Golden-ratio hierarchy of critical wave-function nodes

Establish the conjectured recursive golden-ratio organization of the zero positions of wave functions in a one-dimensional quasicrystal chain: primary nodes should occur at positions x_n = N alpha^{r_n} or x_n = N(1 - alpha^{r_n}), where alpha is the irrational quasiperiodic parameter and r_n are positive integers, while zeros between adjacent primary nodes and at higher orders should obey the corresponding recursively defined self-similar rules.

Background

At the localization-critical point of the nonreciprocal Aubry-André-Harper model, the authors observe a deterministic fractal structure in the real-space wave-function profile. The propagating wavepacket front consists of nested triangular segments, and the associated zeros or nodes appear to be arranged according to the golden-ratio quasiperiodicity.

The authors formulate a conjecture for the node positions in a chain of length N. Primary nodes are proposed to occur at normalized positions determined by powers of the irrational parameter alpha, with secondary and higher-order nodes generated recursively inside the intervals between neighboring primary nodes. Numerical data for N = 987 are reported to agree with the conjectured primary and secondary node positions, but the recursive structure is presented explicitly as a conjecture.

References

Based on the above numerical observations, we propose the following conjecture for the zero positions of the wave function in a 1D quasicrystal chain of length N: x_n = N \alpha{r_n} \quad \text{or} \quad x_n = N (1 - \alpha{r_n}), where \alpha is the irrational number in Eq.~eq4 and r_n are positive integers. These zeros are termed primary nodes. In light of the fractal nature of the wave function, the distribution of zeros within any interval between two adjacent primary nodes follows the same rule, i.e., x_{nm} = x_n + N_n \alpha{r_m} \quad \text{or} \quad x_{nm} = x_n + N_n (1 - \alpha{r_m}), with N_n = |x_{n+1} - x_n| being the distance between neighboring primary nodes. Such zeros are referred to as secondary nodes. Higher-order nodes can be defined recursively in an analogous manner, thereby fully characterizing the multifractal structure of critical wave functions.

— Quench dynamics in nonreciprocal Aubry-André-Harper model  (2609.10342 - Pei et al., 9 Sep 2026) in Section “Anomalous critical dynamical scaling,” paragraph beginning “Based on the above numerical observations, we propose the following conjecture”