Effect of slowly decaying long-range interactions on topological qubits

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 $1/r^\alpha$ where $\alpha$ 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 $H = -\sum_i \sigma^z_i \sigma^z_{i+1} + \lambda \sum_{ij} |i-j|^{-\alpha} \sigma^x_i \sigma^x_j$ with $\lambda > 0$ and $\alpha < 1$. In one variant, the power-law interactions are replaced by all-to-all interactions, $\frac{\lambda}{4 L^\alpha}\sum_{ij} \sigma^x_i \sigma^x_j$, where $L$ is the system size, while the other variant has true power-law interactions but is built out of quantum rotors rather than Ising spins. 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 $\delta$ scales like a stretched exponential $\delta \sim \exp(-C L^{\frac{1+\alpha}{2}})$. 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.