Finding periodic orbits in projected quantum many-body dynamics

TL;DR

This work introduces a general algorithm to locate and characterize periodic orbits in TDVP dynamics projected onto matrix product state manifolds for quantum many-body systems. By constraining gradient optimization to the iMPS tangent space and using a fidelity-based objective, the method uncovers both stable and unstable periodic orbits in the kicked Ising model, along with surrounding KAM tori whose dimensionality scales as $\chi^2$. The orbits persist across coupling values, with low-leakage states in the prethermal regime corresponding to approximate eigenstates of the one-period propagator, while increasing entanglement drives higher leakage in the chaotic regime. Overall, the study demonstrates that TDVP-projected quantum dynamics exhibit classical-chaos structures, offering a bridge between quantum many-body dynamics and classical phase-space intuition and providing a route to identifying non-thermal eigenstates and quantum scars.

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 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.

Figures (11)

• Figure 1: (a) Dynamics of local expectation values $\langle \sigma^{x,y}\rangle$ along periodic orbits of different bond dimensions. Two different orbits for $\chi = 1$ are represented by violet colors, and three unique orbits for $\chi = 2$, and $4$ are shown in different shades of blue and red, respectively, finally 4 orbits for $\chi = 3$ are shown in shades of green. (b) All orbits for $\chi>1$ are characterized by non-vanishing entanglement that is a periodic function of time. Entanglement is shown only during the half-period since it is not affected by the evolution with the single site $x$-magnetic field during the second half-period. Dots correspond to the dynamics of entanglement in full (numerically exact) unitary dynamics obtained with TEBD. All orbits are obtained for $J = 1.09$. The two orbits in the boxes are marked as the ground state (GS) and the ceiling state (CS), the reason for which we discuss in the next section.
• Figure 2: (a) Distance between the time-evolved initial state on the periodic orbit and the trajectory obtained by a small deformation of the orbit, quantified by $1 - \mathcal{D}(A_L(t), A_L'(t))$. Trajectories and their color coding coincide with orbits in Fig. \ref{['fig:traj1']} for $J = 1.09$. Both $\chi = 1$ orbits and one of the $\chi = 2$ orbits are stable, while the remaining orbits are unstable, and small deformations lead to an exponential increase in distance with time. Floquet exponents for periodic orbits starting from $\chi = 2$ have the following values $[10^{-5}, 0.3, 0.72, 0.35, 0.17, 0.45, 0.35, 0.29, 0.35, 0.48]$. (b) Stroboscopic dynamics of expectation values of local spin projections for the GS perturbed by a random perturbation with strength $\delta = 0.01$ may be understood as a projection of the KAM torus. The blue star here corresponds to the original periodic orbit values of the local observables. (c) Fourier spectra (without the $\omega = 0$ component) of the $\langle \sigma^x(t) \rangle$ expectation values have sharp peaks that can be interpreted as the frequencies associated with the motion on the four-dimensional KAM torus. Black dashed lines correspond to phases of the four Floquet multipliers defined in Eq. \ref{['Eq:F-mult']} match the location of the peaks (we ignore the small peak at $\omega \approx 3.05$ as it is perturbation-dependent and does not appear for all observables).
• Figure 3: (a) Smooth evolution of the low leakage GS orbit with respect to the propagator parameter $J$, visualized via dynamics of local spin expectation values. (b) Dynamics of entanglement entropy for the first half period of the orbit and from exact unitary dynamics for $L = 50$ TEBD. As we increase $J$, the orbit moves to a more entangled region of the MPS manifold, and the leakage along the orbit increases. (c) The difference between the entanglement entropy in exact unitary and TDVP dynamics ($\delta S_\text{ent}$) at $t = T$ (top panel), and the corresponding Floquet exponent (bottom panel). We observe that both quantities behave in a similar way as functions of $J$. At $J \approx 1.27$, the orbit becomes unstable, and for similar values of $J$, $\delta S_\text{ent}$ starts increasing more rapidly.
• Figure 4: (a) IPR of MPS states corresponding to periodic orbits from Fig. \ref{['fig:traj1']} for $J=1.09$ over the basis of eigenstates of the unitary from Eq. \ref{['eq:Kick']}. For all cases, IPR decays exponentially with $L$, but two orbits with the lowest leakage show parametrically slower decay of IPR. (b) IPR scaling with $L$ for the lowest leakage $\chi=2$ orbit (GS) at 5 different values of the coupling $J = 0.6, 1.0, 1.2, 1.4, \pi/2$. For $J = 0.6$, the IPR is close to one and stays nearly constant, whereas at the dual unitary point $J=\pi/2$, the IPR decays as $\sim e^{-0.55L}$, in agreement with expectations for a random state, $\sim e^{-0.56L}$. Due to MPS being translation- and inversion-symmetric for this family of orbits, we consider the zero momentum sector with parity $+1$.
• Figure 5: Performance of the GD algorithm using the tangent space of iMPS. The algorithm terminates when $\delta F$ reaches $10^{-15}$. The data is for the GS orbit from Fig. \ref{['fig:traj1']}.