Universal signatures of Majorana zero modes in critical Kitaev chains

Many topological or critical aspects of the Kitaev chain are well known, with several classic results. In contrast, the study of the critical behavior of the strong Majorana zero modes (MZM) has been overlooked. Here we introduce two topological markers which, surprisingly, exhibit non-trivial signatures over the entire (1+1) Ising critical line. We first analytically compute the MZM fidelity ${\cal{F}}_{\rm MZM}$--a measure of the MZM mapping between parity sectors. It takes a universal value along the (1+1) Ising critical line, ${\cal{F}}_{\rm MZM}=\sqrt{8}/\pi$, independent of the energy. We also obtain an exact analytical result for the critical MZM occupation number ${{\cal N}}_{\rm MZM}$ which depends on the Catalan's constant ${\cal G}\approx 0.91596559$, for both the ground-state (${{\cal N}}_{\rm MZM}=1/2-4{\cal{G}}/\pi^2\approx 0.12877$) and the first excited state (${{\cal N}}_{\rm MZM}=1/2+(8-4{\cal{G}})/\pi^2\approx 0.93934$). We further compute finite-size corrections which identically vanish for the special ratio $\Delta/t=\sqrt{2}-1$ between pairing and hopping in the critical Kitaev chain.