Topological Genesis of Quantum Chaos: Multifractality and the Poisson–GOE–Poisson Crossover in Anderson Lattices

Abstract

𝐃𝐞𝐬𝐜𝐫𝐢𝐩𝐭𝐢𝐨𝐧 We present numerical evidence that macroscopic quantum chaos can emerge without nonlinear dynamics in the evolution equation. Instead, it appears as a continuous phase transition driven solely by the topological degradation of the underlying spatial substrate. 𝐊𝐞𝐲 𝐟𝐞𝐚𝐭𝐮𝐫𝐞𝐬 • Two-dimensional toroidal mesh with N = L × L sites (L = 20–50).• Tight-binding Hamiltonian with uncorrelated Anderson disorder (W as the primary control parameter).• Independent 10% bond-percolation channel confirming that the observed behavior is a generic consequence of connectivity loss rather than a disorder-model artifact. 𝐌𝐚𝐢𝐧 𝐫𝐞𝐬𝐮𝐥𝐭𝐬 The perfect torus displays a symmetry-induced spectral degeneracy, numerically close to Poisson statistics. Weak-to-moderate disorder (W ≈ 1–4) drives the spectrum into the Gaussian Orthogonal Ensemble (GOE) with ⟨r⟩ ≈ 0.530–0.534. Strong disorder (W ≳ 8) fragments the topology, isolates nodes, and restores genuine Poisson statistics (⟨r⟩ → 0.41–0.39). Finite-size scaling of the participation radius supports an Anderson-transition-like critical region near W ≈ 4–6. In this crossover regime, eigenstates become strongly multifractal, with monotonically decreasing generalized dimensions Dq and a broad asymmetric f(α) spectrum peaking near α ≈ 2.2–2.3. These findings suggest that Wigner–Dyson level repulsion and algorithmic chaos are thermodynamic consequences of spatial frustration. Because the same phenomenology is reproduced by a hardware-realistic percolation channel, the identified critical threshold (W ≈ 4–6) provides a quantitative failure boundary for distributed tensor-processing interconnects, with direct implications for the design of fault-tolerant AI accelerator hardware.