Quantum Geometry of Many-Body Dynamics with Continuous Symmetries

Speaker: ​Sanjay Moudgalya (Tata Institute of Fundamental Research)
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Symmetries in quantum many-body systems are typically characterized by their algebraic structure. In this talk, I will show that for continuous symmetries such as U(1) and SU(2), there are underlying quantum geometric structures that play central roles in many-body dynamics. We demonstrate this in noisy Brownian models of unitary quantum dynamics, focusing on entanglement entropies and correlation functions. By mapping the disorder-averaged late-time dynamics to the low-energy physics of effective replica Hamiltonians, we show that the dynamics is governed by the quantum geometry of their ground-state manifolds. This geometry is, in turn, directly related to that of the k-commutants—the symmetry algebras acting on k replicas of the system—and is independent of microscopic details of the noisy evolution. This perspective enables simple geometric derivations, based on the time-dependent variational principle (TDVP), of several characteristic dynamical phenomena, including sub-ballistic Rényi entanglement growth and the anomalous decay of non-hydrodynamic correlators in interacting systems with continuous symmetries. We compare these behaviors across interacting systems with Abelian and non-Abelian continuous symmetries, as well as free-fermion systems, whose distinct dynamics can be traced to differences in the geometry of their k-commutants. More broadly, these results establish a geometric framework for systematically understanding observables and universal dynamical behavior in noisy quantum systems with continuous symmetries.