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 $1/r^{\hat{I}}\pm$ where $\hat{I}\pm$ 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 $H=-{\sum }_i\ddot{I}{ƒ}_i^z\ddot{I}{ƒ}_{i+1}^z+\hat{I}»{\sum }_{ij}|i-j{|}^{-\hat{I}\pm }\ddot{I}{ƒ}_i^x\ddot{I}{ƒ}_j^x$ with $\hat{I}»>0$ and $\hat{I}\pm <1$. In one variant, the power-law interactions are replaced by all-to-all interactions, $\frac{\hat{I}}{»}4L^{\hat{I}}\pm {\sum }_{ij}\ddot{I}{ƒ}_i^x\ddot{I}{ƒ}_j^x$, 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 $\hat{I}´$ scales like a stretched exponential $\hat{I}´\sim \exp (-C{L}^{\frac{1+\hat{I}\pm }{2}})$. 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.

20 pages