Diagnosing Strong-to-Weak Symmetry Breaking via Wightman Correlators

Symmetry plays a fundamental role in quantum many-body physics, and a central concept is spontaneous symmetry breaking, which imposes crucial constraints on the possible quantum phases and their transitions. Recent developments have extended the discussion of symmetry and its breaking to mixed states, enhancing our understanding of novel quantum phases that have no counterpart in pure states. Specific attention has been paid to scenarios where a strongly symmetric density matrix exhibits spontaneous symmetry breaking to weak symmetry, characterized by the fidelity correlator. In this work, we propose the Wightman correlator as an alternative diagnostic tool. This construction relies on the introduction of the thermofield double state for a generic density matrix, which maps the strong symmetry of the density matrix to the doubled symmetry of the pure state, allowing the Wightman correlator to emerge naturally as a standard probe of symmetry breaking. We prove the equivalence between the Wightman function and the fidelity correlator in defining strong-to-weak symmetry breaking, and examine explicit examples involving spin glasses, thermal density matrices, and the decohered Ising model. Additionally, we discuss a susceptibility interpretation of the Wightman correlator.