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.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.

arXiv:2603.05313Mar 5, 20260 citations

Strong zero modes and eigenstate phase transitions in the XYZ/interacting Majorana chain

Paul Fendley

Abstract I explicitly construct a strong zero mode in the XYZ chain or, equivalently, Majorana wires coupled via a four-fermion interaction. The strong zero mode is an operator that pairs states in different symmetry sectors, resulting in identical spectra up to exponentially small finite-size corrections. Such pairing occurs in the Ising/Majorana fermion chain and possibly in strongly disordered many-body localized phases. The proof here shows that the strong zero mode occurs in a clean interacting system, and that it possesses some remarkable structure—despite being a rather elaborate operator, it squares to the identity. Eigenstate phase transitions separate regions with different strong zero modes.Abstract I explicitly construct a strong zero mode in the XYZ chain or, equivalently, Majorana wires coupled via a four-fermion interaction. The strong zero mode is an operator that pairs states in different symmetry sectors, resulting in identical spectra up to exponentially small finite-size corrections. Such pairing occurs in the Ising/Majorana fermion chain and possibly in strongly disordered many-body localized phases. The proof here shows that the strong zero mode occurs in a clean interacting system, and that it possesses some remarkable structure—despite being a rather elaborate operator, it squares to the identity. Eigenstate phase transitions separate regions with different strong zero modes.

arXiv:1512.03441Journal of Physics A Mathematical and TheoreticalJun 15, 2016148 citations

Ising chain with random short- and long-range interactions

L.L. Gonçalves, André P. Vieira

Journal of Magnetism and Magnetic MaterialsJan 1, 19984 citations

Spin correlations in the Ising chains

Baohua Teng, H.K. Sy

Solid State CommunicationsFeb 12, 20046 citations

The Ising dual-reflection interface : Z4 symmetry and Majorana strong zero modes

Juliane Graf, Federica Maria Surace, Marcus Berg +1 more

We investigate an interface in the transverse field quantum Ising chain connecting an ordered ferromagnetic phase and a disordered paramagnetic phase that are KramersWannier duals of each other. Unlike prior studies focused on non-invertible defects, this interface exhibits a symmetry that combines Kramers-Wannier transformation with spatial reflection. We demonstrate that, under open boundary conditions, this setup gives rise to a discrete Z4 symmetry, encompassing the conventional Z2 Ising parity as a subgroup, while in a closed geometry a non-invertible symmetry emerges. Using the Jordan-Wigner transformation, we map the spin chain onto a solvable quadratic Majorana fermion system. In this formulation, the Z4 symmetry is realized manifestly as a parity-dependent reflection with respect to a Majorana site, in contrast to the conventional reflection which mirrors with respect to the central link of the Majorana chain. Additionally, we construct Majorana strong zero modes that retain the Z4 symmetry, ensure degeneracies of all energy eigenstates, and are robust under generic local symmetrypreserving perturbations of the fermion model, including interactions. Finally, we develop quantum circuit realizations of our model paving the way towards the creation of exact Majorana strong zero modes with digital quantum hardware.We investigate an interface in the transverse field quantum Ising chain connecting an ordered ferromagnetic phase and a disordered paramagnetic phase that are KramersWannier duals of each other. Unlike prior studies focused on non-invertible defects, this interface exhibits a symmetry that combines Kramers-Wannier transformation with spatial reflection. We demonstrate that, under open boundary conditions, this setup gives rise to a discrete Z4 symmetry, encompassing the conventional Z2 Ising parity as a subgroup, while in a closed geometry a non-invertible symmetry emerges. Using the Jordan-Wigner transformation, we map the spin chain onto a solvable quadratic Majorana fermion system. In this formulation, the Z4 symmetry is realized manifestly as a parity-dependent reflection with respect to a Majorana site, in contrast to the conventional reflection which mirrors with respect to the central link of the Majorana chain. Additionally, we construct Majorana strong zero modes that retain the Z4 symmetry, ensure degeneracies of all energy eigenstates, and are robust under generic local symmetrypreserving perturbations of the fermion model, including interactions. Finally, we develop quantum circuit realizations of our model paving the way towards the creation of exact Majorana strong zero modes with digital quantum hardware.

Jan 1, 20260 citations

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.