Symmetry and its Signatures in Quantum Many-Body Dynamics

Abstract

Symmetry has long been a defining feature in our understanding of statistical or manybody systems. By making appeals to universal properties associated with global symmetries and topology, one may describe universal properties of “typical” states and dynamics in equilibrium, even when keeping track of the precise dynamics of a particular many-body system is impossible. This challenge of tracking allowable states and dynamical transitions is only exacerbated for non-equilibrium systems, where one cannot rely on the same notions of typicality. Further, when driven out of equilibrium by external interactions, quantum orders constructed from highly sensitive correlations between states are liable to vanish. Despite these conceptual and practical difficulties, the rise of quantum technologies and accompanying theoretical developments has motivated a surge of interest in dynamical quantum phenomena. The recent developments in the field of quantum many-body dynamics provide satisfactory accounts of many interesting phenomena, including failures of the Eigenstate Thermalization Hypothesis, various dynamical and mixed-state phases of matter, and measurement-induced dynamics and phase transitions. Many of these results are explained for specific systems or within different conceptual frameworks, however these results rarely generalize. In this thesis, I attempt to unify many aspects of quantum many-body dynamics under the same conceptual framework through an investigation of the universal signatures of symmetry in quantum dynamical systems. This is accomplished via a mapping between the averaged dynamics and the low-energy spectrum of an effective Hamiltonian in a “doubled Hilbert space,” comprised of two copies of the original space. This provides a general and versatile framework to qualitatively understand both familiar and novel universal properties of dynamical phenomena like charge diffusion, sub(super)-diffusion of multipole moments in systems with short and long-range interactions, charge and multipole, and even measurement-induced phase transitions. By expanding into a doubled Hilbert space, one may capture the subtleties of non-equilibrium physics, and particularly dynamical phases, within the framework of equilibrium physics and phases. In this work, we examine how to understand various symmetry-constrained dynamical phases and phase transitions using through a dual description of symmetry-constrained equilibrium phases and symmetry-breaking transitions in an enlarged Hilbert space.