Logarithmically Correlated Landscapes and Localization in Non-Hermitian Quasicrystals

Abstract

We study a one-dimensional Hatano-Nelson ring whose nonreciprocal hopping is quasiperiodically modulated through zero. A gauge transformation maps every eigenstate onto a single spatial envelope, whose logarithm becomes a deterministic, logarithmically correlated field once the hopping vanishes along the quasiperiodic orbit. We derive an exact Fourier representation and prove that the landscape variance grows logarithmically with system size, with a stiffness set by the modulation power and the arithmetic of the incommensurate frequency. The extended phase terminates at an algebraic boundary given by Jensen's formula. Inside the singular regime, localization requires the stiffness to exceed a critical threshold: above it, the wavefunction weight concentrates on the few highest landscape peaks and the state is localized; below it, the weight spreads over too many competing peaks and the state is a critical multifractal. The extreme-value statistics are anomalous: peak gaps grow as a power of the logarithm of rank, with an exponent that encodes the continued-fraction type of the frequency, distinct from the Anderson, Aubry-André, and random-gauge universality classes. The stiffness adds across channels in multiband lattices, and all signatures survive percent-level component disorder, placing the mechanism within reach of nonreciprocal topolectrical circuits.