Lieb-Robinson bounds, automorphic equivalence and LPPL for long-range interacting fermions

We prove a Lieb-Robinson bound for lattice fermion models with polynomially decaying interactions, which can be used to show the locality of the quasi-local inverse Liouvillian. This allows us to prove automorphic equivalence and the local perturbations perturb locally (LPPL) principle for these systems. The proof of the Lieb-Robinson bound is based on the work of Else et al. (2020), and our results also apply to spin systems. We explain why some newer Lieb-Robinson bounds for long-range spin systems cannot be used to prove the locality of the quasi-local inverse Liouvillian, and in some cases may not even hold for fermionic systems.

💡 Research Summary

The paper establishes a Lieb‑Robinson bound for lattice fermion systems whose interactions decay polynomially with distance, and then uses this bound to prove two fundamental results: automorphic equivalence of gapped ground states and the local perturbations perturb locally (LPPL) principle. The authors work on finite, surface‑regular graphs of dimension (D) and consider even, self‑adjoint interactions (\Phi) satisfying (|\Phi(Z)|\le C(1+\operatorname{diam}Z)^{-\alpha}) with (\alpha>D). They introduce a norm (|\Phi|_{\alpha,n}) that weights both the size of the support and its diameter, which is essential for controlling long‑range contributions.

The core technical advance is an improvement of existing Lieb‑Robinson bounds for long‑range systems. Starting from the finite‑range bound of