Quantum three-rotor problem in the identity representation

TL;DR

This work analyzes the quantum three-rotor system, a minimal model of coupled Josephson junctions, to uncover quantum signatures of the classical order-chaos-order dynamics. By canonical quantization, separating center-of-mass and relative variables, and exploiting the $S_3\times\mathbb{Z}_2$ symmetry, the authors desymmetrize the spectrum and study universal spectral statistics within the identity representation. They derive Weyl-type DOS estimates, construct a group-averaged basis for the identity sector, and perform large-scale numerical diagonalization to reveal Poisson-to-Wigner-Dyson transitions in spacings, along with nonuniversal features such as oscillator-like pockets at low energies and saturation in the number variance. The results highlight how symmetry-aware desymmetrization and energy-window analysis yield clear quantum hallmarks of chaos in a fully interacting, finite system, with implications for chaos diagnostics in other symmetry-rich quantum models.

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.

Figures (10)

• Figure 1: (a) Variational ground state energies of $\hat{H}$ (blue dots) as a function of $\tilde{g}$ based on the ansatz $\psi \propto e^{-aV}$ are very close to the numerical values (red squares). Optimal values of the variational parameter $a$ are also shown (black stars). (b) The variational ground state normalized probability density on the $-\pi \leq \varphi_1, \varphi_2 \leq \pi$ torus for the intermediate value $\tilde{g} \approx 0.3$ where $a = 0.3$. The state is weakly localized around the minimum $(0,0)$ of the potential $V$ (\ref{['e:Hamiltonian-total-CM-rel']}).
• Figure 2: Eigenstates of kinetic energy $T$, represented as $(m, n)$ pairs on the $\mathbb{Z} + \omega \mathbb{Z}$ lattice on the complex plane for $\omega = e^{2\pi i/3}$. Rotations by elements of the cyclic group $C_6 = \langle e^{i \pi/3} \rangle$ are symmetries of the lattice that preserve the distance $|m + n \omega|$ from the origin (and hence preserve the eigenvalues of $T$). So the cardinality of each orbit must be divisible by $6$ implying that the degeneracies are multiples of 6. Dashed (red) and solid (blue) arrows point to two such orbits corresponding to energies $T = 1$ and $T=3$.
• Figure 3: Lowest 100 numerically obtained eigenvalues of $\hat{H}$ (red, lower) follow a pattern similar to that of the 2d harmonic oscillator (blue, higher) provided $\tilde{g}$ is large.
• Figure 4: Numerically computed number of energy levels $N(\hat{E}, \tilde{g})$ below energy $\hat{E}$ for $\tilde{g} = 300$ (blue dots) agrees well with semiclassical estimate of Eq. (\ref{['e:weyl-formula-phi12-integ']}) (black dashes for $\hat{E} > 4.5 \tilde{g}$ and black circles for $\hat{E} < 4.5 \tilde{g}$). Asymptotically, $N \sim (2\pi/\sqrt{3})(\hat{E} - 3 \tilde{g})$.
• Figure 5: Nearest neighbor spacing histograms (binwidth 0.1 for (a-b) and 0.3 for (c-g)) in various energy windows for $\tilde{g} = 6000$ [in (g) and (h) we take $\tilde{g} = 1000$ and $\tilde{g} = 300$ due to inadequate high energy data for $\tilde{g} = 6000$]. At low energies the spacing distribution in (b) is a perturbation to that of a desymmetrized/purified 2d harmonic oscillator (see Section \ref{['s:harmonic-approx-purification-spacing']}) shown in (a) which displays a large number of nearly degenerate levels. In (c-h) the spacing distributions are compared with exponential (black dotted), Wigner surmise (red dashed) and Brody (blue dot-dashed) distributions showing an order-chaos-order transition. The fitted Brody parameter $\nu$ to one significant digit is indicated. Semiclassical expectations are seen to hold even when $\tilde{g}$ is reduced to $300$.