Testing Entanglement and Quantum Chaos Operator in Dirac and Graphene Quantum Systems with SU(3) Casimir Measurement

Abstract

We develop a unified theoretical framework for symmetry-resolved entanglement, its geometric formulation in curved spacetime, and its dynamical relation to quantum chaos. We formulate the entanglement asymmetry model and establish a replica-based path-integral construction for the exact evaluation of symmetry-resolved entanglement measures. The framework is extended to quantum field theory in curved spacetime, where the interplay between spacetime geometry, internal symmetries, and quantum correlations is incorporated systematically. In particular, we construct the SU(3) symmetry-resolved decomposition using irreducible representations, quadratic and higher-order Casimir invariants, and Casimir-weighted projection operators, providing an exact representation-theoretic characterization of the entanglement structure. We then investigate the dynamical evolution of entanglement and its connection with the emergence of quantum chaos by introducing a quantum chaos operator and deriving its exact form in \(\mathcal {D}\) dimensions. The operator is subsequently incorporated into the Dirac equation in \(\mathcal {D}\) dimensions and evaluated explicitly for graphene Dirac fermions, establishing a direct connection between relativistic quantum dynamics, symmetry-resolved entanglement, and chaotic evolution. We further examine the domain of applicability of the construction under strong disorder, strong interactions, finite temperature, and finite density, and discuss its extension to nonequilibrium states and open quantum systems. These results provide a general framework for analyzing symmetry-resolved quantum correlations and their dynamical degradation in relativistic and condensed-matter quantum systems, with potential applications to quantum field theory in curved spacetime, Dirac materials, and quantum information in strongly correlated and chaotic regimes.