Data and Code for: Quenched Properties of the Spectral Form Factor

Abstract

These files serve to comply to the data availability statement for the publicpation of (https://arxiv.org/abs/2509.14406) to PRE. The 'Local Minima' python notebook is the code used to (after choosing/generating the desired spectrum) compute the SFF and the corresponding Y variable defined in Eq. (16) of the text, from which its empirical distribution shown or various cases (Figures 1,2,6,7,10,11 and A3 in the text) is readily obtained. Finally, the notebook also includes the computation of the local minima discussed in the beginning of Section V. The data can be then plotted as is done in Figures 8 and 12 in the text. The .npy files '3003_FORSFF.npy', '792_FORSFF.npy', etc. are examples of generated spectra for the Hermitian Model defined in Eq. (23) of the text. Each file is a NumPy array containing energy spectra for many independent disorder realizations of the model at fixed system size. The array has shape (n_samples, D), where n_samples is the number of realizations and D is the Hilbert-space dimension. Each row corresponds to one realization, and each column corresponds to an energy level; within each row, the eigenvalues are real and sorted in ascending order. From these spectra, the distribution of the local energy gap ratios defined in Eq. (19) can be empirically obtained as is shown in Fig. 4 of the text. Using these spectra, the Spectral Form Factor (Eq (1) and its quenched and annealed counterparts (Eq.s (7) and (8) respectively) are obtained. The corresponding data examples are given in 'D3003_sff_L_14_logtimes' and D792_sff_L_12_logtimes''. Each CSV file contains ensemble-averaged spectral form factor data computed from the corresponding eigenvalue file. Rows correspond to different time values, and columns store the time (time), the averaged spectral form factor (K), its disconnected part (K_disc), the connected contribution (K_conn), and the ensemble average of the logarithm of the spectral form factor (mean_lnSFF). The results for the quenched SFF are shown in Fig. 5 of the text The various DSFFs (Eq, (5)) for the Dissipative XY model defined in Eq. (25) are obtained using samples found in the file 'XY_spin_L_14_dim_3003_g_0.4.npy'. The file contains complex energy spectra of a bosonic lattice Hamiltonian for many independent disorder realizations at fixed model parameters. The stored array has shape (n_samples, dim), where n_samples is the number of disorder realizations and dim is the Hilbert-space dimension of the fixed-particle-number sector. Each row corresponds to one realization of the random onsite potential, and each column corresponds to one eigenvalue of the Hamiltonian; the eigenvalues are complex and are not sorted. With these, the empirical distributions of the Complex Spacing ratios (Figures 13 and 14) can be obtained. The DSFF data is saved in the file 'sff_data_XY_3003_g_0.4.npz'. It stores a one-dimensional array of time values (times), a one-dimensional array of projection angles (phis), and the Hilbert-space dimension (D). For each angle, the file includes the sample-averaged spectral form factor (mean_SFF_label), the sample-averaged complex partition sum (mean_Z_label), and the sample average of the logarithm of the spectral form factor (mean_log_SFF_label), all stored as two-dimensional arrays with rows corresponding to different angles and columns corresponding to different times. Similar Constructions are available for the standard RMT spectra and the Random Energy Model