Diffraction induced quantum chaos in a one-dimensional Bose gas

TL;DR

This work studies how a localized delta impurity breaks the integrability of the one-dimensional Lieb–Liniger Bose gas and induces quantum chaos. Using exact diagonalization in a truncated Bethe basis and analytic impurity form factors, the authors analyze level-spacing statistics, participation ratios, and eigenstate structure for two- and three-particle sectors. They find that the low-energy spectrum exhibits random-matrix statistics, with a parity-dependent pattern for two particles (odd sector effectively integrable, even sector chaotic) and chaotic behavior in both parity sectors for three particles, while high-energy behavior can revert toward quasi-integrable or Brody-like statistics. The central mechanism is diffraction off the impurity, which qualitatively alters scattering and violates integrability, providing a diffraction-driven route to chaos with implications for thermalization, transport, and entanglement in 1D ultracold gases.

Abstract

We investigate the Lieb--Liniger model of interacting one-dimensional bosons coupled to a localized impurity, modeled by a delta barrier. While the Lieb--Liniger gas is integrable, the impurity breaks integrability and induces a transition towards quantum chaos. We show that the low-energy spectrum exhibits random-matrix statistics, in striking contrast to the Bohigas--Giannoni--Schmit conjecture, where chaotic behavior typically emerges at high energy. For two bosons, the odd-parity sector remains integrable, whereas the even-parity sector displays clear signatures of chaos at low energy and a crossover back to quasi-integrable behavior at higher energies. For three bosons, both parity sectors exhibit spectral statistics close to chaos at low energy. We argue that this unconventional form of many-body quantum chaos originates from diffractive processes induced by the impurity.

Figures (9)

• Figure 1: Energy spectrum (first 20 levels in units of $\hbar^2/mL^2$) for two particles and $\gamma=10$ as a function of the barrier coupling strength $\gamma_B$.
• Figure 2: Level spacing distribution of the unfolded spectrum for two particles. The histogram is computed over the lowest $N_\ell=40$ energy levels for different $\gamma$ and $\gamma_B$ in the range [5-10]. In total 4400 energy levels are used to compute these histograms. The levels are sorted by parity and the numerical histograms are compared to the Poisson and Wigner--Dyson distributions.
• Figure 3: Level spacing distribution of the unfolded spectrum for two particles. The histogram is computed over the lowest $N_\ell=40$ energy levels for $\gamma=10$ and $\gamma_B$ in the range [0.1-0.3]. In total 1000 energy levels are used to compute these histograms. The levels are sorted by parity and the numerical histograms are compared to the Poisson and Wigner--Dyson distributions.
• Figure 4: Participation ratio of the eigenstates of hamiltonian (\ref{['eq_H_LL']}) for $N=2$, $\gamma=10$ and $\gamma_B=6$. Red circles correspond to the odd parity states and blue diamonds to the even parity states. Energy is in units of $\hbar^2/mL^2$.
• Figure 5: Modulus square of the wave function $|\alpha^{(n)}_{\vec{\lambda}}|^2$ for two neighboring states in energy but with different parity. State $n=16$ is odd and is mainly localized on a single Bethe state. State $n=17$ is even and spreads uniformly over the constant energy shell defined as $E=\frac{\hbar^2}{2m}\left(\frac{2\pi}{L}\right)^2 (I_1^2+I_2^2)$ (red dashed line) in Bethe space (here $E\simeq 55\hbar^2/mL^2$). The gray line displays the bosonic symmetry line under permutation of Bethe numbers.