Excited-state quantum phase transitions and chaos in a three-level Lipkin model

Abstract

Excited-state quantum phase transitions (ESQPTs) have been extensively studied in two-level models, but their characterization remains challenging in systems displaying mixed regular and chaotic dynamics. In this work, we investigate ESQPTs within the three-level Lipkin-Meshkov-Glick model, where an enlarged Hilbert space and multiple separatrices give rise to rich spectral structures strongly influenced by chaos. To investigate the different dynamical regions, we have calculated Poincaré sections and Peres lattices. In addition, by combining chaos-sensitive measures with standard ESQPT diagnostics, we provide a static analysis of ESQPT signatures in this model and establish a robust framework for future studies of its dynamical behavior. The degree of chaos and the Kullback-Leibler divergence are found to be very effective chaos-sensitive measures, which are complementary to ESQPT diagnostics such as the mean field limit and the participation ratio. Hence we provide a standard framework to work with ESQPTs in chaotic three-level systems.

Figures (10)

• Figure 1: Schematic image of the 3-level Lipkin model used in this work, described by Hamiltonian (\ref{['hamU3old']}).
• Figure 2: Energy density (energy per particle $E/N$) of the symmetric representation of the 3-level LMG model as a function of the control parameter $\lambda$, for $N=15$ (top) and $N=100$ (bottom) particles. In both pictures, the thin lines represent the numerical energies, whose colors describe the parity sector in which they have been calculated. The dashed-dotted black line is the variational ground state \ref{['energysym']}, whose convergence to the numerical GS increases with $N$. The thick colored lines in the bottom spectrum represent the most relevant ESQPT separatrices \ref{['Separatrix1']}, \ref{['Separatrix2']}, while the rest are given in thin black lines \ref{['Separatrices']}.
• Figure 3: Histogram of 100 bins representing the density of states of the 3-level LMG model for $N=100$ particles and for some representative values of the control parameter $\lambda$ in the three QPT regions. The solid black line is the smoothed DOS with a MA window size of 15 elements. The dashed vertical lines represent the ESQPT separatrices \ref{['Separatrix1']}, and \ref{['Separatrix2']}.
• Figure 4: Poincaré sections of the energy surface in the position and momentum space \ref{['eq:classical_H']}, for a fixed value of the control parameter $\lambda=0.98$ and some representative values of the energy $E$ inside each dynamical region \ref{['DynamicalRegions']}. The sections are plotted in the $(q_2, p_2)$ space for crossings with the $q_1=0.8165$ plane, the equilibrium value of $q_1$. Quasi-integrable trajectories are depicted in the diagonal subplots of the grid. Quasichaotic and chaotic trajectories are shown in the bottom left and top right subplots respectively.
• Figure 5: Peres lattices for the expected values $|\langle S_{11}\rangle/N|$ vs the different excitation energies of the 3-level LMG model. We choose $N=100$ particles, some representative values of the control parameter $\lambda$ in each column, and we restrict to the parity sector $p=(+,+,+)$ for the sake of clarity. The dashed vertical lines represent the ESQPT separatrices \ref{['Separatrix1']}, \ref{['Separatrix2']}. Similar results are obtained for $\langle S_{22}\rangle$, $\langle S_{00}\rangle$ and $\langle S_{22}-S_{00}\rangle$.