[Paper Review] Stability of Majorana Edge Zero Modes against Interactions

Tohru Koma|arXiv (Cornell University)|May 23, 2022

Cold Atom Physics and Bose-Einstein Condensates4 citations

TL;DR

This paper proves the stability of Majorana edge zero modes and the associated spectral gap in interacting Majorana chains under generic weak interactions with even fermion parity. Using a series expansion and Lieb-Robinson bounds, it establishes that the zero mode remains localized and the gap persists, with the edge mode's topological protection preserved under perturbations, and extends the result to ladders, showing a Z₂ index stability for odd numbers of legs.

ABSTRACT

We study an interacting Majorana chain with an open boundary condition. In the case without interactions, the system shows a prototypical Majorana edge zero mode in the sector of the ground state with a spectral gap above the sector. We prove that both of the Majorana edge zero mode and the non-vanishing spectral gap are stable against generic weak interactions whose fermion parity is even. We also deal with the corresponding ladder systems, and discuss the $\mathbb{Z}_2$ index for the Majorana edge zero modes.

Motivation & Objective

• To establish the stability of Majorana edge zero modes in interacting Majorana chains under generic weak interactions.
• To prove that the non-vanishing spectral gap above the ground state remains stable under such interactions.
• To extend the analysis to Majorana ladder systems and define a Z₂ topological index for edge modes.
• To demonstrate that the even-odd distinction of Majorana edge modes is preserved under weak even-parity perturbations.

Proposed method

• Constructs a series expansion for the Majorana edge zero mode operator γ in the presence of weak interactions.
• Applies a unitary transformation U(g) generated by a time-ordered exponential to map the interacting Hamiltonian to a free form.
• Uses the Lieb-Robinson bound with subexponentially decaying functions to prove locality of the transformed Majorana operators.
• Implements a local averaging procedure Π≥2m−1 to approximate the transformed operators and control non-locality.
• Derives a local decomposition Δm(c2ℓ−1) of U(g)c2ℓ−1U(g)† with subexponentially decaying norms.
• Relies on the fact that only operators with even fermion parity contribute to the commutator bounds, ensuring stability.

Experimental results

Research questions

• RQ1 Can Majorana edge zero modes survive weak interactions that preserve fermion parity in a one-dimensional chain?
• RQ2 Is the spectral gap above the ground state stable against such interactions?
• RQ3 How does the presence of multiple chains (ladders) affect the stability of edge modes, especially when interactions are introduced?
• RQ4 What topological index characterizes the stability of Majorana edge modes in interacting ladder systems?
• RQ5 Does the even-odd distinction of the number of edge modes remain invariant under weak even-parity perturbations?

Key findings

• The Majorana edge zero mode γ0 remains stable and localized under generic weak interactions with even fermion parity, as proven via a convergent series expansion.
• The spectral gap above the ground state remains non-vanishing under the same class of interactions, ensuring topological protection.
• For L-leg Majorana ladders, when L is odd, a single unpaired Majorana edge zero mode persists under weak even-parity interactions, indicating Z₂ topological stability.
• The Z₂ index for the edge mode is robust against weak perturbations with even fermion parity, implying that the even-odd character of the number of zero modes is preserved.
• The locality of the transformed Majorana operator U(g)c2ℓ−1U(g)† is proven via Lieb-Robinson bounds, with norm decay subexponential in distance from the edge.
• The proof relies on a decomposition of the transformed operator into local components Δm(c2ℓ−1), each with subexponentially decaying norm, ensuring physical realizability.