Detecting Quantum Anomalies in Open Systems

Symmetries and quantum anomalies serve as powerful tools for constraining complicated quantum many-body systems, offering valuable insights into low-energy characteristics based on their ultraviolet structure. Nevertheless, their applicability has traditionally been confined to closed quantum systems, rendering them largely unexplored for open quantum systems described by density matrices. In this Letter, we introduce a novel and experimentally feasible approach to detect quantum anomalies in open systems. Specifically, we claim that, when coupled with an external environment, the mixed 't Hooft anomaly between spin rotation symmetry and lattice translation symmetry gives distinctive characteristics for half-integer and integer spin chains in measurements of exp(iθS_{tot}^{z}) as a function of θ. Notably, the half-integer spin chain manifests a topological phenomenon akin to the "level crossing" observed in closed systems. To substantiate our assertion, we develop a lattice-level spacetime rotation to analyze the aforementioned measurements. Based on the matrix product density operator and transfer matrix formalism, we analytically establish and numerically demonstrate the unavoidable singular behavior of exp(iθS_{tot}^{z}) for half-integer spin chains. Conceptually, our work demonstrates a way to discuss notions like "spectral flow" and "flux threading" in open systems not necessarily with a Hamiltonian.