Random-matrix theory of Majorana fermions and topological superconductors

Beenakker, C. W. J.

doi:10.1103/revmodphys.87.1037

Cited by 272 publications

(306 citation statements)

References 181 publications

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“…Using this approach in combination with a scaling argument [24,30], we confirm the predicted universality of the exponent α [24,31,32] characterizing the asymptotic behavior of the ensemble averaged density of states ρ ∼ E α in the limit E → 0 for each of the ten symmetry classes, respectively. This approach makes transparent why the exponent α does not depend on the choice of the random matrix ensemble.…”

Section: Introductionsupporting

confidence: 60%

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Existence of zero-energy impurity states in different classes of topological insulators and superconductors and their relation to topological phase transitions

Kimme

Hyart

2016

Phys. Rev. B

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Existence of zero-energy impurity states in different classes of topological insulators and superconductors and their relation to topological phase transitions Kimme, Lukas; Hyart, Timo Kimme, L., & Hyart, T. (2016 We consider the effects of impurities on topological insulators and superconductors. We start by identifying the general conditions under which the eigenenergies of an arbitrary Hamiltonian H belonging to one of the Altland-Zirnbauer symmetry classes undergo a robust zero energy crossing as a function of an external parameter which can be, for example, the impurity strength. We define a generalized root of det H and use it to predict or rule out robust zero-energy crossings in all symmetry classes. We complement this result with an analysis based on almost degenerate perturbation theory, which allows a derivation of the asymptotic low-energy behavior of the ensemble averaged density of states ρ ∼ E α for all symmetry classes and makes it transparent that the exponent α does not depend on the choice of the random matrix ensemble. Finally, we show that a lattice of impurities can drive a topologically trivial system into a nontrivial phase, and in particular we demonstrate that impurity bands carrying extremely large Chern numbers can appear in different symmetry classes of two-dimensional topological insulators and superconductors. We use the generalized root of det H (k) to reveal a spiderweblike momentum space structure of the energy gap closings that separate the topologically distinct phases in p x + ip y superconductors in the presence of an impurity lattice.

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Section: Introductionsupporting

confidence: 60%

“…[24,31,32]. α relates to q(H ) in the following way: α > 1 in classes C and CII where q cannot be defined.…”

Section: Level Repulsion and Density Of States In Random Matrix mentioning

confidence: 99%

Existence of zero-energy impurity states in different classes of topological insulators and superconductors and their relation to topological phase transitions

Kimme

Hyart

2016

Phys. Rev. B

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“…The probability density for such a random matrix M is [53] (N dot + N). For large N dot + N the distribution of wave-function components is asymptotically Gaussian [48,54]:…”

mentioning

confidence: 99%

Approximating the Sachdev-Ye-Kitaev model with Majorana wires

2017

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The Sachdev-Ye-Kitaev (SYK) model describes a collection of randomly interacting Majorana fermions that exhibits profound connections to quantum chaos and black holes. We propose a solid-state implementation based on a quantum dot coupled to an array of topological superconducting wires hosting Majorana zero modes. Interactions and disorder intrinsic to the dot mediate the desired random Majorana couplings, while an approximate symmetry suppresses additional unwanted terms. We use random-matrix theory and numerics to show that our setup emulates the SYK model (up to corrections that we quantify) and discuss experimental signatures. DOI: 10.1103/PhysRevB.96.121119 Introduction. Majorana fermions provide building blocks for many novel phenomena. As one notable example, Majorana-fermion zero modes [1,2] capture the essence of non-Abelian statistics and topological quantum computation [3,4], and correspondingly now form the centerpiece of a vibrant experimental effort [5][6][7][8][9][10][11][12][13][14][15][16]. More recently, randomly interacting Majorana fermions governed by the "Sachdev-YeKitaev (SYK) model" [17][18][19] were shown to exhibit sharp connections to chaos, quantum-information scrambling, and black holes-naturally igniting broad interdisciplinary activity (see, e.g., [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]). The goal of this Rapid Communication is to exploit hardware components of a Majorana-based topological quantum computer for a tabletop implementation of the SYK model, thus uniting these very different topics.The SYK Hamiltonian reads

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“…Also, magnetic chains have their own characteristic properties such as long-range hopping between the sites [8,10,17,22,29]. Since the early work on the subject [30][31][32][33], the effects of disorder in the nanowire systems [34] and Kitaev's toy model [35] have been studied extensively. However, the special properties of magnetic chains have received relatively little attention [36,37].…”

Section: Introductionmentioning

confidence: 99%

Topological superconductivity and anti-Shiba states in disordered chains of magnetic adatoms

2016

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Regular arrays of magnetic atoms on a superconductor provide a promising platform for topological superconductivity. In this work, we study the effects of disorder in these systems, focusing on vacancies realized by missing magnetic atoms. We develop approaches that allow treatment of ferromagnetic dense chains as well as long-range hopping ferromagnetic and helical Shiba chains at arbitrary subgap energies. Vacancies in magnetic chains play an analogous role to magnetic impurities in a clean s-wave superconductor. A single vacancy in a topological chain gives rise to a low-lying "anti-Shiba" state below the band edge of a regular magnetic chain. Proliferation of the anti-Shiba band formed by a finite density of hybridized vacancy states leads to deterioration of the topological phase, which exhibits unusual fragility in a particular parameter region in dilute chains. We also consider local fluctuation in the Shiba coupling and discuss how vacancy states could contribute to experimental verification of topological superconductivity.

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Random-matrix theory of Majorana fermions and topological superconductors

Cited by 272 publications

References 181 publications

Existence of zero-energy impurity states in different classes of topological insulators and superconductors and their relation to topological phase transitions

Existence of zero-energy impurity states in different classes of topological insulators and superconductors and their relation to topological phase transitions

Approximating the Sachdev-Ye-Kitaev model with Majorana wires

Topological superconductivity and anti-Shiba states in disordered chains of magnetic adatoms

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