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Kagome lattice structures withcharge degrees of freedom
Aroon OBrienMax Planck Institute for the Physics of
Complex Systems, Dresden Frank Pollmann, University of California, Berkeley
Masaaki Nakamura , MPI-PKS, Dresden Peter Fulde, MPI-PKS, Dresden, Asian Pacic Center for the Theoretical
Physics, Pohang Michael Schreiber , TU Chemnitz
NTZ CompPhys08, Nov28 2008
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Outline
Introduction-Frustration andFractionalization
A theoretical model of frustration Analysing the model Current approaches and Outlook
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Fractionalization Similarly - removed bond would with charge -e would give rise to
fractional charges with charge e/2+!
Similarly - add/remove one charged particles on a frustrated lattice -gives two fractionally charged excitations
Fractionalization-observed experimentally in FractionalQuantum Hall Effect [3]
[1]W.P.Su,J.R.Schrieffer, and A.J.Heeger,Phys.Rev.Lett.,v 42 ,p1968,1979.[2]W.P.Su and J.R. Schrieffer, Phys. Rev. Lett.,v 46, p738,1981.[3]D.C.Tsui,H.L.Stormer, and A.C.Gossard, Phys.Rev.Lett. 53, 722-723(1984)
One excitation-decays into two collective excitations
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Geometric Frustration Fractional charges -arise also in theoretical
models of geometrically frustrated systems[1] Occur in lattice structures where it is
impossible to minimize the energy of all localinteractions:
Characterised by a macroscopic ground-
state degeneracy -> high density of low-lyingexcitations:
[1] P.Fulde,K.Penc,N.Shannon,Ann.Phys.,v 11,892(2002)
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Geometric Frustration in nature Spinel minerals form pyrochlore structures:
M3 H(XO 4) forms a kagome lattice structure:
Samarian Spinel
(Iranian Crown Jewels)
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Fractionalised charges due togeometrical frustration
There are models of 2D lattice structuressupporting fractional excitations [5].
These approaches so far yield fractional
excitations that are confined [6]. 3D lattices have been shown to support
deconfined phases[7,8]
What we know already
[5]P.Fulde, K.Penc, N. Shannon, Ann.Phys.,v 11 ,892 (2002).[6]F.Pollmann and P.Fulde, EurophysLett.,v 75 ,133 (2006)[7]Bergmann, G. Fiete, and L.Balents, Phys.Rev. Lett. v 96 (2006)[8]Olga Sikora et. Al, to be published
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Fractionalised charges due togeometrical frustrationWhat we would like to know!
Kagome lattice models-can we investigate the dynamics of systems exhibiting charge fractionalization? Can we determinethe confinement/deconfinement of the excitations?Do these fractionalized excitations exhibit fractionalisedstatistics ? What are they?Can we use such models to explain experimentalobservations in real materials with such structures?
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A model of fractionalizationConsider a model of spinless fermions on
the kagome lattice
Extended Hubbard model with chargedegrees of freedom
Consider 1/3 filling
At t =0, V >0, macroscopic number of ground states
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A model of fractionalization
Strong correlation limit (large nearest-neighbour repulsions V ) -> localconstraint of 1 particle per triangle onthe lattice -> triangle rule
Finite hopping of fractional charges instrongly correlated limit where
Add one particle -> increase system
energy by 2V
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A model of fractionalization
One particle with charge e is added to thesystem - it can decay into two defects eachcarrying the charge e/2 -> 2 fractionalcharges are created
One excitation-decays into two collective excitations
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A model of fractionalization Large Hilbert space sizes -> limit
numerical investigation
Lowest order hopping process liftingdegeneracy - particle hopping aroundhexagons :
Derive an effective model Hamiltonianencapsulating behaviour in the strongcorrelation limit
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A model of fractionalization
Where g =
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Effective model Exact in the limit of infinitely large V
Reduces drastically Hilbert space size
Has no fermionic sign problem!
Example: No. of configurations for a 147-sitecluster at 1/3 filling:
No. of configurations for a 147-site cluster at1/3 filling subject to the triangle rule:
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Mapping to Quantum Dimer Model
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Mapping to Quantum Dimer Model
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Mapping to Quantum Dimer Model
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Mapping to Quantum Dimer Model
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Mapping to Quantum Dimer Model
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Quantum Dimer Mapping
Mapping-effective Hamiltonian to plaquettephase (mu=0) of known system [8]:
[8] R. Moessner, S.L.Sondhi,P.Chandra, Phys.Rev.Lett.53,722-723 (2001)
Fractional charges confinedNumerically confirmed - exactdiagonalisation gives ground-statesenergiesDistance between defects1/# flippable hexagons
Ground-state energies for a 147-site kagome lattice model
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Investigating dynamical properties
With a doped system-consider dynamical properties - add extra term toHamiltonian
Projected hopping operator
Original effective Hamiltonian
Describes a system at 1/3 filling +/- one particle
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Numerical Methods Model Hamiltonian basis transformation
-> Lanczos recursion method [9] Analyse finite clusters from 25-75 sites Direct insight into system dynamics-
from spectral function calculations
[9] C. Lanczos J. Res. Natl. Bur. Stand. 45 , 255 (1950)
Spectral function - gives probability for adding (+) or removing (-) a particle with momentum k and energy to thesystem
Density of states- sum over all k - space contributions:
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How good is the model? Exact and effective models on a 27-site
cluster are comparedDensity of States - a comparison
Hole contribution Particle contribution
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Results
Density of states figuresshow thatfinite-size
effectsdecreasemarkedlywith systemsize:
48-site cluster 75-site cluster Hole contribution
Particle contribution
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Results Hole contribution is symmetric; the
eigenspectrum for the 1/3 filled system in thepresence of one hole defect is symmetric:
Underlying bipartiteness for the particle hoppingin the presence of one hole defect!
A gauge transformation that changes the sign of each hopping process must exist!
Hole contribution to the densityof states
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Results
A sitesB sites
expressible in terms of agauge transformation
Example - 2D Square Lattice
Eigenspectrum symmetryBipartite hopping on
kagome lattice
Bipartiteness
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Results Large peak in particle
contribution - at zero
momentum- full spectralweight of flat bandcontained in a singledelta peak:
GS wavefunction exacteigenfunction of the
effective Hamiltonian, inthe limit of .
This can be shownanalytically
Particle contribution to the spectral
function for the three energy bandsat k=(0,0), 75- site cluster
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Do such models model realsystems?
Materials which may provide theanswerMH 3(XO 4)2
Here protons act as particles at 1/3filling
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Do such models model realsystems?
Model gives three possible charge-orderedstates - material shows just two of these atdifferent temperatures!
Goal-to obtain a phase diagram of the modelto compare with that of corresponding realmaterials
Apply Random Phase Approximation tocalculate charge susceptibities; calculatespectral functions in the limit of small V
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Conclusion and Outlook With exact diagonalisation on finite size clusters
we are able to analyse the dynamics of kagomelattice models at specific fillings
RPA treatment of Hubbard model/spectralfunction calculations - hope to compare theresults of our theoretical model with real materials
Understand most prominent features of spectrum - what is the physicalinterpretation?
Compare -bosonic and fermionic dynamics Effective model is bipartite in nature-how can we understand this through a
gauge transformation?
QDM mapping -> we have a confined ground state- evidence of this in thespectral function results?
Thank you!
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Fractionalization
Fractional excitations exhibitfractional statistics [a]:
[a]D.Arovas,J.R.Schrieffer, and F. Wilczek, Phys. Rev. Lett. 53 ,722-723 (1984) )
3D -> fermionic/bosonic statistics2D -> possibility of anyonic statistics!