Strong and weak symmetries and their spontaneous symmetry breaking in mixed states emerging from the quantum Ising model under multiple decoherence

In realistic settings, quantum systems often evolve from pure states to mixed states due to unintended interactions with the environment, a process known as quantum decoherence. Recently, distinct mixed phases of matter induced by decoherence have attracted substantial interest across various fields. A notable finding in this context is the existence of two types of symmetry in the density matrix: strong and weak symmetries.

In this study, we investigate the spontaneous symmetry breaking of both strong and weak symmetries, using decohered transverse-field Ising model as a case study. By employing the doubled Hilbert formalism, we can reformulate decoherence as an imaginary-time evolution. This reformulation enables us to establish a connection between decoherence and the filtering operation used to obtain an approximate ground state under perturbations in matrix product state formalism. In light of this connection, we further explore the potential relationship between the decohered transverse-field Ising model and the quantum Ashkin-Teller model, suggesting possible similarities between their phase diagrams.