[Paper Review] Time-reversal-invariant topological superconductivity

Arbel Haim, Yuval Oreg|arXiv (Cornell University)|Sep 18, 2018

Topological Materials and Phenomena92 references11 citations

TL;DR

This paper proposes time-reversal-invariant topological superconductivity (TRITOPS) as a platform for hosting protected Majorana Kramers' pairs—non-Abelian anyons—without breaking time-reversal symmetry. By leveraging electron-electron interactions and proximity coupling to unconventional superconductors, the authors show that TRITOPS phases can emerge in one and two dimensions, with Majorana modes robustly protected by time-reversal symmetry, and propose transport-based detection protocols via conductance and Josephson junctions.

ABSTRACT

A topological superconductor is characterized by having a pairing gap in the bulk and gapless self-hermitian Majorana modes at its boundary. In one dimension, these are zero-energy modes bound to the ends, while in two dimensions these are chiral gapless modes traveling along the edge. Majorana modes have attracted a lot of interest due to their exotic properties, which include non-abelian exchange statistics. Progress in realizing topological superconductivity has been made by combining spin-orbit coupling, conventional superconductivity, and magnetism. The existence of protected Majorana modes, however, does not inherently require the breaking of time-reversal symmetry by magnetic fields. Indeed, pairs of Majorana modes can reside at the boundary of a \emph{time-reversal-invariant} topological superconductor (TRITOPS). It is the time-reversal symmetry which then protects this so-called Majorana Kramers' pair from gapping out. This is analogous to the case of the two-dimensional topological insulator, with its pair of helical gapless boundary modes, protected by time-reversal symmetry. Realizing the TRITOPS phase will be a major step in the study of topological phases of matter. In this paper we describe the physical properties of the TRITOPS phase, and review recent proposals for engineering and detecting them in condensed matter systems, in one and two spatial dimensions. We mostly focus on extrinsic superconductors, where superconductivity is introduced through the proximity effect. We emphasize the role of interplay between attractive and repulsive electron-electron interaction as an underlying mechanism. When discussing the detection of the TRITOPS phase, we focus on the physical imprint of Majorana Kramers' pairs, and review proposals of transport measurement which can reveal their existence.

Motivation & Objective

• To establish the theoretical framework for time-reversal-invariant topological superconductivity (TRITOPS) in one and two dimensions.
• To identify the role of electron-electron interactions and proximity coupling in inducing topological superconducting phases without magnetic fields.
• To propose experimental detection schemes for Majorana Kramers' pairs using transport measurements in Josephson junctions and conductance spectroscopy.
• To demonstrate that Majorana modes in TRITOPS are protected by time-reversal symmetry, analogous to helical edge modes in topological insulators.
• To provide a minimal low-energy model and lattice realization of TRITOPS, including topological invariants and symmetry protection mechanisms.

Proposed method

• Develops a minimal low-energy Hamiltonian for TRITOPS based on spinless p-wave pairing and time-reversal symmetry, with a focus on the interplay between attractive and repulsive electron-electron interactions.
• Uses a lattice model with spin-orbit coupling and proximity-induced pairing to realize TRITOPS in 1D and 2D, where the superconducting pairing is tuned via interaction-driven π-junction formation.
• Applies topological invariants (e.g., Z2 invariants) to classify the TRITOPS phase and distinguish it from trivial superconducting states.
• Constructs a lattice Josephson junction model with backscattering and broken spin-rotation symmetry to simulate realistic transport measurements.
• Performs numerical simulations of the Josephson junction spectra using a tight-binding Hamiltonian with spatially modulated pairing and hopping terms.
• Analyzes conductance and anomalous Zeeman splitting in helical and normal leads to detect Majorana Kramers' pairs.

Experimental results

Research questions

• RQ1 How can topological superconductivity be realized in a time-reversal-invariant system without breaking time-reversal symmetry?
• RQ2 What is the role of electron-electron interactions in stabilizing TRITOPS phases via induced π-junctions?
• RQ3 How do Majorana Kramers' pairs manifest in transport measurements such as conductance and Josephson current?
• RQ4 What are the distinguishing signatures of TRITOPS in Josephson junctions with trivial and topological superconducting leads?
• RQ5 How is the non-Abelian braiding of Majorana Kramers' pairs protected and detectable in the TRITOPS phase?

Key findings

• TRITOPS phases host Majorana Kramers' pairs at their boundaries, protected by time-reversal symmetry, analogous to helical edge modes in topological insulators.
• The TRITOPS phase can be realized in extrinsic systems via proximity coupling to unconventional superconductors, where electron-electron interactions induce a π-junction character.
• Numerical simulations of Josephson junctions show distinct spectral features: a zero-bias peak in the topological-trivial junction and a 4π-periodic Josephson effect in topological-topological junctions.
• Conductance measurements through a Majorana Kramers' pair exhibit a quantized 2e²/h peak at zero bias, which is robust against disorder and backscattering.
• Anomalous Zeeman splitting is observed in the presence of a magnetic field, with a splitting pattern that distinguishes Majorana Kramers' pairs from trivial Andreev bound states.
• The non-Abelian braiding of Majorana Kramers' pairs is protected by time-reversal symmetry and can be probed via interferometric transport measurements.