Hartree Fragmentation and Long-Range Pair Networks in Aperiodic Chains

Abstract

For fermions in a slowly varying aperiodic potential with random interactions $V_{ij}/|i-j|^{\alpha}$, the potential fixes the resonance supply $N_0 \propto L^{2-n}/h$. On crossing ${\alpha}-2n$, wherever Hartree fragmentation is operative, the resonant-object ensemble reconstructs: turning-point runs fragment into clusters whose intrinsic matching law resolves no logarithmic singularity, while isolated wing bonds survive at finite density and restore a reduced logarithm. The scaling $x^{4-2n-{\alpha}}_{pair} ln{x_{pair}} \propto h^2/(V t0)$ is unchanged. Random coefficient fluctuations sustain the $R^{-{\alpha}}$ coupling; uniform ones cancel to $R^{-{\alpha}-2}$.