1. 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

2. 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: Model: atoms at the vertex of the tiling, 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 nodal lines have a fractal structure.There are “confined states” at E=0 (10%). The 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 than2, where =4, as in a square lattice.The bandwidth is bigger than 2, where =4, as in a square lattice. From computer simulations, it is belived that there are critical, extended and localized statesFrom computer simulations, it is belived that there are critical, extended and localized states

3. 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,    and  B =  where  tends to infinity.

5. The DOS is symmetric around E=0.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.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.A gap is formed around E=0, EVEN WHEN A-ATOMS PERCOLATE. States tend to be more localized around E=0.States tend to be more localized around E=0. The bandwidth is bigger than2.The bandwidth is bigger than 2. In 2D, all states are localized (scaling theory of Abrahams), although power-law decaying states can change the picture.In 2D, all states are localized (scaling theory of Abrahams), although power-law 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,

6. 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,

7. P(Z) is a BINOMIAL distribution. Symmetric DOS, BIPARTITE LATTICE

9. Frustration in a renormalized Hamiltonian RENORMALIZATION Since H produces a hop between sublattices:

10. FRUSTRATION + + + Bonding + - + - + - + - + Anti- Bonding + 0 - 0 + 0 - 0 + E2E2 + + + Bonding state Lifshitz tail Degenerate states Compression of the band + - + - + - + - + Anti- Bonding + + Frustrated bond Rises the energy +1 - E=-1-1+1

11. If c i (E) is the amplitude at site i for an energy E, from the equation of motion: sum of all negative bonds sum of all positive bonds Statistical Bounds

12. 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.

13. Where f 0 (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)