Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1806.02993 · Physical review. B./Physical review. B

Resilience of Majorana fermions in the face of disorder

Alireza Habibi, Somaye Jafari, Shahin Rouhani

Published Jul 26, 2018• 20 citations
arXivHTMLPDF

Abstract

We elucidate the reduction of the winding number caused by the on-site disorder in a next-nearest-neighbor XY model. When disorder becomes strong enough, Majorana edge modes become critically extended, beyond which they collapse into Anderson localized (AL) states in the bulk, resulting in a topological Anderson insulating state. We identify a resilience threshold ${W}_{t}$ for every pair of Majorana fermions (MFs). In response to increasing disorder, every pair of MFs collapse into AL states in the bulk beyond their resilience threshold. For very strong disorder, all Majorana fermions collapse, and a topologically trivial state is obtained. We show that the threshold values are related to the localization length of Majorana fermions, which can be efficiently calculated by an appropriate modification of the transfer-matrix method. At the topological transition point, the localization length of the zero modes diverges, and the system becomes scale invariant. The number of peaks in the localization length as a function of disorder strength determines the number of zero modes in the clean state before disorder is introduced. This finding elevates the transfer-matrix method to the level of a tool for the determination of the topological index of both clean and disordered systems.

DOI
10.1103/physrevb.98.035142

Related papers

Topological phase diagram of the disordered 2XY model in presence of generalized Dzyaloshinskii–Moriya interaction

Alireza Habibi, Rasoul Ghadimi, S A Jafari

Abstract The topological index of a system determines its edge physics. However, in situations such as strong disorder where due to level repulsion the spectral gap closes, the topological indices are not well-defined. In this paper, we show that the localization length of zero modes determined by the transfer matrix method reveals much more information than the topological index. The localization length can provide not only information about the topological index of the Hamiltonian itself, but it can also provide information about the topological indices of the ‘relative’ Hamiltonians. As a case study, we study a generalized XY model (2XY model) further augmented by a generalized Dziyaloshinskii–Moriya-like (DM) interaction parameterized by that after fermionization breaks the time-reversal invariance. The parent Hamiltonian at which belongs to the BDI class is indexed by an integer winding number while the daughter Hamiltonian which belongs to class D is specified by a Z 2 index . We show that the localization length, in addition to determining Z 2 , can count the number of Majorana zero modes leftover at the boundary of the daughter Hamiltonian—which are not protected by the winding number anymore. Therefore the localization length outperforms the standard topological indices in two respects: (i) it is much faster and more accurate to calculate and (ii) it can count the winding number of the parent Hamiltonian by looking into the edges of the daughter Hamiltonian.

arXiv:1807.01339•Journal of Physics Condensed Matter•Aug 30, 2019•5 citations

Majorana Zero Modes and Bulk-Boundary Correspondence at Quantum Criticality

S Rahul, Ravi Kumar, Y R Kartik +1 more

Majorana zero modes are well studied in the gapped phases of topological systems. We investigate Majorana zero modes at the topological quantum criticality in one dimensional topological superconducting model with longer range interaction. We identify stable localized Majorana zero modes appearing at criticality under certain conditions. Topological invariant number for these non-trivial criticalities is obtained from zeros of a complex function associated with the Hamiltonian. Behavior of parametric curve at criticalities validate the invariant obtained and account for the appearance of Majorana zero modes at criticality. Trivial and non-trivial topological nature of criticality due to the presence of multicritical point cause an unusual topological transition along the critical line. We observe and investigate this unique transition in terms of eigenvalue spectrum. Appearance of MZMs at criticality demands integer value of topological invariant number in order to validate the concept of bulk-boundary correspondence. Hence we propose a scheme to separate the invariant number into fractional and integer contribution to establish bulk-boundary correspondence at criticality.

arXiv:1906.04462•Journal of the Physical Society of Japan•Aug 11, 2021•12 citations

Majorana wave-function oscillations, fermion parity switches, and disorder in Kitaev chains

Suraj Hegde, Smitha Vishveshwara

We study the decay and oscillations of Majorana fermion wave functions and ground-state (GS) fermion parity in one-dimensional topological superconducting lattice systems. Using a Majorana transfer matrix method, we find that Majorana wave-function properties are encoded in the associated Lyapunov exponent, which in turn is the sum of two independent components: a ``superconducting component,'' which characterizes the gap induced decay, and the ``normal component,'' which determines the oscillations and response to chemical potential configurations. The topological phase transition separating phases with and without Majorana end modes is seen to be a cancellation of these two components. We show that Majorana wave-function oscillations are completely determined by an underlying nonsuperconducting tight-binding model and are solely responsible for GS fermion parity switches in finite-sized systems. These observations enable us to analytically chart out wave-function oscillations, the resultant GS parity configuration as a function of parameter space in uniform wires, and special parity switch points where degenerate zero energy Majorana modes are restored in spite of finite size effects. For disordered wires, we find that band oscillations are completely washed out leading to a second localization length for the Majorana mode and the remnant oscillations are randomized as per Anderson localization physics in normal systems. Our transfer matrix method further allows us to (i) reproduce known results on the scaling of midgap Majorana states and demonstrate the origin of its log-normal distribution, (ii) identify contrasting behavior of disorder-dependent GS parity switches for the cases of even versus odd number of lattice sites, and (iii) chart out the GS parity configuration and associated parity switch points as a function of disorder strength.

arXiv:1603.03394•Physical review. B./Physical review. B•Sep 30, 2016•83 citations

First-order topological phase transitions and disorder-induced Majorana modes in interacting fermion chains

Shruti Agarwal, Shreekant Gawande, Satoshi Nishimoto +2 more

Using a combination of the mean-field Bogoliubov--de Gennes approach and the density matrix renormalization group method, we discover first-order topological transitions between topological superconducting and trivial insulating phases in a sawtooth lattice of intersite attractive fermions. The topological characterization of different phases is achieved in terms of winding numbers, Majorana edge modes, and entanglement spectra. By studying the effect of disorder on first-order topological phase transitions, we establish disorder-induced topological phase coexistence as a mechanism for generating a finite density of Majorana particles.

arXiv:2204.06306•Physical review. B./Physical review. B•Mar 20, 2023•1 citation

Strong zero modes in random Ising-Majorana chains

Saurav Kantha, Nicolas Laflorencie

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:2603.05313•Mar 5, 2026•0 citations

Topologically nontrivial multicritical points

Ravi Kumar, Pasquale Marra

Recently, the intriguing interplay between topology and quantum criticality has been unveiled in one-dimensional topological chains with extended nearest-neighbor couplings. In these systems, topologically distinct critical phases emerge with localized edge modes despite the vanishing bulk gap. In this work, we study the topological multicritical points at which distinct gapped and critical phases intersect. Specifically, we consider a topological chain with coupling up to the third nearest neighbors, which shows stable localized edge modes at the multicritical points. These points possess only nontrivial gapped and critical phases around them and are also characterized by the quadratic dispersion around the gap-closing points. We characterize the topological multicritical points in terms of the topological invariant obtained from the zeros of the complex function associated with the Hamiltonian. Further, we analyze the nature of zeros in the vicinity of the multicritical points by calculating the discriminants of the associated polynomial. The discriminant uniquely identifies the topological multicritical points and distinguishes them from the trivial ones. Moreover, we identify the underlying physical mechanism in terms of kinetic inversion in higher-order terms. We finally study the robustness of the zero-energy modes at the multicritical points at weak disorder strengths, and reveal the presence of a topologically nontrivial gapless Anderson-localized phase at strong disorder strengths.

arXiv:2507.11120•Physical review. B./Physical review. B•Jun 15, 2026•0 citations
We elucidate the reduction of the winding number caused by the on-site disorder in a next-nearest-neighbor XY model. When disorder becomes strong enough, Majorana edge modes become critically extended, beyond which they collapse into Anderson localized (AL) states in the bulk, resulting in a topological Anderson insulating state. We identify a resilience threshold Wt{W}_{t}Wt​ for every pair of Majorana fermions (MFs). In response to increasing disorder, every pair of MFs collapse into AL states in the bulk beyond their resilience threshold. For very strong disorder, all Majorana fermions collapse, and a topologically trivial state is obtained. We show that the threshold values are related to the localization length of Majorana fermions, which can be efficiently calculated by an appropriate modification of the transfer-matrix method. At the topological transition point, the localization length of the zero modes diverges, and the system becomes scale invariant. The number of peaks in the localization length as a function of disorder strength determines the number of zero modes in the clean state before disorder is introduced. This finding elevates the transfer-matrix method to the level of a tool for the determination of the topological index of both clean and disordered systems.
Abstract The topological index of a system determines its edge physics. However, in situations such as strong disorder where due to level repulsion the spectral gap closes, the topological indices are not well-defined. In this paper, we show that the localization length of zero modes determined by the transfer matrix method reveals much more information than the topological index. The localization length can provide not only information about the topological index of the Hamiltonian itself, but it can also provide information about the topological indices of the ‘relative’ Hamiltonians. As a case study, we study a generalized XY model (2XY model) further augmented by a generalized Dziyaloshinskii–Moriya-like (DM) interaction parameterized by that after fermionization breaks the time-reversal invariance. The parent Hamiltonian at which belongs to the BDI class is indexed by an integer winding number while the daughter Hamiltonian which belongs to class D is specified by a Z 2 index . We show that the localization length, in addition to determining Z 2 , can count the number of Majorana zero modes leftover at the boundary of the daughter Hamiltonian—which are not protected by the winding number anymore. Therefore the localization length outperforms the standard topological indices in two respects: (i) it is much faster and more accurate to calculate and (ii) it can count the winding number of the parent Hamiltonian by looking into the edges of the daughter Hamiltonian.
Majorana zero modes are well studied in the gapped phases of topological systems. We investigate Majorana zero modes at the topological quantum criticality in one dimensional topological superconducting model with longer range interaction. We identify stable localized Majorana zero modes appearing at criticality under certain conditions. Topological invariant number for these non-trivial criticalities is obtained from zeros of a complex function associated with the Hamiltonian. Behavior of parametric curve at criticalities validate the invariant obtained and account for the appearance of Majorana zero modes at criticality. Trivial and non-trivial topological nature of criticality due to the presence of multicritical point cause an unusual topological transition along the critical line. We observe and investigate this unique transition in terms of eigenvalue spectrum. Appearance of MZMs at criticality demands integer value of topological invariant number in order to validate the concept of bulk-boundary correspondence. Hence we propose a scheme to separate the invariant number into fractional and integer contribution to establish bulk-boundary correspondence at criticality.
We study the decay and oscillations of Majorana fermion wave functions and ground-state (GS) fermion parity in one-dimensional topological superconducting lattice systems. Using a Majorana transfer matrix method, we find that Majorana wave-function properties are encoded in the associated Lyapunov exponent, which in turn is the sum of two independent components: a ``superconducting component,'' which characterizes the gap induced decay, and the ``normal component,'' which determines the oscillations and response to chemical potential configurations. The topological phase transition separating phases with and without Majorana end modes is seen to be a cancellation of these two components. We show that Majorana wave-function oscillations are completely determined by an underlying nonsuperconducting tight-binding model and are solely responsible for GS fermion parity switches in finite-sized systems. These observations enable us to analytically chart out wave-function oscillations, the resultant GS parity configuration as a function of parameter space in uniform wires, and special parity switch points where degenerate zero energy Majorana modes are restored in spite of finite size effects. For disordered wires, we find that band oscillations are completely washed out leading to a second localization length for the Majorana mode and the remnant oscillations are randomized as per Anderson localization physics in normal systems. Our transfer matrix method further allows us to (i) reproduce known results on the scaling of midgap Majorana states and demonstrate the origin of its log-normal distribution, (ii) identify contrasting behavior of disorder-dependent GS parity switches for the cases of even versus odd number of lattice sites, and (iii) chart out the GS parity configuration and associated parity switch points as a function of disorder strength.
Using a combination of the mean-field Bogoliubov--de Gennes approach and the density matrix renormalization group method, we discover first-order topological transitions between topological superconducting and trivial insulating phases in a sawtooth lattice of intersite attractive fermions. The topological characterization of different phases is achieved in terms of winding numbers, Majorana edge modes, and entanglement spectra. By studying the effect of disorder on first-order topological phase transitions, we establish disorder-induced topological phase coexistence as a mechanism for generating a finite density of Majorana particles.
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.
Recently, the intriguing interplay between topology and quantum criticality has been unveiled in one-dimensional topological chains with extended nearest-neighbor couplings. In these systems, topologically distinct critical phases emerge with localized edge modes despite the vanishing bulk gap. In this work, we study the topological multicritical points at which distinct gapped and critical phases intersect. Specifically, we consider a topological chain with coupling up to the third nearest neighbors, which shows stable localized edge modes at the multicritical points. These points possess only nontrivial gapped and critical phases around them and are also characterized by the quadratic dispersion around the gap-closing points. We characterize the topological multicritical points in terms of the topological invariant obtained from the zeros of the complex function associated with the Hamiltonian. Further, we analyze the nature of zeros in the vicinity of the multicritical points by calculating the discriminants of the associated polynomial. The discriminant uniquely identifies the topological multicritical points and distinguishes them from the trivial ones. Moreover, we identify the underlying physical mechanism in terms of kinetic inversion in higher-order terms. We finally study the robustness of the zero-energy modes at the multicritical points at weak disorder strengths, and reveal the presence of a topologically nontrivial gapless Anderson-localized phase at strong disorder strengths.