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Modulated superconductivity due to vacancy and magnetic order in $A_x$Fe$_{2-x/2}$Se$_2$ [$A=$Cs, K, (Tl,Rb), (Tl,K)] iron-selenide superconductors

We present a calculation of a 'modulated' superconducting state in iron-selenide superconductors. The zero-momentum $d-$wave pairing breaks the translational symmetry of the conventional BaFe$_2$Se$_2$-like crystal of $I4/mmm$ space group. This pairing state becomes compatible when the Fe vacancies form an ordered state and the crystal symmetry changes to a low-temperature $I4/m$ one. For the specific case of an incommensurate vacancy order at $Q_v=(1/5, 3/5)$ in K$_{0.82(2)}$Fe$_{1.626(3)}$Se$_2$, we find that it induces a block checkerboard antiferromagnetic phase at wavevector ${\bf Q}_m=4{\bf Q}_v$. The coexistence of vacancy order and magnetic order leads to a reconstructed ground state which naturally couples to the $d-$wave superconductivity in a uniform phase in what we propose will be a general coupling for all iron-selenide superconductors. Our results agree with numerous experimental data available to date. We thus suggest that the incommensurability leads to uniform coexistence of multiple phases as a viable alternative to a nanoscale phase separation in high-$T_c$ superconductors and play an important role in the enhancement of superconductivity.