  One indication of thermalization time is subsystem entanglement reaching
thermal values. Recent studies on local quantum circuits reveal two exponential
stages with decay rates $r_1$ and $r_2$ of the purity before and after
thermalization. We provide an entanglement membrane theory interpretation, with
$r_1$ corresponding to the domain wall free energy. Circuit geometry can lead
to $r_1 < r_2$, producing a ``phantom eigenvalue". Competition between the
domain wall and magnon leads to $r_2 < r_1$ when the magnon prevails. However,
when the domain wall wins, this mechanism provides a practical approach for
measuring entanglement growth through local correlation functions.
