Quantifying robustness and locality of Majorana bound states in interacting systems

TL;DR

This work develops a general framework to quantify the robustness and locality of Majorana bound states in interacting systems by defining ground-state MBSs within the two-level sector spanned by |e⟩ and |o⟩ and extracting regionally localized operators via the fermionic partial trace γ_R. It derives rigorous bounds on environment-induced couplings and energy splitting that depend on the localization of the ground-state MBSs, and introduces gauge-aware quality measures, including a many-body Majorana polarization, to optimize locality bounds. The framework applies to both interacting and non-interacting systems and is illustrated with an interacting Kitaev chain, showing how sweet-spot localization, spatial profiles, and dot-coupling bounds align with theoretical predictions. These results provide practical criteria for assessing the feasibility of coupling-based braiding and topological protection in realistic, strongly interacting platforms, with potential applications in experimental Kitaev-chain setups and beyond.

Abstract

Protecting qubits from perturbations is a central challenge in quantum computing. Topological superconductors with separated Majorana bound states (MBSs) provide a strong form of protection that only depends on the locality of perturbations. While the link between MBS separation, robust degeneracy, and protected braiding is well understood in non-interacting systems, recent experimental progress in short quantum-dot-based Kitaev chains highlights the need to establish these connections rigorously for interacting systems. We do this by defining MBSs from many-body ground states and show how their locality constrains their coupling to an environment. This, in turn, quantifies the protection of the energy degeneracy and the feasibility of non-abelian braiding.

Figures (5)

• Figure 1: We show that the separation of MBSs implies energy protection and braiding, using methods applicable in interacting systems. (a). We consider a system $S$, described by two MBSs. It interacts with an environment $B$, with a coupling acting in a subregion $R$. (b). The couplings to the environment are bounded by the localization of the MBSs, quantified by the partial trace. (c). A weakly coupled MBS almost commutes with the Hamiltonian, implying stability of the energy spectrum. The localization of the MBSs also determines the feasibility of non-abelian braiding.
• Figure 2: Quality measures and locality of ground-state MBSs for an 8-site interacting Kitaev chain. The spatial profile of the norm of the reduced MBSs is plotted in (a), for the two sets of parameters marked in (b). The heatmaps in (b) show the Majorana polarization and the local parity for a single region covering the left half of the system. $\delta\mu$ is a global detuning of the chemical potential away from the interacting sweet spot in \ref{['eq:sweetspot']}. The quality measures quantify the protection of the system against perturbations acting on half the chain, and give a hint of the phase diagram of this model.
• Figure 3: The ground-state energy splitting $\delta E$ (solid) of an (initially degenerate) interacting Kitaev chain when coupled to a quantum dot, varying the dot level $\varepsilon_d$. The dashed and dotted lines represent the bounds in \ref{['eq:non_perturbative_bound_ex', 'eq:perturbative_bound_ex']}, respectively, normalized by $\lambda$. We use the parameters $t=\Delta=U/2$, $\mu \approx -2.7321$ at the edges and twice that in the bulk, $\phi = \pi/6$ and $\lambda = t/100$. The chemical potential is chosen according to Ref. katsuraExactGroundStates2015 to get degenerate ground states.
• Figure 4: When coupling multiple systems, each with a low-energy fermionic mode, we can formulate an effective theory involving only the ground-state MBSs. With three systems, non-abelian braiding can be implemented if the desired couplings (solid lines) can be tuned and the undesired couplings (dashed lines) stay small. We put bounds on the undesired couplings and specify how to maximize the desired ones.
• Figure 5: In the gauge that maximizes Majorana separation within each coupling region, the odd part of the coupling Hamiltonian generates four terms for every pair of coupled systems. This is illustrated to the left. To reduce the number of terms, we perform a gauge rotation, leading to the simplified structure to the right.