# Eigenstate entanglement in integrable collective spin models

Meenu Kumari1 and Álvaro M. Alhambra2

1Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada
2Max-Planck-Institut fur Quantenoptik, D-85748 Garching, Germany

### Abstract

The average entanglement entropy (EE) of the energy eigenstates in non-vanishing partitions has been recently proposed as a diagnostic of integrability in quantum many-body systems. For it to be a faithful characterization of quantum integrability, it should distinguish quantum systems with a well-defined classical limit in the same way as the unequivocal classical integrability criteria. We examine the proposed diagnostic in the class of collective spin models characterized by permutation symmetry in the spins. The well-known Lipkin-Meshov-Glick (LMG) model is a paradigmatic integrable system in this class with a well-defined classical limit. Thus, this model is an excellent testbed for examining quantum integrability diagnostics. First, we calculate analytically the average EE of the Dicke basis $\{|j,m\rangle \}_{m=-j}^j$ in any non-vanishing bipartition, and show that in the thermodynamic limit, it converges to $1/2$ of the maximal EE in the corresponding bipartition. Using finite-size scaling, we numerically demonstrate that the aforementioned average EE in the thermodynamic limit is universal for all parameter values of the LMG model. Our analysis illustrates how a value of the average EE far away from the maximal in the thermodynamic limit could be a signature of integrability.

Classical systems can be categorized on the basis of their integrability. While integrable systems are perfectly predictable, nonintegrable systems often exhibit chaotic behaviour that renders the long-term evolution of such systems unpredictable. The distinction between integrable and nonintegrable systems is well-characterized in classical physics but this is a long-standing open problem in quantum physics. Every measure studied so far for such a characterization appears to fall short in some way.

A new perspective on this issue is given by quantum information, and in particular through the concept of entanglement entropy, which measures quantum correlations. It is widely believed that if the entanglement in a particular model is very close to maximal possible value, the model should be quantum chaotic. This leads to the question whether entanglement being far from maximal value could be associated with quantum integrability. Here, we analyse this through a model known to have a classically integrable limit making it an ideal candidate of a quantum integrable system. We show that the amount of entanglement that our model contains is very far from maximal, but also a universal quantity independent of the model’s coupling parameters. This exact value should thus be set by a fundamental feature of the model, that is, its integrability.

Our result shows how entanglement entropy is able to characterize when a quantum system is integrable or not, depending on the amount of quantum correlations between the different parts that form it. In doing so, we provide strong evidence that this concept can be crucial in solving the long-standing question of characterizing quantum integrability and chaos.

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[2] D. J. Nader, C. A. González-Rodríguez, and S. Lerma-Hernández, "Avoided crossings and dynamical tunneling close to excited-state quantum phase transitions", Physical Review E 104 6, 064116 (2021).

[3] Budhaditya Bhattacharjee, Pratik Nandy, and Tanay Pathak, "Eigenstate capacity and Page curve in fermionic Gaussian states", Physical Review B 104 21, 214306 (2021).

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