Effect of slowly decaying long-range interactions on topological qubits

TL;DR

The authors address how slowly decaying long-range interactions with $f(r)\sim r^{-\alpha}$ ($0<\alpha<1$) affect the ground-state degeneracy of topological qubits. Using Coleman-style instanton techniques, they compute the ground-state splitting $\delta$ for several toy models—an all-to-all Ising chain, a quantum rotor chain with power-law couplings, and a nonlocal toy model—finding a universal stretched-exponential scaling $\log\delta\sim -C\,L^{(1+\alpha)/2}$. The results indicate that non-summable long-range perturbations can preserve a robust, yet weaker, form of ground-state splitting suppression compared to short-range cases, with precise leading-order constants derived in each model. These findings offer insight into the resilience of topological qubits under nonlocal perturbations and motivate extensions to 2D systems and Coulomb-like interactions. They also illustrate how instanton methods can yield controlled predictions for nonlocal perturbations where standard stability theorems fail.

Abstract

We study the robustness of topological ground state degeneracy to long-range interactions in quantum many-body systems. We focus on slowly decaying two-body interactions that scale like a power-law $1/r^α$ where $α$ is smaller than the spatial dimension; such interactions are beyond the reach of known stability theorems which only apply to short-range or rapidly decaying long-range perturbations. Our main result is a computation of the ground state splitting of several toy models, which are variants of the 1D Ising model $H = -\sum_i σ^z_i σ^z_{i+1} + λ\sum_{ij} |i-j|^{-α} σ^x_i σ^x_j$ with $λ> 0$ and $α< 1$. These models are also closely connected to the Kitaev p-wave wire model with power-law density-density interactions. In these examples, we find that the splitting $δ$ scales like a stretched exponential $δ\sim \exp(-C L^{\frac{1+α}{2}})$ where $L$ is the system size. Our computations are based on path integral techniques similar to the instanton method introduced by Coleman. We also study another toy model with long-range interactions that can be analyzed without path integral techniques and that shows similar behavior.

Figures (1)

• Figure 1: Minimal path $\bar{\theta}_n(\tau; \tau^*)$ for large but finite $L$.