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Recently, KFe$_2$As$_2$ was shown to exhibit a structural phase transitionfrom a tetragonal to a collapsed tetragonal phase under applied pressure ofabout $15~\mathrm{GPa}$. Surprisingly, the collapsed tetragonal phase hosts asuperconducting state with $T_c \sim 12~\mathrm{K}$, while the tetragonal phaseis a $T_c \leq 3.4~\mathrm{K}$ superconductor. We show that the key differencebetween the previously known non-superconducting collapsed tetragonal phase inAFe$_2$As$_2$ (A= Ba, Ca, Eu, Sr) and the superconducting collapsed tetragonalphase in KFe$_2$As$_2$ is the qualitatively distinct electronic structure.While the collapsed phase in the former compounds features only electronpockets at the Brillouin zone boundary and no hole pockets are present in theBrillouin zone center, the collapsed phase in KFe$_2$As$_2$ has almost nestedelectron and hole pockets. Within a random phase approximation spin fluctuationapproach we calculate the superconducting order parameter in the collapsedtetragonal phase. We propose that a Lifshitz transition associated with thestructural collapse changes the pairing symmetry from $d$-wave (tetragonal) to$s_\pm$ (collapsed tetragonal). Our DFT+DMFT calculations show that effects ofcorrelations on the electronic structure of the collapsed tetragonal phase areminimal. Finally, we argue that our results are compatible with a change ofsign of the Hall coefficient with pressure as observed experimentally.

Origin of the superconducting state in the collapsed tetragonal phase of KFe2As2