Density of states and frustration in the quantum percolation problem

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: • 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

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

5. 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.

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. 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

11. 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:

12. 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.

13. (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.