Pure gapped ground states of spin chains are short-range entangled
Wojciech De Roeck, Martin Fraas, Bruno de O. Carvalho
TL;DR
The paper proves that any pure gapped ground state $\psi$ of a finite-range one-dimensional spin chain is short-range entangled by constructing a locally generated automorphism $\alpha$ with exponentially decaying tails such that $\psi = \phi \circ \alpha$ for some pure product state $\phi$. The approach combines exponential clustering, Hastings factorization, and the split property to first obtain exponential decay of mutual correlations, then cut and purify the state to produce locally factorized pure states, and finally assemble an infinite sequence of local unitaries into an overall LGA. The main technical contributions are the quantitative bounds on mutual correlations, the construction of exponentially anchored unitaries that connect half-chains to intermediate states, and the convergence of infinite products of LGAs to yield a global disentangler. This work solidifies the lack of intrinsic topological order in 1D gapped systems without symmetry and provides a rigorous bridge to matrix product state representations and operator-algebraic classifications of gapped phases.
Abstract
We consider spin chains with a finite range Hamiltonian. For reasons of simplicity, the chain is taken to be infinitely long. A ground state is said to be a unique gapped ground state if its GNS Hamiltonian has a unique ground state, separated by a gap from the rest of the spectrum. By combining some powerful techniques developed in the last years, we prove that each unique gapped ground state is short-range entangled: It can be mapped into a product state by a finite time evolution map generated by a Hamiltonian with exponentially quasi-local interaction terms. This claim makes precise the common belief that one-dimensional gapped systems are topologically trivial in the bulk.