Topological phases with parafermions: theory and blueprints (1504.02476v1)

Abstract: We concisely review the recent evolution in the study of parafermions -- exotic emergent excitations that generalize Majorana fermions and similarly underpin a host of novel phenomena. First we illustrate the intimate connection between Z_3-symmetric "spin" chains and one-dimensional parafermion lattice models, highlighting how the latter host a topological phase featuring protected edge zero modes. We then tour several blueprints for the laboratory realization of parafermion zero modes -- focusing on quantum Hall/superconductor hybrids, quantum Hall bilayers, and two-dimensional topological insulators -- and describe striking experimental fingerprints that they provide. Finally, we discuss how coupled parafermion arrays in quantum Hall architectures yield topological phases that potentially furnish hardware for a universal, intrinsically decoherence-free quantum computer.

• The paper demonstrates that parafermion chains exhibit topological phases with edge-zero modes and non-Abelian fractional statistics.
• It outlines experimental blueprints using quantum Hall setups, bilayers, and 2D topological insulators to realize parafermions.
• The work emphasizes that chirality and strong interactions are crucial for stabilizing parafermion zero modes, paving the way for advanced quantum computation.

Overview of Topological Phases with Parafermions: Theory and Blueprints

The paper by Alicea and Fendley reviews theoretical advancements and experimental proposals related to parafermions—quasi-particle excitations that offer generalizations of Majorana modes. These quasi-particles emerge from strongly interacting systems and promise features advantageous for the development of quantum computing devices that could operate without decoherence.

Parafermions originate as solutions to interacting models, with the simplest examples being the $\mathbb{Z}_3$ parafermions obtained from $\mathbb{Z}_3$ symmetric spin chains. These lattice models exhibit topological phases with edge-localized, zero-energy modes reminiscent of Majorana zero modes but are inherently more complex due to their fractionalized nature. The topological robustness of parafermion zero modes in such systems is contingent on the interplay between interactions (induced by the chirality of the model), which provides a non-trivial distinction absent in the Majorana case.

Key Theoretical Insights

The work elaborates on the parafermion chain, explicitly showing how its Hamiltonian connects to clock models with $n$-fold symmetry. The Hamiltonian in the parafermion framework reveals a topological phase when the interaction terms create protected edge states. These zero modes at system edges exhibit non-Abelian fractional statistics leading to degenerate ground states. However, unlike Majorana zero modes, strong zero modes only prevail under specific chiral conditions; otherwise, the system may only admit weak zero modes which do not yield degeneracies in the entire energy spectrum.

Experimental Proposals

A significant portion of the discussion is devoted to experimental "blueprints" that leverage known physical systems to realize parafermion zero modes:

1. Quantum Hall/Superconductor Hybrids: A setup utilizing $\nu = 2/3$ fractional quantum Hall states with superconducting proximity effects presents one method. Incompatible gapping processes at the interface between direct tunneling and singlet pairing can lead to parafermions at domain walls.
2. Quantum Hall Bilayers: Proposals are made for quantum Hall bilayer systems with crossed tunneling for parafermion realization. Here, non-local anyon condensation plays a critical role, notably through processes that alter the topological charge configurations across regions, allowing for gapless modes at domain separations.
3. 2D Topological Insulator Edges: Interacting edges of a 2D topological insulator can support parafermion zero modes when combined with superconductivity and careful management of intrinsic electron-electron interactions, such as reinforcing stability against certain perturbations.

Implications and Future Directions

The analysis in the paper suggests that parafermions hold the potential for both advancing basic understanding of topological phases in strongly-correlated systems and contributing to advancements in quantum computing technology. Those findings are particularly intriguing given the scope for these zero modes in potentially implementing gates that are richer than those provided by systems limited to Majorana zero modes. Importantly, Fibonacci anyons—a more exotic quasi-particle believed crucial for universal quantum computation—can emerge via specifically designed parafermion networks, thereby offering pathways to universal topological quantum computation.

In summary, the body of work surveyed by Alicea and Fendley sets a strong foundation for continued exploration and holds promise of functional utility in robust quantum information systems. The theoretical insights into the chirality and interaction requirements help clarify the conditions necessary to experimentally realize and manipulate these complex topological states.