Classical fixed-point bifurcations and quantum crossovers in coupled-top systems with standard and nonstandard symmetries

We study the classical and quantum dynamics of coupled-top systems both in the absence and in the presence of nonlinear torsion acting on the individual tops. In the torsion-free case, the system reduces to two identical coupled tops, known as the Feingold-Peres (FP) model, which preserves both permutation and chiral symmetries and belongs to the BDI (chiral-orthogonal) symmetry class, a representative of the nonstandard symmetry classes. For nonzero torsion (NZT), we consider two configurations: (I) identical torsional directions for the two tops and (II) opposite torsional directions. Configuration (I) preserves permutation symmetry but breaks chiral symmetry and therefore falls within the standard Wigner–Dyson symmetry classes, whereas configuration (II) breaks permutation symmetry while preserving chiral symmetry and thus belongs to the nonstandard BDI symmetry class. At the classical level, all these models exhibit bifurcations of fixed points as the coupling between the tops is varied. We investigate how these classical bifurcations manifest in the quantum regime through crossovers in several quantum-mechanical quantities, including the energy eigenvalues, the localization properties of the eigenstates characterized by the Shannon entropy, and the bipartite entanglement between the two tops. Our results show that, for models in the BDI symmetry class, the crossovers between the ground and excited states occur symmetrically, reflecting the underlying chiral symmetry. In contrast, in a model lacking chiral symmetry, the corresponding crossovers occur at different coupling strengths. These results demonstrate that classical fixed-point bifurcations leave clear signatures in the quantum properties of coupled-top systems and highlight the important role of symmetry in shaping the resulting quantum crossovers.