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Borel Complexity of the Gapped-Phase Relation for Quantum Spin Systems

Zongmin Liu

Published Dec 21, 2024• 0 citations
Publisher

Abstract

We study the descriptive complexity of the relation that two finite-range translation-invariant quantum spin interactions belong to the same gapped phase. The key observation is that this relation is exactly an increasing union of arc-connection relations of closed uniformly gapped sets, bringing the classification problem into classical continuum topology. On the Polish space of finite-range interactions, the gapped set is Borel of rank at most four under standard definitions of the spectral gap, and the gap functional is upper semicontinuous with attained supremum. On the uniformly gap-stable interior, the phase relation is smooth: its classes are the countably many open connected components, with a locally constant complete invariant. Standard frustration-free models satisfying local topological quantum order lie in this interior, so any descriptive complexity is confined to the unstable boundary. Homogeneity of the gap yields a radial reduction for finite-dimensional coupling families. Consequently, two-coupling families have a smooth phase relation; for three couplings, every fixed-gap-level relation is Borel reducible to the eventual-equality relation $E_0$, and the full phase relation is Borel and therefore not $\Sigma^1_1$-complete; four independent interaction directions are the first dimension in which general analytic arc-connection complexity is no longer topologically excluded. Finite levels of the defining gap constraints are semialgebraic.

DOI
10.5281/zenodo.22050715

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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
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