Bootstrapping Symmetries in Quantum Many-Body Systems from the Cross Spectral Form Factor

Quantum Physics [Submitted on 1 Apr 2026] Bootstrapping Symmetries in Quantum Many-Body Systems from the Cross Spectral Form Factor Chen Bai, Zihan Zhou, Bastien Lapierre, Shinsei Ryu 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. Comments: 47 pages, 10 figures Subjects: Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th) Cite as: arXiv:2604.01296 [quant-ph]   (or arXiv:2604.01296v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.01296 Focus to learn more Submission history From: Chen Bai [view email] [v1] Wed, 1 Apr 2026 18:02:36 UTC (3,815 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 Change to browse by: cond-mat cond-mat.stat-mech cond-mat.str-el hep-th References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)