MERLIN SCIENCE — Decidable Spectral Gap for Frustration-Free Spin-1/2 Chains — E8 Intelligence Research

Here is the finding. For frustration-free spin-1/2 chains, the spectral gap question is decidable. We can completely classify these systems into gapped and gapless phases. This holds even though the general problem for translationally invariant Hamiltonians is undecidable, as shown by Cubitt, Perez-Garcia, and Wolf in 2015. Here is the context. We are talking about Hamiltonians written as a sum of local terms, each positive semidefinite, with a ground state annihilated by every term. That is the frustration-free condition. The spectral gap is the energy difference between the first excited state and the ground state. For a general quantum many-body system, determining whether that gap closes in the thermodynamic limit is notoriously hard. But for this restricted class, the mathematics gives us a clean answer. Here is the mechanism. The key is the matrix product state representation. The ground state of a gapped phase has finite correlation length. Its MPS transfer matrix is injective Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com