Effect of slowly decaying long-range interactions on topological qubits

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 $α< 1$. In one variant, the power-law interactions are replaced by all-to-all interactions, $\fracλ{4 L^α}\sum_{ij} σ^x_i σ^x_j$, 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 $δ$ scales like a stretched exponential $δ\sim \exp(-C L^{\frac{1+α}{2}})$. 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.