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

TL;DR

This paper introduces a bootstrap method using spectral correlations to systematically uncover hidden symmetries and their representation theory in quantum many-body systems, without prior knowledge of the full symmetry group.

Contribution

The authors develop a novel bootstrap framework that reconstructs the full symmetry group and its representations solely from spectral data and a known subgroup, applicable to various quantum systems.

Findings

Successfully recovers the full symmetry group and its representations from spectral data.

Identifies the $ ext{Z}_4$ symmetry in the quantum torus chain example.

Detects the $ ext{SO}(4)$ symmetry in the Fermi-Hubbard model.

Abstract

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.