vary
• Maybe optical Lattices can be DM free

Future Directions: Pyrochlore Lattice
• Corner Sharing
Tetrahedra
• From perfect hexagons
to super-tetrahedra
(Large-N also Dimer
expansions for S=1/2)

THE END

Phase Diagram of some organic materials

Striking Spectra of Cs2CuCl4

Shastry-Sutherland Lattice
Exact singlet GS with no broken symmetry
No Fluidity

Fluidity: Isolated spin-half objects must be free to flow
Deconfinement is the key property
May (must) have Topological Degeneracies

Spin-Liquids in 1D (Special Case)
• 1D QHM has a spinliquid
ground state
• As does the
Majumdar-Ghosh
Model
Heisenberg
Ising
MG model

Dip in Spin-Wave spectra at (pi,0)
Square-Lattice With Frustration

Spectra of TLM

Anisotropic Triangular-Lattice

Layered Molecular Crystals (Shimizu et al)

Zheng et al
J=250K (TLM)

Gapless Spin-Liquid? Projected (Spinon) Fermi Liquid?

Dip is absent in the Cuprates

Scaled (Parameter Free Plots)

Anisotropy Parameter

Elstner, Singh, Young JAP 1993