Chaos, thermalization and breakdown of quantum-classical correspondence in a collective many-body system

Abstract

We investigate thermalization and the quantum-classical correspondence in the fully-connected Bose-Hubbard model, focusing on the four-site case. Our analysis of the classical phase-space structure and its excited-state quantum phase transitions leads us to three dynamical regimes: symmetry-breaking low-energy states, an intermediate region where quantum and classical equilibrium states markedly disagree, and a high-energy regime with restored correspondence. The observed classical intermittency above the first excited-state quantum phase transition contrasts with quantum dynamics, which remains trapped in symmetry-breaking sectors despite the existence of a classically connected phase space. This mismatch originates from the population of imbalance-carrying eigenstates and persists even for relatively large number of particles. Our results reveal unexpectedly slow convergence to the classical limit, signaling robust finite-size effects in collective many-body dynamics.

Figures (7)

• Figure 1: Real and imaginary parts of the classical generalized imbalance function, $I(\mathbf{q},\mathbf{p})$, in Eq. \ref{['eq:imbalanceclassical']} for initial conditions $(\mathbf{q},\mathbf{p})\in\Omega$ at different energies, $\epsilon=H(\mathbf{q},\mathbf{p})$. (a) $-6.17\leq \epsilon\leq -6.13$. (b) $-6.07\leq \epsilon\leq -6.03$. (c) $-5.37\leq \epsilon\leq -5.33$. (d) $-4.57\leq \epsilon\leq -4.53$. All results are for $\mathcal{N}=4$, $J=1$ and $U=-10$.
• Figure 2: (a)-(d) Real and imaginary parts of the generalized imbalance operator, Eq. \ref{['eq:imbalanceop']}, diagonalized in the respective energy eigenspaces, for $J=1$, $U=-10$ and $N=75$. The eigenvalues $\epsilon=E/N$ are chosen in the following energy ranges: (a) $-7.1\leq \epsilon\leq -6.9$. (b) $-5.9\leq \epsilon\leq -5.7$. (c) $-5.3\leq \epsilon\leq -5.1$. (d) $-3.9\leq \epsilon\leq -3.7$. (e) Number of eigenvalues with non-zero absolute value for different $N$. The threshold $|i_{n}|^{2}>10^{-3}$ is taken for the numerical computation.
• Figure 3: Eigenvalues of the generalized imbalance operator $\hat{I}$, defined in Eq. \ref{['eq:imbalanceop']}, in the $4\times 4$ reduced eigenspaces $\{E_{n,r}\}$ with $r=0,\pi/2,\pi,3\pi/2$, as a function of energy, $i_{n}$. Dashed orange lines represent the various ESQPT critical energies, together with the ground-state (first line) and the maximum energies (last line). Model parameters are $J=1$, $U=-10$, and $N=55$.
• Figure 4: Quantitative analysis of chaos in the 4-site BH model. The blue line shows the classical fraction of regularity, $f_{\textrm{reg}}$, obtained from the Lyapunov exponent, Eq. \ref{['eq:lyapunov']}, calculated for $10^{4}$ initial conditions at each given energy. The green lines shows the fraction of regularity as obtained from the Berry-Robnik distribution, Eq. \ref{['eq:psberryrobnik']} for different system sizes: $N=75$ (light green), $101$ (green) and $125$ (dark green). Dashed orange lines represent the different ESQPT critical energies as well as the ground-state and maximum energy values.
• Figure 5: Time evolution of the occupation number for classical trajectories with initial conditions $(\mathbf{q},\mathbf{p})$ at fixed energies $\epsilon=H(\mathbf{q},\mathbf{p})$ defined by Eq. \ref{['eq:classicalH']} (blue) and time evolution for the coherent state Eq. \ref{['eq:coherentstate']} (red) with the same $(\mathbf{q},\mathbf{p})$. For each panel we represent the time evolution $\langle \hat{n}_{\max}(t)\rangle$ of the number particle operator of the well with the initially largest number of particles. The green lines represent the average of 2000 classical trajectories with initial coordinates $(\mathbf{q},\mathbf{p})$ taken from a Gaussian distribution centered at the same coordinate values as the quantum coherent states and with a width $\sigma=1/\sqrt{2N}$ with $N=55$. Three different cases are selected: (a) $\epsilon=-7.05<\epsilon_{c1}$ below the first ESQPT, with $(\mathbf{q},\mathbf{p})\approx ((0.086,-0.171,-0.9,0.151),(-0.38,0.392,0.911,-0.048))$, (b) $\epsilon=-5.85>\epsilon_{c1}$ slightly above the first ESQPT, with $(\mathbf{q},\mathbf{p})\approx ((0.211,0.393,-0.221,0.224),(-0.623,-0.022,0.045,-1.145))$, and (c) $\epsilon=-3.65$ significantly above it, with $(\mathbf{q},\mathbf{p})\approx ((0.306,-0.948,0.289,-0.01),(-0.58,0.652,-0.373,0.151))$. Model parameters are $J=1$, $U=-10$, $N=55$.