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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

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
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
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.

slide5

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.
s parameter tendency for a gap opening at the middle of the spectrum
S parameter= tendency for a gap opening at the middle of the spectrum

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,

slide7

P(Z) is a BINOMIAL distribution.

Symmetric DOS, BIPARTITE LATTICE

slide9

Frustration in a renormalized Hamiltonian

RENORMALIZATION

Since H produces a hop between sublattices:

slide10

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

slide11

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:

slide12

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.

slide13

(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.