We investigate a mixed-state quantum criticality in the Ising model under $X+ZZ$ decoherence. In the doubled Hilbert-space formalism, the decohered state resides on the self-dual critical line of the quantum Ashkin-Teller (qAT) model, as a result of the specific choice of the decoherence channel. On the other hand, since the mixed state under $X+ZZ$ decoherence satisfies the Kramers-Wannier self-duality in a weak sense, the Ising criticality of the pure state can be partially preserved in the mixed system. By making use of the combination of the doubled Hilbert-space formalism and matrix product states, a numerical study is carried out. We numerically find that up to moderate decoherence strength, the behavior of the mixed states on the critical line is consistent with Ising-like scaling within the accessible system sizes, with the effective central charge and critical exponents close to the Ising–conformal field theory (CFT) values $c=1/2, \eta =1/4$, and ${\eta }_X=2$. Our results of the central charge and critical exponents contradict the ones in the $c=1$ orbifold boson CFT describing the critical state of the qAT model. In addition, we also numerically observe the flow of the central charge up to strong decoherence. The strong decoherence washes out the remnant Ising criticality and induces strong-to-weak spontaneous symmetry breaking.

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• Conformal field theory
• Critical phenomena
• Density matrix renormalization group
• Open quantum systems & decoherence
• Quantum phase transitions