In variants of the 1D Ising model with power-law or all-to-all interactions for alpha less than 1, the topological ground state splitting scales as exp(-C L to the power (1+alpha)/2).
Quasi-locality bounds for quantum lattice systems. part ii. per- turbations of frustration-free spin models with gapped ground states,
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Effect of slowly decaying long-range interactions on topological qubits
In variants of the 1D Ising model with power-law or all-to-all interactions for alpha less than 1, the topological ground state splitting scales as exp(-C L to the power (1+alpha)/2).