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Conjugate linear perturbations of Dirac operators and Majorana fermions

Ákos Nagy

TL;DR

We address the spectral analysis of perturbed Dirac operators by introducing generalized Jackiw–Rossi operators $H_{\\nabla,\\Phi}$ on Clifford-module bundles, where $H_{\\nabla,\\Phi}=\\slashed{D}_\\nabla+\\mathcal{A}_\\Phi$. A central result is that any conjugate-linear perturbation of a Dirac-type operator can be realized as half of such an operator via a doubling construction, enabling a Weitzenböck identity and concentration results around zeros of $\\Phi$. The framework yields kernel-conformal invariance and a canonical 2D model on $\\mathbb{C}$ that yields Fredholm indices tied to the zero index $k$ of $\\Phi$, and it provides a blueprint for higher-dimensional and Bogoliubov-de Gennes (BdG) generalizations. Overall, the work gives a geometric, modular approach to Majorana-like perturbations of Dirac operators across dimensions, with potential applications to topological phases and superconductors.

Abstract

We study a canonical class of perturbations of Dirac operators that are defined in any dimension and on any Hermitian Clifford module bundle. These operators generalize the 2-dimensional Jackiw-Rossi operator, which describes electronic excitations on topological superconductors. We also describe the low energy spectrum of these operators on complete surfaces, under mild hypotheses.

Conjugate linear perturbations of Dirac operators and Majorana fermions

TL;DR

We address the spectral analysis of perturbed Dirac operators by introducing generalized Jackiw–Rossi operators on Clifford-module bundles, where . A central result is that any conjugate-linear perturbation of a Dirac-type operator can be realized as half of such an operator via a doubling construction, enabling a Weitzenböck identity and concentration results around zeros of . The framework yields kernel-conformal invariance and a canonical 2D model on that yields Fredholm indices tied to the zero index of , and it provides a blueprint for higher-dimensional and Bogoliubov-de Gennes (BdG) generalizations. Overall, the work gives a geometric, modular approach to Majorana-like perturbations of Dirac operators across dimensions, with potential applications to topological phases and superconductors.

Abstract

We study a canonical class of perturbations of Dirac operators that are defined in any dimension and on any Hermitian Clifford module bundle. These operators generalize the 2-dimensional Jackiw-Rossi operator, which describes electronic excitations on topological superconductors. We also describe the low energy spectrum of these operators on complete surfaces, under mild hypotheses.

Paper Structure

This paper contains 1 section, 1 theorem, 17 equations.

Table of Contents

  1. Generalized Jackiw--Rossi theory on pseudo-Riemannian manifolds

Key Result

Lemma 1.5

Let $E_1$ and $E_2$ be Hermitian vector bundles over a smooth, closed manifold, $X$. Let be a Dirac-type operator, and be a conjugate linear bundle map. Then $\slashed{E} \mathrel{\vcenter{\hbox{.}\hbox{.}}}= E_1 \oplus E_2$ is a Hermitian Clifford module bundle, where the Clifford multiplication, $c$, is induced by the symbol of $D$. Furthermore, let is a generalized Jackiw--Rossi operator as

Theorems & Definitions (5)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Definition 1.4
  • Lemma 1.5