Quantum three-rotor problem in the identity representation

Abstract

The quantum three-rotor problem concerns the dynamics of 3 equally massive particles moving on a circle subject to pairwise attractive cosine potentials and can model coupled Josephson junctions. Classically, it displays order-chaos-order behavior with increasing energy. The quantum system admits a dimensionless coupling with semiclassical behavior at strong coupling. We study stationary states with periodic `relative' wave functions. Perturbative and harmonic approximations capture the spectrum at weak coupling and that of low-lying states at strong coupling. More generally, the cumulative distribution of energy levels obtained by numerical diagonalization is well-described by a Weyl-like semiclassical estimate. However, the system has an $S_3 \times Z_2$ symmetry that is obscured when working with relative angles. By exploiting a basis for invariant states, we obtain the spectrum restricted to the identity representation. To uncover universal quantum hallmarks of chaos, we partition the spectrum into energy windows where the classical motion is regular, mixed or chaotic and unfold each separately. At strong coupling, we find striking signatures of transitions between regularity and chaos: spacing distributions morph from Poisson to Wigner-Dyson while the number variance shifts from linear to logarithmic behavior at small lengths. Some nonuniversal features are also examined. For instance, for strong coupling, the number variance saturates and oscillates at large lengths while the spectral form factor displays a nonuniversal peak at short times. Moreover, deviations from Poisson spacings at asymptotically low and high energies are well-explained by quantum harmonic and free-rotor spectra projected to the identity representation at strong and weak coupling. Interestingly, the degeneracy of free-rotor levels admits an elegant formula that we deduce using properties of Eisenstein primes.