-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
1/48
Kagome Lattice
Antiferromagnets
RRP Singh UC DAVIS
M. Rigol Georgetown Univ.
D. Huse Princeton Univ.
PRL 2007 and cond-mat 2007
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
2/48
Motivation
Frustration is a key concept in studyingcomplex systems
Magnetism, Glasses, Protein-folding, .. Competing Tendencies can be resolved in
many wayscompeting phases
New Physics emerges from thiscompetition
Challenge to Computational Methods
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
3/48
Triangular-Kagome Lattice Magnets
Triangular-Lattice:
Edge sharing triangles
Kagome-Lattice:
Corner sharing triangles
Site-depletion makes Kagome-Lattice more frustrated
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
4/48
Classic example of Frustration
Ising ModelA Triangle: 6 out of 8 states are groundstates)
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 criticalpoint)
KLM: Ground State Entropy ( finitecorrelation length at T=0)
?
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
5/48
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
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
6/48
Quantum Heisenberg Model
Spin is a good quantum number
Pair of spins like to form rotaionallyinvariant singlets entangled state
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
7/48
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?
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
8/48
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
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
9/48
Is there a VBC?: SU(N) Large N:
Many Possibilities Here Too
Large N: Max-Perfect Hexagons
Honeycomb StripesMarston
Zeng
Nikolic
Senthil
36-siteunit cell
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
10/48
Dimer Expansion for spin-halfEmpty Triangles are Key
The rest are in local ground state
Ka ome Lattice Shastry-Sutherland Lattice
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
11/48
Series Expansion around arbitrary
Dimer Configuration
Graphs
defined by
triangles
All graphs
to 5th order
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
12/48
Degeneracy Lifts in 3rd/4th Order
But Not Completely
3rd Order: Bind 3Es
into H4th Order:
Honeycomb over
Stripe
Leftover: Pinwheels
2^(N/36) Low
energy states
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
13/48
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
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
14/48
36-site PBC too many wraps
New graphs start contributing in 4th order
Closed Loops of 4 triangles
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
15/48
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 somany singlets below lowest triplet?)
VBC provides an explanation
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
16/48
Spin-gap is zero or small?
Exact Diagonalization upto 36-site
Most robust message
From finite clusters
Lots of singlets below
triplet
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
17/48
Can we calculate the spin spectra
18 by 18 matrix
Left for
Future
work
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
18/48
Experimental Status
New material:Herbertsmithite
ZnCu_3(OH)_6Cl_2
Cu atoms carry spin-half
Kagome-layers of Cu
Separated by layers of
Zn
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
19/48
Some experimental properties
Curie-Weiss T=300K
No LRO down to
50mKBUT
Susceptibility turns up
at low THelton et al PRL
Ofer et al cond-mat
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
20/48
Specifc heat sublinear at low-T
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
21/48
Is the upturn due to impurity?
Misguich+sindzingre
Rigol+
RRPS
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
22/48
But muSR tracks bulk susceptibility
suggests it is intrinsic!
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
23/48
Interesting Crossover (Classical
Dimer Liquid)
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
24/48
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)
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
25/48
Clusters for finite-size studies with
Periodic Boundary Conditions
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
26/48
Susceptibility with Dp and Dz
XY ORDER CANTING
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
27/48
Entropy
Misguich and SinzindgreLowering of entropy due to
DM Interactions
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
28/48
DM Interactions
Finite-T conclusions D_z: Reduces entropy, reduces isotropic
susceptibilityLeads 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 highlyanisotropiccould be the answer
induced Ising anisotropy Must have D_p greater than D_z to match withexperiments
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
29/48
Conclusion for the Material
DM Interactions with Dp>Dz present.
Impurities also present but not simple
additivemust be embedded
DM+Impurities+Dilution (stoichiometry
requires Zn/Cu to substitute each other)
Single Crystals can measure anisotropy
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
30/48
Summary and Conclusions
Kagome Lattice appears to have a VBC
ground state (Debate is not over)
Spectra and spin-gap calculations areessential to further understand it
DM interactions are allowed- Will be there
only magnitudes can vary
Maybe optical Lattices can be DM free
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
31/48
Future Directions: Pyrochlore Lattice
Corner Sharing
Tetrahedra
From perfect hexagons
to super-tetrahedra
(Large-N also Dimer
expansions for S=1/2)
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
32/48
THE END
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
33/48
Phase Diagram of some organic materials
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
34/48
Striking Spectra of Cs2CuCl4
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
35/48
Shastry-Sutherland Lattice
Exact singlet GS with no broken symmetry
No Fluidity
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
36/48
Fluidity: Isolated spin-half objects must be free to flow
Deconfinement is the key property
May (must) have Topological Degeneracies
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
37/48
Spin-Liquids in 1D (Special Case)
1D QHM has a spin-
liquid ground state
As does theMajumdar-Ghosh
Model
MG model
Heisenberg
Ising
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
38/48
Dip in Spin-Wave spectra at (pi,0)
Square-Lattice With Frustration
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
39/48
Spectra of TLM
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
40/48
Anisotropic Triangular-Lattice
Layered Molecular Crystals (Shimizu et al)
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
41/48
Layered Molecular Crystals (Shimizu et al)
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
42/48
J=250K (TLM)
Zheng et al
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
43/48
Gapless Spin-Liquid? Projected (Spinon) Fermi Liquid?
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
44/48
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
45/48
Dip is absent in the Cuprates
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
46/48
Scaled (Parameter Free Plots)
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
47/48
Anisotropy Parameter
Elstner, Singh, Young JAP 1993
-
8/3/2019 RRP Singh, M. Rigol and D. Huse- Kagome Lattice Antiferromagnets
48/48
, g , g