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  • Open Access

Finding Periodic Orbits in Projected Quantum Many-Body Dynamics

Elena Petrova1,*, Marko Ljubotina1,2,3, Gökhan Yalnız1, and Maksym Serbyn1

  • *Contact author: ptrva1998@gmail.com
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PRX Quantum 6, 040333 – Published 12 November, 2025

DOI: https://doi.org/10.1103/tldp-kvkd

Abstract

Describing general quantum many-body dynamics is a challenging task due to the exponential growth of the Hilbert space with system size. The time-dependent variational principle (TDVP) provides a powerful tool to tackle this task by projecting quantum evolution onto a classical dynamical system within a variational manifold. In classical systems, periodic orbits play a crucial role in understanding the structure of the phase space and the long-term behavior of the system. However, finding periodic orbits is generally difficult, and their existence and properties in generic TDVP dynamics over matrix product states have remained largely unexplored. In this work, we develop an algorithm to systematically identify and characterize periodic orbits in TDVP dynamics. Applying our method to the periodically kicked Ising model, we uncover both stable and unstable periodic orbits. We characterize the Kolmogorov-Arnold-Moser tori in the vicinity of stable periodic orbits and track the change of the periodic orbits as we modify the Hamiltonian parameters. We observe that periodic orbits exist at any value of the coupling constant of the kicked Ising model between prethermal and fully thermalizing regimes, but their relevance to quantum dynamics and imprint on quantum eigenstates diminishes as the system leaves the prethermal regime. Our results demonstrate that periodic orbits provide valuable insights into the TDVP approximation of quantum many-body evolution and establish a closer connection between quantum and classical chaos.

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Popular Summary

Quantum many-body systems are notoriously difficult to study because their Hilbert space grows exponentially with system size. A widely used approach to mitigate this challenge is the variational principle, first formulated by Dirac, which projects unitary quantum dynamics onto a classical dynamical system. This projection dramatically reduced the complexity at the cost of potentially distorting the true dynamics. Our central question is: What aspects of the original quantum many-body system can still be understood from such projected dynamics?

To address this, we project the dynamics of the periodically driven spin-1/2 Ising model onto a variational manifold of matrix product states that include short-range entanglement. We find and characterize periodic orbits in the resulting classical system, identifying both stable and unstable orbits. In the so-called prethermal regime (a transient state where the system behaves as if governed by an effective, nearly conserved Hamiltonian), the classical periodic orbits correspond closely to approximate eigenstates of the true quantum evolution. As the model enters the fully chaotic regime, the classical orbits persist but move into regions of high entanglement where the projection becomes increasingly inaccurate and trajectories imprinted on quantum eigenstates fade.

Our work connects quantum many-body dynamics for systems far from any semiclassical limit and the structures of classical chaos, allowing us to understand quantum dynamics using classical structures. Specifically, our orbit-finding method can be used to uncover quantum many-body scars, to approximate eigenstates in nonergodic periodically driven systems, and to detect quantum counterparts of classical phenomena such as bifurcations of periodic orbits.

See Also

ScarFinder: A Detector of Optimal Scar Trajectories in Quantum Many-Body Dynamics

Jie Ren, Andrew Hallam, Lei Ying, and Zlatko Papić
PRX Quantum 6, 040332 (2025)

Article Text

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PRX Quantum 6, 040333– Published 12 November, 2025
  • Received 23 April 2025
  • Revised 31 August 2025
  • Accepted 25 September 2025
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