  In this paper, we mainly focus on the rigorous convergence analysis for two
fully decoupled, unconditional energy stable methods of the two-phase
magnetohydrodynamics (MHD) model, which described in our previous work
\cite{2022Highly}. The two methods consist of semi-implicit stabilization
method/invariant energy quadratization (IEQ) method \cite{2019EfficientCHEN,
Yang2016Linear, Yang2017Efficient, 2019EfficientYANG} for the phase field
system, the pressure projection correction method for the saddle point MHD
system, the exquisite implicit-explicit treatments for nonlinear coupled terms,
which leads to only require solving a sequence of small elliptic equations at
each time step. As far as we know, it's the first time to establish the optimal
convergence analysis of fully decoupled and unconditional energy stable methods
for multi-physics nonlinear two-phase MHD model. In addition, several numerical
examples are showed to test the accuracy and stability of the presented
methods.
