Helical liquids and Majorana bound states in quantum wires (1003.1145v2)

Abstract: We show that the combination of spin-orbit coupling with a Zeeman field or strong interactions may lead to the formation of a helical liquid in single-channel quantum wires. In a helical liquid, electrons with opposite velocities have opposite spin precession. We argue that zero-energy Majorana bound states are formed in various situations when the wire is situated in proximity to a conventional s-wave superconductor. This occurs when the external magnetic field, the superconducting gap, or, in particular, the chemical potential vary along the wire. We discuss experimental consequences of the formation of the helical liquid and the Majorana bound states.

• The paper introduces a detailed Hamiltonian model that shows how spin-orbit coupling, Zeeman fields, and superconducting proximity create a helical liquid supporting Majorana bound states.
• It establishes that a quantum phase transition occurs when B² exceeds Δ² + μ², a critical condition for the emergence of zero-energy Majorana states.
• Spatial modulation of magnetic and superconducting parameters is demonstrated as a practical strategy for manipulating non-Abelian Majorana states in quantum computing applications.

Helical Liquids and Majorana Bound States in Quantum Wires

The paper "Helical liquids and Majorana bound states in quantum wires" by Oreg, Refael, and von Oppen investigates the conditions under which a helical liquid forms in quantum wires and outlines the implications for generating and manipulating Majorana bound states, which hold significant potential for quantum computation. The paper considers system setups that include spin-orbit coupling, external magnetic fields, and superconducting and interacting effects in one-dimensional systems like quantum wires.

The central thesis concerns the formation of helical liquids in quantum wires characterized by strong spin-orbit coupling and their capability to host Majorana bound states when brought into proximity with conventional $s$-wave superconductors. A helical liquid is a state in which electrons with opposite linear momenta have opposite spin orientations. The combination of a helical liquid and the presence of a non-uniform magnetic field, superconducting gap, or chemical potential can lead to zero-energy Majorana bound states, an essential aspect for the development of fault-tolerant quantum memory and computation due to their non-local nature.

Key Findings

The paper concludes that when a quantum wire exhibits specific conditions—namely, when it is subjected to spin-orbit coupling in the presence of a Zeeman field or electron-electron interactions—a unique state of matter emerges. This state is described as helical liquid, where the characterization of velocities and spin orientations potentially leads to the generation of Majorana fermions.

Crucial results from the model include:

• The paper provides a detailed Hamiltonian formulation for the quantum wire system, considering spin-orbit coupling, magnetic fields, and the influence of superconductors.
• It denotes the conditions for the emergence of Majorana states, emphasizing the dependence on magnetic field strength $B$, superconducting gap $\Delta$, and chemical potential $\mu$.
• A central finding is that zero-energy Majorana bound states can form at localized regions of a wire when $B^2$ exceeds $\Delta^2 + \mu^2$, marking a quantum phase transition.
• The authors demonstrate that varying these parameters spatially along a wire can effectively induce and manipulate Majorana states, laying out specific cases where a change in magnetic field, superconducting gap, or chemical potential fosters the formation of these states.

Specific configurations allow the realization of experimental setups conducive to these phenomena, such as a wire bent into a ring, where the appropriate modulation of chemical potential via micrometer-sized gates could simplify Majorana state manipulation. Additionally, the manuscript provides analytic solutions and energy spectra indicative of the feasible realization of Majorana bound states in quantum wires.

Implications

From a theoretical standpoint, this research advances the understanding of Majorana fermions within condensed matter—especially in one-dimensional systems—thereby bridging a gap between complex theoretical predictions and practical implementations. The implications for quantum computing are profound, primarily because Majorana bound states offer non-local storage of information, potentially leading to enhanced fault tolerance in quantum computation schemes.

Practically, the configurations discussed, particularly those involving networks of quantum wires, suggest a scalable path with which to explore Majorana fermions’ non-Abelian statistics, a crucial feature for implementing topological quantum computations. Subsequent scientific work could expand on the experimental techniques, such as using gate and Josephson effects directly on the wire systems, for robust and observable manipulation of Majorana states.

This paper serves as a theoretical framework for efforts to catalyze the next era of quantum computational technologies, offering a plausible and experimentally viable route to harnessing the elusive and promising properties of Majorana bound states. Continued research in this domain is necessary to translate these theoretical insights into experimental realizations capable of advancing quantum information processing.