Dirac and Weyl Superconductors in Three Dimensions (1402.7070v1)

Abstract: We introduce the concept of 3D Dirac (Weyl) superconductors (SC), which have protected bulk four(two)-fold nodal points and surface Andreev arcs at zero energy. We provide a sufficient criterion for realizing them in centrosymmetric SCs with odd-parity pairing and mirror symmetry, e.g., the nodal phases of Cu$_x$Bi$_2$Se$_3$. Pairs of Dirac nodes appear in a mirror-invariant plane when the mirror winding number is nontrivial. Breaking mirror symmetry may gap Dirac nodes producing a topological SC. Each Dirac node evolves to a nodal ring when inversion-gauge symmetry is broken. A Dirac node may split into a pair of Weyl nodes, only when time-reversal symmetry is broken.

Dirac and Weyl Superconductors in Three Dimensions: An Expert Analysis

This paper introduces the concept of Dirac and Weyl superconductors (SCs) in three-dimensional (3D) space, expanding upon the paper of topological states of matter. By exploring the realization of Dirac (Weyl) SCs with protected bulk nodal points and unique surface phenomena, the paper draws attention to the conditions under which these SC phases occur and their stability under symmetry alterations.

A significant contribution of the paper is providing a criterion for the realization of 3D Dirac SCs in centrosymmetric SCs with odd-parity pairing and robust mirror symmetry. Notably, it discusses how nodal phases like Cu$_x$Bi$_2$Se$_3$ exemplify these conditions, resulting in Dirac nodes appearing in a mirror-invariant plane when the mirror winding number is non-trivial. The authors derive an effective Hamiltonian exhibiting these nodes' behavior, revealing a complex interplay of symmetries that stabilizes the Dirac points under stringent conditions.

Numerically, the paper explores the features of Majorana Kramers arcs arising on surfaces due to these nodes. The presence and specific characteristics of the Majorana modes on the surface depend crucially on maintaining key symmetries like time-reversal (TRS), the product of TRS and inversion-gauge symmetry, and mirror symmetry. However, when these symmetries are altered individually, the nodal structure changes predictably:

1. Breaking Mirror Symmetry: Fully gapping the bulk Dirac nodes can lead to topological SCs, shown by specific numerically simulated surface states, where Majorana arcs become helical Majorana surface states.
2. Breaking Inversion-Gauge Symmetry: This extends a Dirac node into a nodal ring within the mirror invariant plane. As supported by simulation results, the bulk forms nodal rings, and surface states show zero energy modes with extended areas inside the projected rings.
3. Breaking TRS: A Dirac node may split into two Weyl nodes, which lack the protection given by Chern numbers in SC contexts unless TRS is broken, permitting them to exist off the plane with strong numerical visualization supporting this evolutionary pathway.

These findings denote that the symmetry conditions are integral to the qualitative behavior of 3D Dirac and Weyl SCs. From a theoretical standpoint, such SCs provide a fertile ground for exploring non-trivial topological surface states and broaden the understanding of topological superconductivity. The presence of nodal phases is evident not only in Cu$_x$Bi$_2$Se$_3$ but potentially in other materials too, suggesting further experimental avenues for investigation using technologies like the angular resolved photoemission spectroscopy or scanning tunneling microscope.

The paper posits that beyond Cu$_x$Bi$_2$Se$_3$, these insights could aid in identifying other candidate materials conjuring odd-parity SCs stabilized by similar topological constraints. Therefore, the haLLMark of Dirac SCs extends the concept of symmetry-protected topological phases into superconductors, potentially influencing future developments in condensed matter physics and conventional and quantum computing technologies.