Translation-Invariant Pure States with Exotic Half-Chain Factors
Abstract
We construct translation-invariant pure quasifree fermionic states on the one-dimensional lattice whose half-chain von Neumann algebras are hyperfinite factors of type II∞ or IIIλ for any 0<λ≤ 1. Via the Jordan-Wigner-Araki transformation, these yield translation-invariant pure spin-chain states with the same half-chain factors, demonstrating that such factors need not be of type I or III. Each state is the unique ground state of a gapless, translation-invariant parent Hamiltonian with superpolynomially decaying interactions. For any fixed factor type, the interval entanglement entropy can exhibit widely varying growth rates, from arbitrarily slow divergence to nearly power-law growth, showing that entropy scaling does not determine the factor type. In contrast, we prove that gauge-invariant quadratic Hamiltonians with finitely many flavours and exponentially decaying hopping exhibit a dichotomy: their ground states either satisfy an area law with a type I half-chain factor, or have logarithmic entropy growth with a type III1 factor. The counterexample was discovered with AI assistance.
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BibTeX
@misc{liu2026translationinvariant,
title = {{Translation-Invariant Pure States with Exotic Half-Chain Factors}},
author = {Ruizhi Liu and Jinmin Yi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24154},
eprint = {2610.24154},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24154},
}
RIS
TY - UNPB
TI - Translation-Invariant Pure States with Exotic Half-Chain Factors
AU - Ruizhi Liu
AU - Jinmin Yi
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.24154
UR - https://arxiv.org/abs/2610.24154
AB - We construct translation-invariant pure quasifree fermionic states on the one-dimensional lattice whose half-chain von Neumann algebras are hyperfinite factors of type II_∞ or III_λ for any 0<λ≤ 1. Via the Jordan-Wigner-Araki transformation, these yield translation-invariant pure spin-chain states with the same half-chain factors, demonstrating that such factors need not be of type I or III. Each state is the unique ground state of a gapless, translation-invariant parent Hamiltonian with superpolynomially decaying interactions. For any fixed factor type, the interval entanglement entropy can exhibit widely varying growth rates, from arbitrarily slow divergence to nearly power-law growth, showing that entropy scaling does not determine the factor type. In contrast, we prove that gauge-invariant quadratic Hamiltonians with finitely many flavours and exponentially decaying hopping exhibit a dichotomy: their ground states either satisfy an area law with a type I half-chain factor, or have logarithmic entropy growth with a type III_1 factor. The counterexample was discovered with AI assistance.
ER -