Abstract
One indication of thermalization time is subsystem entanglement reaching thermal values. Recent studies on local quantum circuits reveal two exponential stages with decay rates r₁ and r₂ of the purity before and after thermalization. We provide an entanglement membrane theory interpretation, with r₁ corresponding to the domain wall free energy. Circuit geometry can lead to r₁ < r₂, producing a ``phantom eigenvalue". Competition between the domain wall and magnon leads to r₂ < r₁ when the magnon prevails. However, when the domain wall wins, this mechanism provides a practical approach for measuring entanglement growth through local correlation functions.