Density of states and frustration in the quantum percolation problem

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Density of states and frustration in the quantum percolation problem . Gerardo G. Naumis* Rafael A. Barrio* Chumin Wang** * Instituto de Física, UNAM, México **Instituto de Materiales, UNAM, México. Density of states (DOS) of a Penrose tiling .

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Density of states and frustration in the quantum percolation problem

Gerardo G. Naumis*

Rafael A. Barrio*

Chumin Wang**

*Instituto de Física, UNAM, México

**Instituto de Materiales, UNAM, México

Density of states (DOS) of a Penrose tiling

Penrose tiling:example of a quasiperiodic potential (LRO without periodicity; it is neither periodic, nor disordered).

Model:atoms at the vertex of the tiling, using an s-band tight-binding Hamiltonian:

• The DOS is symmetric around E=0.
• There are “confined states” at E=0 (10%). The nodal lines have a fractal structure.
• A gap is formed around E=0.
• States tend to be more localized around E=0.
• The bandwidth is bigger than2<Z>, where <Z>=4, as in a square lattice.
• From computer simulations, it is belived that there are critical, extended and localized states
DOS of random binary alloy in the split-band limit (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.

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,

eA=0, andeB=d, where d tends to infinity.

Two bands are formed. For the A band,

the B atoms can be removed.

We get a quantum percolation problem,

• The DOS is symmetric around E=0.
• There are “confined states” at E=0. The fraction depends on x, and was calculated by Kirkpatrick et. al.
• A gap is formed around E=0, EVEN WHEN A-ATOMS PERCOLATE.
• States tend to be more localized around E=0.
• The bandwidth is bigger than2<Z>.
• In 2D, all states are localized (scaling theory of Abrahams), although power-law decaying states can change the picture.

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.

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

P(Z) is a BINOMIAL distribution.

Symmetric DOS, BIPARTITE LATTICE

Frustration in a renormalized Hamiltonian

RENORMALIZATION

Since H produces a hop between sublattices:

Degenerate states

+ - +

- + -

+ - +

Anti- Bonding

+ + +

+ + +

+ + +

Bonding

state

Compression

of the band

+1

+

-1

Frustrated bond

Rises the energy

-

+

-1

E=-1-1+1

FRUSTRATION

Lifshitz tail

+ + +

+ + +

+ + +

Bonding

+ - +

- + -

+ - +

Anti- Bonding

+ 0 -

0 + 0

- 0 +

E2

sum of all positive bonds

sum of all negative bonds

Statistical Bounds

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

Statistical 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.

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

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