Asymptotic form of the lattice Green's function of the square lattice

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Abstract

Asymptotic form of the lattice Green's function of the square lattice for large distance at fixed energy is obtained. The general forms outside and inside the band are shown to be I ∼ (σ1r)-1 2exp(-κ1r) or I ∼ (σ2r)-1 2exp {i[( π 4) + κ2r]}, respectively. Explicit dependences of σ1,2and κ1,2on the energy and the angle are obtained. They take on simple form for (11) and (10) directions. The reciprocal of the correlation length, κ1, goes like a1 2as a approaches 2. It is shown that the line of a constant amplitude outside the band is a circle for a ∼ 2, and a diamondshape for a ∼ ∞ and that of a constant phase inside the band is a circle for a ∼ 2, and a square for a ∼ 0. © 1973.

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Katsura, S., & Inawashiro, S. (1973). Asymptotic form of the lattice Green’s function of the square lattice. Annals of Physics, 77(1–2), 206–215. https://doi.org/10.1016/0003-4916(73)90416-8

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