Finite-Size Scaling Analysis of the Poisson-to-GUE Transition in Finite Quantum Systems: Numerical Determination of the Critical Exponent and Thermodynamic Limit

We present a comprehensive numerical investigation of the transition from integrable (Poisson) toquantum chaotic (Gaussian Unitary Ensemble, GUE) spectral statistics in a finite-dimensional quan-tum model. Using the nearest-neighbor gap ratio r as a probe, we analyze the dependence of spectralcorrelations on a coupling parameter χ. A rigorous Finite-Size Scaling (FSS) analysis is performedacross system dimensions ranging from D = 512 to D = 16384, with 30 independent realizations perconfiguration. The raw numerical data obtained from extensive matrix diagonalizations reveal thatthe critical coupling χc(D) scales as χc(D) = χc(∞) + A · D−ν . Through a non-linear power-lawfit to the simulation data, we extract a thermodynamic critical point of χc(∞) = 28.14 ± 0.99 anda critical exponent ν = 0.693 ± 0.123. The derived exponent places the transition within the 3DAnderson localization universality class. This work establishes a robust, data-driven methodology foridentifying quantum phase transitions through spectral statistics.