AI Uncovers Hidden Symmetries in Quantum Systems

Symmetries are fundamental to understanding quantum many-body systems, governing everything from energy spectra to phases of matter. However, many symmetries remain hidden, not directly apparent in the Hamiltonian, making them challenging to identify systematically. A new bootstrap framework now offers a practical route to uncover these hidden symmetries directly from dynamical spectral observables, bridging the gap between coarse detection and precise algebraic identification.

Researchers have developed a that reconstructs the representation theory of hidden finite group symmetries using only a known symmetry subgroup and spectral correlations between its sectors. This approach introduces a novel variant of the spectral form factor, called the cross spectral form factor (xSFF), computed via exact diagonalization. By combining constraints from this data with algebraic conditions of fusion rules, the bootstrap procedure sharply restricts candidate symmetry groups, recovering key representation-theoretic data such as irreducible representation dimensions and character tables.

Ology relies on the xSFF, which generalizes the standard spectral form factor from auto-correlations within a single symmetry sector to cross-correlations between distinct sectors. The late-time averaged plateaus of the xSFF are governed by branching rules, providing numerical constraints on branching multiplicities. These constraints, alongside fusion algebra conditions, seed an algorithm that systematically extracts the full symmetry group's structure without prior assumptions about the group itself.

From applying this bootstrap to various quantum many-body lattice models demonstrate its effectiveness. For instance, in an S3-invariant chain with only a Z3 subgroup known, uniquely identified the full S3 symmetry, including its branching matrix and character table. Similarly, in a Kennedy-Tasaki transformed spin-1 model, it recovered the hidden D4 symmetry from a manifest V4 subgroup, even when the symmetry was encoded in a non-local transformation. The approach also handled cases with higher branching multiplicities, such as an extended Ashkin-Teller model at the Potts point, correctly identifying S4 symmetry.

Of this work are significant for both theoretical and experimental physics. By enabling the systematic identification of hidden symmetries from spectral data, it provides a tool for exploring complex quantum systems where symmetries are not manifest. This could aid in classifying phases of matter, understanding phase transitions, and analyzing low-energy behavior in strongly correlated systems. applies equally well to chaotic and integrable systems and accommodates both unitary and anti-unitary symmetries, broadening its applicability.

Despite its strengths, the framework has limitations. It currently focuses on finite groups, though extensions to compact Lie groups are discussed but remain computationally challenging due to the infinite number of irreps. Additionally, while can detect projective representations, extracting their full fusion algebra is an open problem. The bootstrap also assumes the known subgroup is part of the full symmetry group, which may not always hold, and unique identification of the group up to isomorphism is not guaranteed in all cases, though it works effectively in the examples studied.