Topological Superconductivity and Superfluidity (0803.3614v1)

Abstract: We construct time reversal invariant topological superconductors and superfluids in two and three dimensions which are analogous to the recently discovered quantum spin Hall and three-d $Z_2$ topological insulators respectively. These states have a full pairing gap in the bulk, gapless counter-propagating Majorana states at the boundary, and a pair of Majorana zero modes associated with each vortex. We show that the time reversal symmetry naturally emerges as a supersymmetry, which changes the parity of the fermion number associated with each time-reversal invariant vortex. In the presence of external T-breaking fields, non-local topological correlation is established among these fields, which is an experimentally observable manifestation of the emergent supersymmetry.

• The paper establishes a framework where fully gapped bulks coexist with gapless counter-propagating Majorana edge states in time-reversal invariant systems.
• The authors employ the Bogoliubov–de Gennes formalism to derive a Z2 classification for both 2D and 3D topological superconductors, highlighting parity flips under time-reversal.
• The work reveals emergent supersymmetry through Majorana zero modes at vortices, offering a pathway for future experimental validation in quantum materials.

Topological Superconductivity and Superfluidity: An Analysis

This paper, authored by Qi et al., explores the conceptualization and characterization of time-reversal invariant topological superconductors and superfluids in dimensions two and three. The research situates itself within the broader context of topological states of quantum matter, drawing analogy to the quantum spin Hall (QSH) and $Z_2$ topological insulators.

Summary of Core Contributions

The authors propose a framework where topological superconductors and superfluids host a full pairing gap in the bulk while supporting gapless counter-propagating Majorana states at the boundary. Furthermore, these states exhibit a pair of Majorana zero modes associated with each vortex. This work identifies time-reversal symmetry as a natural emergent supersymmetry altering the parity of the fermion number at time-reversal-invariant vortices.

Analytical Approach and Results

The authors employ the Bogoliubov-de Gennes (BdG) formalism to describe the systems and provide a $Z_2$ classification for both two-dimensional (2D) and three-dimensional (3D) cases. They highlight how the edge states in 2D topological superconductors form a helical Majorana liquid, analogous to those in QSH insulators but with significant differences stemming from their constituent Majorana fermions.

The haLLMark of the proposed topological superconductors is their robust edge states, which remain gapless under time-reversal invariant perturbations. This is characterized through a $Z_2$ classification, analogously derived by the transformation properties of the fermion number parity under time-reversal in the presence of topological defects. The parity operator $(-1)^{N_F}$ is shown to change sign under time-reversal operations in nontrivial states—a phenomenon indicative of an emergent supersymmetry.

Numerical and Theoretical Implications

The research presents a comprehensive derivation of the $Z_2$ topological classification criterion for TRI superconductors, establishing that fermion number parity flips occur around topologically nontrivial defects. This insight not only provides a refined method to classify such superconducting states but also impacts the broader understanding of topological matter where supersymmetry can be naturally realized without microscopic fine-tuning.

Future Directions

While the paper primarily focuses on theoretical advancements, it also prompts considerations for future experimental verification, especially pertaining to supersymmetry in condensed matter systems. The presence of non-local topological correlations induced by time-reversal breaking fields provides a pathway for potential experimentation in superconductors with pinned vortices in magnetic fields.

Conclusion

In conclusion, this work offers a rigorous theoretical framework and classification for understanding time-reversal invariant topological superconductors and superfluids. The implications of this research extend towards experimental observations of emergent supersymmetry, displaying a fundamental advancement in the field of condensed matter physics. Moving forward, the integration of these findings with complementary experimental studies promises to deepen the exploration of topological phases in quantum materials.