Usually, they are called supersymmetric states because they preserve all the supercharges, i.e., $Q\mathrm{\Psi }=Q^{†}\mathrm{\Psi }=0$, but we call them BPS states in a sense that they satisfy the BPS bound. Generally, when there are multiple supercharges, not all of them can be preserved at the same time. In fact, in a suitable basis, supercharges ${\tilde{Q}}_i$ should follow the relation $\{{\tilde{Q}}_i,{\tilde{Q}}_i^{†}\}=H-q_i\ge 0$, with central charges $q_i$. Then the BPS bound becomes $H={\max }_i(q_i)$, and states that saturate this bound are called BPS states.

For the classification of general pairs of $(N,f)$, see (6.13) in [20].

The exponential $N$ dependence of the coefficient is chosen in analogy with the generalized Rosenzweig-Porter model [46, 47], where the strength of off-diagonal mixing scales as a power of the Hilbert-space dimension. In our numerics, the smooth transition occurs around $g=1$ due to this $N$ dependence.

If one further resolves the Hilbert space using the additional conserved block quantum numbers of $H_{\mathrm{com}}$, the remaining level statistics are expected to become Poisson. However, these block quantum numbers are not preserved once the $Q_{\mathrm{SYK}}$ is turned on, and hence the same refinement cannot be employed for the deformed model (15).

For $H=\{Q,Q^{†}\}$, the Hodge decomposition implies that every nontrivial $Q$-cohomology class has a unique representative annihilated by both $Q$ and $Q^{†}$. Hence the $Q$-cohomology is isomorphic to the BPS subspace. The same statement holds for $Q^{†}$-cohomology.

Here, BPS chaos is diagnosed through the spectral statistics of the restricted operator $O_{\mathrm{BPS}}$. This diagnostic is distinct from operator growth under Hamiltonian time evolution, which is not studied in this work.

The randomness of $A_i$ alone does not generate random-matrix statistics, since the same probe ensemble yields Poisson-like statistics at the integrable endpoint. A fixed probe such as $O={\psi }_1^{†}{\psi }_1$ gives the same qualitative crossover, although its degeneracies make the spectral statistics less clean.