Density of states and Density of states and frustration in the frustration in the

quantum percolation quantum percolation problem problem

Gerardo G. Naumis*Gerardo G. Naumis*Rafael A. Barrio*Rafael A. Barrio*Chumin Wang**Chumin Wang**

**Instituto de Física, UNAM, MéxicoInstituto de Física, UNAM, México**Instituto de Materiales, UNAM, México**Instituto de Materiales, UNAM, México

Density of states (DOS) of a Penrose tiling Density of states (DOS) of a Penrose tiling

),(

||ji

jitH

Penrose tiling:Penrose tiling: example of a quasiperiodic example of a quasiperiodic potential (LRO without periodicity; it is potential (LRO without periodicity; it is neither periodic, nor disordered).neither periodic, nor disordered).Model:Model: atoms at the vertex of the tiling, atoms at the vertex of the tiling, using an s-band tight-binding Hamiltonian:using an s-band tight-binding Hamiltonian:

•The DOS is symmetric around E=0.The DOS is symmetric around E=0.•There are “confined states” at E=0 (10%). The There are “confined states” at E=0 (10%). The nodal lines have a fractal structure.nodal lines have a fractal structure.• A gap is formed around E=0.A gap is formed around E=0.•States tend to be more localized around E=0.States tend to be more localized around E=0.•The bandwidth is bigger thanThe bandwidth is bigger than 2<Z>, where 2<Z>, where <Z>=4, as in a square lattice.<Z>=4, as in a square lattice.•From computer simulations, it is belived that From computer simulations, it is belived that there are critical, extended and localized statesthere are critical, extended and localized states

DOSDOS of random binary alloy in the split-band limit of random binary alloy in the split-band limit (akin to the quantum percolation problem)(akin to the quantum percolation problem)

Model of a random binary alloy in a square lattice (quoted in Ziman’s book “Models of disorder”), studied by S. Kirkpatrick and P. Eggarter, Phys. Rev. B6, 3598, 1972.

),(

||||jii

i jiiiH

The model is defined in a square lattice, where two kinds of atoms, A and B, have concentrations x and 1-x respectively. The corresponding self energies are,

andB=where tends to infinity.

•The DOS is symmetric around E=0.The DOS is symmetric around E=0.•There are “confined states” at E=0. The There are “confined states” at E=0. The fraction depends on x, and was calculated fraction depends on x, and was calculated by Kirkpatrick et. al.by Kirkpatrick et. al.•A gap is formed around E=0, EVEN A gap is formed around E=0, EVEN WHEN A-ATOMS PERCOLATE.WHEN A-ATOMS PERCOLATE.•States tend to be more localized around States tend to be more localized around E=0.E=0.•The bandwidth is bigger thanThe bandwidth is bigger than 2<Z>.2<Z>.•In 2D, all states are localized (scaling In 2D, all states are localized (scaling theory of Abrahams), although power-law theory of Abrahams), although power-law decaying states can change the picture.decaying states can change the picture.

Two bands are formed. For the A band, the B atoms can be removed. We get a quantum percolation problem,

AjiAi

iAA jiiiH),(

||||

S parameter= tendency for a gap opening at the middle of the S parameter= tendency for a gap opening at the middle of the spectrumspectrum

,)(

)()(3

2

33

3224

S

Ai

inni

nin dEEE ,)(

,...,

2211

21

||...|||||||jj

AAAAAAn

AAni

n iHjjHjjHiiHi

Where the moments are defined as,

S>1, the DOS is UNIMODAL, S<1 BIMODAL (SQL S=1.25, Honeycomb=0.67)

We calculate the moments via de Cyrot-Lackmann theorem, which states that the n-th moment is given by the number of paths with n-hops that start and end in a given site.

,...,

221221

21

||...|||),...,,,(jj

AAAAnensamblein iHjjHjjjjiP

With disorder, certain paths are block by B atoms, and,

xZxxCZZP ZZZ

Z

Z

Z

Z

4)1()( 444

0

4

02

P(Z) is a BINOMIAL distribution.

03 Symmetric DOS, BIPARTITE LATTICE

4x

)1(4 3 xx

4

04 )()()()(

Z

Zcba ZNZNZNZP

2)( ZZNa

3

0

33 3)1()(l

lllb xZlxxCZZN

38

)2(,0)1(x

NN cc

Z

l

lllc xlxxCN

0

22 4)1(2)3(

4

0

44 8)1(2)4(l

lllc xlxxCN

Frustration in a renormalized Hamiltonian

ii

iiPiiP ||,||

|)()|)(( PPEPPH AA

|| PPH AA

RENORMALIZATION

Since H produces a hop between sublattices:

|| PPH AA

|||| 22 PEPEHPHHPH AAAAAAAA

FRUSTRATIONFRUSTRATION

+ + + + + + + + + Bonding

+ - + - + - + - + Anti- Bonding

+ 0 - 0 + 0 - 0 +

E2

+ + + + + + + + + Bondingstate

Lifshitz tailDegenerate states

Compression of the band

+ - + - + - + - + Anti- Bonding

+

+-1 Frustrated bondRises the energy

+1

--1

E=-1-1+1

Aji

ijAAAiiAA jiHiiZH

,),(

2

,

2 ||)(||

ji

jiijAAii

i EcEcHEcZE,

*222 )()()()(

If ci(E) is the amplitude at site i for an energy E, from the equation of motion:

)()()( 23

22

21

2 ECECECE

sum of all negative bonds sum of all positive bonds

)(

)(

)(

22

21

23

EC

EC

EC

Statistical Bounds

)(

)()()()()(

22

2222221

EcZZ

EcEcZZEcZEC

ii

i

iiii

ii

i

)1(44)()1(44 21 xxxECxxx

0)()()( 23

22

21

2 ECECECE

Statistical BoundsStatistical Bounds

The correlation amplitude-local coordination is estimated using the standard desviation of the binomial distribution, the normalization condition and two extreme cases:

Example, for x=0.65 the maximum value is 3.56; in the simulations was 3.58.

)1(44)()( 23

22 xxxECEC

)1(44)1(314122 2 xxxxxxxE

1)1(326)( 20 xxxxxf

)()()( 23

22

21

2 ECECECE

Where f0(x) is the number of confined states for a given x.

(for x=0.65, the calculated bandwidth is W=6.60, while in the simulations was 6.65)