Berry connection and quantum geometry in time-dependent systems with instantaneous quantum integrable field theory

We study many-body quantum geometric effects in a time-dependent system with emergent quantum integrable field theory instantaneously. We establish a theorem stating that the Berry connection matrix, and thus all associated geometric and topological quantities of the system, can be precisely characterized by excitations up to two particles from the initial emergent quantum integrable field theory. To illustrate the many-body geometric influence, we analyze an Ising chain subjected to both a small longitudinal field and a slowly rotating transverse field, whose low-energy physics in the scaling limit is instantaneously governed by the quantum ${E}_{8}$ integrable field theory. Focusing on the quantum geometric potential (QGP), we show the QGP continuously suppresses the instantaneous energy gaps with decreasing longitudinal field, thereby enhancing many-body Landau-Zener tunneling as evidenced by the Loschmidt echo and its associated spectral entropy. The critical threshold for the longitudinal field strength is determined, where the spectral entropy linearly increases with system size and exhibits hyperscaling behavior when approaching the threshold. As the longitudinal field passes the threshold and decreases toward zero, the QGP continuously leads to vanishing instantaneous energy gaps involving more low-energy excitations, resulting in increasing spectral entropy indicative of many-body Landau-Zener tunneling. Our results unveil telltale quantum geometric signatures in time-dependent many-body systems, elucidating the intricate interplay between quantum geometry, integrability, and dynamics.