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 designed 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 symmetry sectors. We introduce a novel variant of the spectral form factor, the cross spectral form factor (xSFF), which we compute via exact diagonalization to seed the bootstrap algorithm. By applying the constraints derived from these data alongside the algebraic conditions of the fusion rules, our bootstrap procedure sharply restricts the set of candidate groups $G$. Remarkably, without any prior assumptions regarding the full symmetry group $G$, our method can systematically recover its representation-theoretic data, including the number and dimensions of the irreducible representations, their branching rules with respect to $N$, the fusion algebra, and the full character table. This framework applies equally well to chaotic and integrable many-body systems and accommodates both unitary and anti-unitary symmetries. Through various examples, we demonstrate that the underlying group $G$ can be uniquely identified. In particular, our bootstrap independently 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 $\eta$-pairing $\mathrm{SO}(4)$ symmetry of the one-dimensional Fermi-Hubbard model. Our framework thus establishes a practical route to identify symmetries directly from dynamical spectral observables.