Detecting Quantum Anomalies in Open Systems

TL;DR

This paper introduces a feasible method to detect quantum anomalies in open quantum systems, revealing distinctive topological features in measurements related to spin chains coupled with environments.

Contribution

It extends the concept of quantum anomalies from closed to open systems, providing a novel approach using matrix product density operators and transfer matrices.

Findings

Half-integer spin chains show topological phenomena similar to level crossing.

Analytical and numerical evidence of singular behavior in measurements for half-integer spins.

Method applicable without requiring a Hamiltonian, enabling analysis of spectral flow and flux threading in open systems.

Abstract

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 work, 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 $\exp(\rm{i}\theta S^z_{\rm tot})$ as a function of $\theta$. 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 $\exp(\rm{i}\theta S^z_{\rm tot})$ 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.