  The mirror energy difference (MED) of the mirror state, especially for states
bearing the Thomas-Erhman shift, serves as a sensitive probe of isospin
symmetry breaking. We employ the Gamow shell model, which includes the
inter-nucleon correlation and continuum coupling, to investigate the MED for
$sd$-shell nuclei by taking the $^{18}$Ne/$^{18}$O and $^{19}$Na/$^{19}$O as
examples. Our GSM provides good descriptions for the excitation energies and
MEDs for the $^{18}$Ne/$^{18}$O and $^{19}$Na/$^{19}$O. Moreover, our
calculations also reveal that the large MED of the mirror states is caused by
the significant occupation of the weakly bound or unbound $s_{1/2}$ waves,
giving the radial density distribution of the state in the proton-rich nucleus
more extended than that of mirror states in deeply-bound neutron-rich nuclei.
Furthermore, our GSM calculation shows that the contribution of Coulomb is
different for the low-lying states in proton-rich nuclei, which significantly
contributes to MEDs of mirror states. Moreover, the contributions of the
nucleon-nucleon interaction are different for the mirror state, especially for
the state of proton-rich nuclei bearing the Thomas-Erhman shift, which also
contributes to the significant isospin symmetry breaking with large MED.
