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 $\sqrt{8}/π$ 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.

Figures (14)

• Figure 1: Symmetrized fidelity for the clean Ising-Majorana (Kitaev) chain. Different curves correspond to different system sizes indicated in the legend. The vertical dotted line indicates the critical point and the horizontal dotted line indicates the universal value $\sqrt{8}/\pi$ that $\mathcal{F}_{\rm SZM}$ takes at the Ising criticality.
• Figure 2: Overview of the phase diagram of clean and random Ising--Majorana chains. (a) Clean system: The (1+1)D Ising critical point (central charge $c=1/2$) separates a topological phase supporting SZMs from a trivial phase without SZMs. In the topological phase, a finite bulk gap coexists with parity-paired many-body spectra and unit fidelity, at large enough sizes $\mathcal{F}_{\rm SZM}\to 1$, while in the trivial phase $\mathcal{F}_{\rm SZM}\to 0$. At criticality, the fidelity takes the universal value $\mathcal{F}_{\rm SZM}^{c}=\sqrt{8}/\pi$laflorencieUniversalSignaturesMajorana2023. In the spin representation, the ordered phase exhibits broken $\mathbb{Z}_2$ symmetry. (b) Random case: Disorder drives the Ising transition to an infinite-randomness fixed point (IRFP) characterized by an effective central charge${\widetilde{c_{\rm eff}}}=\ln \sqrt{2}$. The topological phase persists in the presence of Anderson localization and extends into a gapless Griffiths regime. Throughout this phase, SZMs remain robust with $\mathcal{F}_{\rm SZM}\to 1$ in the thermodynamic limit, while the trivial phase remains non-topological with zero SZM fidelity. At criticality, the IRFP displays non-trivial fidelity distributions, depending on the disorder ensemble, as discussed in the text, yielding a finite fidelity at the IRFP.
• Figure 3: Microcanonical ensemble results for the symmetrized fidelity for various system sizes, averaged over $2\times 10^4$ samples. (I) Average ${\overline{{\cal{F}}_{\rm SZM}}}$ plotted against the control parameter $\delta$. The vertical dotted line marks the quantum critical point. Trivial and topological regions correspond to $\delta<0$ and $\delta>0$ respectively. Inset (a): log-log plot of $\overline{\mathcal{F}_{\rm SZM}}$ vs inverse system size, plotted for $\delta=-0.5$ (deep in the trivial region) showing a power-law decay to zero Eq. \ref{['eq:Ftrivialpower-law']}. Inset (b): $1-\overline{\mathcal{F}_{\rm SZM}}$ plotted against $L$ on a semi-log plot for various points in the topological region, showing exponential corrections to $\overline{\mathcal{F}_{\rm SZM}}$ as it approaches $1$. (II) Same as plot I, for typical values ${\cal{F}}_{\rm SZM}^{\rm typ}$. Inset (c): Semi-log plot of $\mathcal{F}^{\rm typ}_{\rm SZM}$ for various values of $\delta$ deep in the trivial region, showing a exponential decay to zero. Inset (d): Semi-log plot of $1-\mathcal{F}^{\rm typ}_{\rm SZM}$ at inside the topological region, showing exponential finite size corrections.
• Figure 4: (a) Exponential decay of the typical fidelity, plotted against $L$ for various values of $\delta<0$ in the trivial regime. (b) Data collapse when rescaling the length $L\to L/\xi$, where $\xi\sim |\delta|^{-1}$, see inset.
• Figure 5: (I) Convergence of $\mathcal{F}_{\rm SZM}^{\rm typ}$ to $1$ in the topological regime, close to criticality, for various values of $\delta$ on a semi-log plot. (II) Collapse of the curves shown in (I) using one parameter scaling $L\rightarrow L/\xi$. Inset (a): Length scale $\xi$ vs $1/|\delta|$ on a log-log plot showing a dependence $\xi \sim |\delta|^{-\nu}$. Inset (b): Values of the cost function $C(\nu)$ plotted against the critical exponent $\nu$. The different colors correspond to number of bins used to calculate $C(\nu)$ and the solid circles mark the minimum of the $C(\nu)$ curve. Both the insets show the value of $\nu\sim 2$. Panels (III-IV) and insets (c) and (d) are the corresponding quantities for the average $\overline{\mathcal{F}_{\rm SZM}}$, which also exhibits a critical exponent $\nu$ close to $2$.