aroon o’brien- kagome lattice structures with charge degrees of freedom

8/3/2019 Aroon OBrien- Kagome lattice structures with charge degrees of freedom

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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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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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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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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!

aroon o’brien- kagome lattice structures with charge degrees of freedom

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