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arXiv:2603.05313

Strong zero modes in random Ising-Majorana chains

Saurav Kantha, Nicolas Laflorencie

Published Mar 5, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2603.05313

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We investigate the fate and robustness of topological strong zero modes (SZMs) in random Ising-Majorana chains using the SZM fidelity, FSZM{\cal F}_{\rm SZM}FSZM​, as a many-body diagnostic that quantifies how accurately SZM operators map the {\it entire} spectrum between opposite parity sectors. In clean systems, FSZM=1{\cal F}_{\rm SZM}=1FSZM​=1 in the topological phase, vanishes in the trivial regime, and takes the universal value 8/π\sqrt{8}/π8​/π at the (1+1)(1+1)(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}\{0.5,1\}{0.5,1}, while the canonical ensemble develops a triple-peak structure at {0,0.5,1}\{0,0.5,1\}{0,0.5,1} with power-law singularities. Our results establish FSZM{\cal F}_{\rm SZM}FSZM​ 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.
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 FMZM{\cal{F}}_{\rm MZM}FMZM​--a measure of the MZM mapping between parity sectors. It takes a universal value along the (1+1) Ising critical line, FMZM=8/π{\cal{F}}_{\rm MZM}=\sqrt{8}/πFMZM​=8​/π, independent of the energy. We also obtain an exact analytical result for the critical MZM occupation number NMZM{\cal N}_{\rm MZM}NMZM​ which depends on the Catalan's constant G≈0.91596559{\cal G}\approx 0.91596559G≈0.91596559, for both the ground-state (NMZM=1/2−4G/π2≈0.12877{\cal N}_{\rm MZM}=1/2-4{\cal{G}}/π^2\approx 0.12877NMZM​=1/2−4G/π2≈0.12877) and the first excited state (NMZM=1/2+(8−4G)/π2≈0.93934{\cal N}_{\rm MZM}=1/2+(8-4{\cal{G}})/π^2\approx 0.93934NMZM​=1/2+(8−4G)/π2≈0.93934). We further compute finite-size corrections which identically vanish for the special ratio Δ/t=2−1Δ/t=\sqrt{2}-1Δ/t=2​−1 between pairing and hopping in the critical Kitaev chain.
In clean and weakly disordered systems, topological and trivial phases having a finite bulk energy gap can transit to each other via a quantum critical point. In presence of strong disorder, both the nature of the phases and the associated criticality can fundamentally change. Here we investigate topological properties of a strongly disordered fermionic chain where the bond couplings are drawn from normal probability distributions which are defined by characteristic standard deviations. Using numerical strong disorder renormalization group methods along with analytical techniques, we show that the competition between fluctuation scales renders both the trivial and topological phases gapless with Griffiths like rare regions. Moreover, the transition between these phases is solely governed by the fluctuation scales, rather than the means, rendering the critical behavior to be determined by an infinite randomness fixed point with an irrational central charge. Our work points to a host of novel topological phases and atypical topological phase transitions which can be realized in systems under strong disorder.
The quantum critical properties of interacting fermions in the presence of disorder are still not fully understood. While it is well known that for Dirac fermions, interactions are irrelevant to the noninteracting infinite randomness fixed point (IRFP), the problem remains largely open in the case of Majorana fermions which further display a much richer disorder-free phase diagram. Here, pushing the limits of density matrix renormalization group simulations, we carefully examine the ground state of a Majorana chain with both disorder and interactions. Building on appropriate boundary conditions and key observables such as entanglement, energy gap, and correlations, we strikingly find that the noninteracting Majorana IRFP is very stable against finite interactions, in contrast with previous claims.
Topological classifications of quantum critical systems have recently attracted growing interest, as they go beyond the traditional paradigms of condensed matter and statistical physics. However, such classifications remain largely unexplored in aperiodic environments, particularly under quasiperiodic modulations. In this Letter, we uncover a family of topological quasiperiodic fixed points with critical scalings intermediate between the clean and infinite-randomness limits. By exactly solving the quasiperiodic cluster-Ising chain, we demonstrate that all phase boundaries separating quasiperiodically modulated phases are governed by topological Ising-like fixed points unique to strongly modulated quasiperiodic systems. Although their local bulk critical properties match those of conventional quasiperiodic Ising critical points reported before, these fixed points host robust topological edge degeneracies and exhibit a distinct symmetry-enriched realization of the quasiperiodic Ising criticality, as further supported by complementary lattice simulations.
Two-dimensional time-reversal-invariant topological superconductors host helical Majorana fermions at their boundary. We study the fate of these edge states under the combined influence of strong interactions and disorder, using the effective 1D lattice model for the edge introduced by Jones and Metlitski [Phys. Rev. B 104, 245130 (2021)]. We specifically develop a strong-disorder renormalization group analysis of the lattice model and identify a regime in which time-reversal is broken spontaneously, creating random magnetic domains; Majorana fermions localize to domain walls and form an infinite-randomness fixed point, identical to that appearing in the random transverse-field Ising model. While this infinite-randomness fixed point describes a fine-tuned critical point in a purely 1D system, in our edge context there is no obvious time-reversal-preserving perturbation that destabilizes the fixed point. Our analysis thus suggests that the infinite-randomness fixed point emerges as a stable phase on the edge of 2D topological superconductors when strong disorder and interactions are present.
Two-dimensional time-reversal-invariant topological superconductors host helical Majorana fermions at their boundary. We study the fate of these edge states under the combined influence of strong interactions and disorder, using the effective one-dimensional (1D) lattice model for the edge introduced by Jones and Metlitski [Phys. Rev. B 104, 245130 (2021)]. We specifically develop a strong-disorder renormalization-group analysis of the lattice model and identify a regime in which time-reversal is broken spontaneously, creating random magnetic domains; Majorana fermions localize to domain walls and form an infinite-randomness fixed point, identical to that appearing in the random transverse-field Ising model. While this infinite-randomness fixed point describes a fine-tuned critical point in a purely 1D system, in our edge context there is no obvious time-reversal-preserving perturbation that destabilizes the fixed point. Our analysis thus suggests that the infinite-randomness fixed point emerges as a stable phase on the edge of two-dimensional topological superconductors when strong disorder and interactions are present.