Strong zero modes in random Ising-Majorana chains

Abstract

We investigate the fate and robustness of topological strong zero modes (SZMs) in random Ising-Majorana chains using the SZM fidelity, ${\cal F}_{\rm SZM}$, as a many-body diagnostic that quantifies how accurately SZM operators map the {\it entire} spectrum between opposite parity sectors. In clean systems, ${\cal F}_{\rm SZM}=1$ in the topological phase, vanishes in the trivial regime, and takes the universal value $\sqrt{8}/\pi$ at the $(1+1)$D Ising critical point. Here we study how quenched disorder modifies this picture across the infinite-randomness fixed point (IRFP) governing the criticality of the random chain. In both microcanonical and canonical ensembles, SZMs persist throughout the topological phase, including the gapless Griffiths regime, with fidelities converging exponentially to unity. At the IRFP, however, the fidelity distributions become ensemble dependent: the microcanonical ensemble displays bimodal peaks at $\{0.5,1\}$, while the canonical ensemble develops a triple-peak structure at $\{0,0.5,1\}$ with power-law singularities. Our results establish ${\cal F}_{\rm SZM}$ as a robust probe of localization-protected topological order and uncover distinctive topological features of infinite-randomness criticality. Unlike the clean Ising CFT, where the finite critical value arises from a cancellation of power laws, the IRFP seems to exhibit an intrinsically stronger topological character. The edge-selective structure of the critical distributions may suggest a boundary manifestation of the average Kramers-Wannier duality symmetry at the IRFP.