Bootstrapping symmetries in quantum many-body systems from the cross spectral form factor

Symmetries play a central role in quantum many-body physics, yet uncovering them systematically remains challenging. We introduce a bootstrap framework to reconstruct the representation theory of hidden finite group symmetries of quantum many-body lattice Hamiltonians, using only a known symmetry subgroup $N$ and spectral correlations between its sectors. We introduce the cross spectral form factor (xSFF) as a subgroup-resolved diagnostic of hidden symmetry, computed via exact diagonalization to seed the bootstrap. Applying constraints from these data alongside the algebraic conditions of the fusion rules, our bootstrap sharply restricts candidate groups $G$. Without prior assumptions about $G$, our method can systematically recover its representation-theoretic data, including the number and dimensions of irreducible representations, their branching rules with respect to $N$, the fusion algebra, and the full character table. This framework applies to chaotic and integrable systems and accommodates unitary and anti-unitary symmetries. Through examples, we demonstrate that $G$ can be uniquely identified. Our bootstrap recovers the $\mathbb{Z}_4$ symmetry at the self-dual point of the three-state quantum torus chain, detects signatures of projective representations in the effective Hamiltonian of the driven Bose-Hubbard model, and rediscovers the $η$-pairing $\mathrm{SO}(4)$ symmetry of the one-dimensional Fermi-Hubbard model. Leveraging the universal random-matrix ramp in chaotic systems, we demonstrate the possibility of ramp-only bootstrap alongside an example with hidden commuting symmetries. We establish the xSFF as a diagnostic for conserved order parameters in a symmetry-breaking spin-$1$ chain and introduce an ancilla-assisted measurement circuit for experimental realization. Our framework establishes a practical route to identify symmetries from dynamical spectral observables.