rrp singh, m. rigol and d. huse- kagome lattice antiferromagnets

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


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


3/48

Triangular-Kagome Lattice Magnets

Triangular-Lattice:

Edge sharing triangles

Kagome-Lattice:

Corner sharing triangles

Site-depletion makes Kagome-Lattice more frustrated


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)

?


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


6/48

Quantum Heisenberg Model

Spin is a good quantum number

Pair of spins like to form rotaionallyinvariant singlets entangled state


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?


8/48

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


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


10/48

Dimer Expansion for spin-halfEmpty Triangles are Key

The rest are in local ground state

Ka ome Lattice Shastry-Sutherland Lattice


11/48

Series Expansion around arbitrary

Dimer Configuration

Graphs

defined by

triangles

All graphs

to 5th order


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


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


14/48

36-site PBC too many wraps

New graphs start contributing in 4th order

Closed Loops of 4 triangles


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


16/48

Spin-gap is zero or small?

Exact Diagonalization upto 36-site

Most robust message

From finite clusters

Lots of singlets below

triplet


17/48

Can we calculate the spin spectra

18 by 18 matrix

Left for

Future

work


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


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


20/48

Specifc heat sublinear at low-T


21/48

Is the upturn due to impurity?

Misguich+sindzingre

Rigol+

RRPS


22/48

But muSR tracks bulk susceptibility

suggests it is intrinsic!


23/48

Interesting Crossover (Classical

Dimer Liquid)


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)


25/48

Clusters for finite-size studies with

Periodic Boundary Conditions


26/48

Susceptibility with Dp and Dz

XY ORDER CANTING


27/48

Entropy

Misguich and SinzindgreLowering of entropy due to

DM Interactions


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


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


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


31/48

Future Directions: Pyrochlore Lattice

Corner Sharing

Tetrahedra

From perfect hexagons

to super-tetrahedra

(Large-N also Dimer

expansions for S=1/2)


32/48

THE END


33/48

Phase Diagram of some organic materials


34/48

Striking Spectra of Cs2CuCl4


35/48

Shastry-Sutherland Lattice

Exact singlet GS with no broken symmetry

No Fluidity


36/48

Fluidity: Isolated spin-half objects must be free to flow

Deconfinement is the key property

May (must) have Topological Degeneracies


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


38/48

Dip in Spin-Wave spectra at (pi,0)

Square-Lattice With Frustration


39/48

Spectra of TLM


40/48

Anisotropic Triangular-Lattice

Layered Molecular Crystals (Shimizu et al)


41/48

Layered Molecular Crystals (Shimizu et al)


42/48

J=250K (TLM)

Zheng et al


43/48

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


44/48


45/48

Dip is absent in the Cuprates


46/48

Scaled (Parameter Free Plots)


47/48

Anisotropy Parameter

Elstner, Singh, Young JAP 1993


48/48

, g , g

rrp singh, m. rigol and d. huse- kagome lattice antiferromagnets

Documents