=============================================================================== HIERARCHICAL ANALYTICAL APPROACH TO UNIVERSAL SPECTRAL CORRELATIONS IN BROWNIAN QUANTUM CHAOS =============================================================================== The Mathematica notebook 'hierarchy.nb' generates the results described in the text. Consequently, the data has been saved in '*.dat' form. The notebook (data) is appropriately organised in chapters/sections (folders/subfolders). The notebook has sufficient detail to be understood in conjunction with the text. It also includes additional material. We briefly review its contents now. In the first notebook chapter, 'SFF --- FIG1', we plot the spectral form factor (SFF) in the three standard symmetry classes using the analytic expressions from J. Phys. A 29, 3641 (1996). This chapter produces FIG1 but does not correspond to any data since we plot the analytic functions. The next chapter 'spectral hierarchy (analytics)' again does not correspond to any data, and is split into two parts. It first organises an analytical treatment of DBM in the unitary symmetry class, with dynamics generated by instantaneous Hamiltonians sampled from the GUE. We write down and solve the system of first-order linear differential equations for n=3, and then generate these equations using a brute-force approach that combines and partitions indices. (Although this implementation is not very tidy, we later numerically reproduce the detailed oscillations of the SFF for n<=6.) Here, we also define the analytic expressions for the out-of-time-ordered correlator (OTOC), and we produce the plots for the uniform asymptotics (FIG3) and all spectral correlators for n=2 (FIG10 in Appendix D). The second part of the chapter is an analytical treatment of DBM in the orthogonal symmetry class, where Hamiltonians are sampled from the GOE. (This is done for low orders of the hierarchy since the number of equations proliferates much more rapidly than for the unitary/GUE counterpart.) The chapter 'DBM numerics' generates spectral data for the DBM process within the three symmetry classes (generated by instantaneous Hamiltonians sampled from the GOE, GSE, and GUE). All data generated in the first four sections is found within 'brownian/dbm/hierarchy/g*e', where the asterisk is either 'o','s', or 'u', accordingly. The dynamics are generated by instantaneous Hamiltonians whose matrix dimensions are chosen according to their (B)SYK symmetry analogues. Here, we produce FIG2 that contrasts numerical data and analytics for DBM dynamics generated by GUE Hamiltonians. This is to be later contrasted with BSYK-generated dynamics in the unitary symmetry class, so dynamics are generated for system size N=16. FIG4 and FIG5 in the main text contrast the analytics for the unitary symmetry class with numerics and analytics for the orthogonal and symplectic symmetry classes, respectively. For a meaningful comparison between these symmetry classes, data is generated for Hamiltonians sampled from the GOE and GSE with N=8. We later use the data generated by Hamiltonians sampled from the GOE for direct comparison with BSYK-generated data for N=8 in FIG7. We must also generate data for the symplectic symmetry class using GSE Hamiltonians with N=32 for later comparison with BSYK-generated dynamics in FIG8. The final (fifth) section of this chapter then calculates the OTOC and produces FIG9(a), saving the respective data within 'brownian/dbm/otoc/g*e'. The chapter 'BSYK numerics' is the BSYK analogue of the previous chapter. Here, we implement the BSYK model as described in the main text and include some additional material, e.g., the DOS and SFF in the regular SYK model and averaging properties of the BSYK Hamiltonian. All SFFs (OTOC functions) calculated using BSYK-generated data are saved within 'brownian/bsyk/hierarchy/*' ('brownian/bsyk/otoc/*'), where the asterisk distinguishes the respective symmetry class, i.e., 'orthogonal', 'symplectic', or 'unitary'. Note that we plot figures as a function of the rescaled time (\kappa t). We take the data for times as required (i.e., up to \kappa t ~ 10) when producing the figures. We then have all the data needed to produce FIG6, FIG7, FIG8, and FIG9(b) in the main text. =============================================================================== RESULTS =============================================================================== There are two folders corresponding to the two chapters that generate numerical data. The corresponding file names are given as comments in the notebook. Ten thousand (10^4) independent random realisations were run to produce all given numerical results unless otherwise indicated (10^3 for the BSYK unitary symmetry class in FIG6, BSYK symplectic symmetry class in FIG8, and BSYK orthogonal symmetry class in FIG9(a), and 10^2 for the BSYK unitary and symplectic symmetry classes in FIG9(b)). All data is generated for time steps of duration dt=0.01 (sqrt(dt)=0.1). The 'hierarchy' and 'otoc' folders contain subfolders corresponding to each of the three symmetry classes. These are then arranged by system size N (or Majorana fermions M for the BSYK OTOC). The data files can be imported appropriately to reproduce the plots (see comments in the notebook). Note the possible discrepancy in length of data files depending on the model/symmetry class.