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Finding suitable characterizations of quantum chaos is a major challenge inmany-body physics, with a central difficulty posed by the linearity of theSchr\"odinger equation. A possible solution for recovering non-linearity is toproject the dynamics onto some variational manifold. The classical chaosinduced via this procedure may be used as a signature of quantum chaos in thefull Hilbert space. Here, we demonstrate analytically a previously heuristicconnection between the Lyapunov spectrum from projection onto the matrixproduct state (MPS) manifold and the growth of entanglement. This growth occursby squeezing a localized distribution on the variational manifold. The processqualitatively resembles the Cardy-Calabrese picture, where local perturbationsto a moving MPS reference are interpreted as bosonic quasi-particles. Takingcareful account of the number of distinct channels for these processes recoversthe connection to the Lyapunov spectrum. Our results rigorously establish thephysical significance of the projected Lyapunov spectrum, suggesting it as analternative method of characterizing chaos in quantum many-body systems, onethat is manifestly linked to classical chaos.

Entanglement growth from squeezing on the MPS manifold