Quasi-Product States and Factor Types for the One-Dimensional Hard-Core Model
Abstract
We study the von Neumann algebra generated by a quasi-product state associated with the one-dimensional hard-core lattice gas. If κ>0 denotes the activity (fugacity), the transfer-matrix construction gives the stationary transition matrix P=p&q 1&0, p+q=1, κ=q/p2. The stationary Markov measure on the hard-core path space induces a faithful diagonal state on the corresponding path AF-algebra. We give a state-preserving identification of its GNS von Neumann algebra with the Feldman--Moore algebra of the measured tail equivalence relation. The relation is hyperfinite and ergodic, and its nonzero ratio set is κℤ. Consequently, the generated algebra is the hyperfinite factor of type II1 when κ=1, and of type IIIλ, where λ=min{κ,κ−1}, when κ≠1. The point κ=1 is not a thermodynamic phase transition: the pressure and particle density are analytic for all κ>0. We also describe the precise support-corner relation with the GNS algebra of the diagonal Gibbs state on the quasi-local spin algebra, and we determine the centralizer and the flow of weights.
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BibTeX
@misc{mukhamedov2026quasiproduct,
title = {{Quasi-Product States and Factor Types for the One-Dimensional Hard-Core Model}},
author = {Farrukh Mukhamedov and Yuri Suhov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.12458},
eprint = {2606.12458},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.12458},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Quasi-Product States and Factor Types for the One-Dimensional Hard-Core Model
AU - Farrukh Mukhamedov
AU - Yuri Suhov
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2606.12458
UR - https://arxiv.org/abs/2606.12458
AB - We study the von Neumann algebra generated by a quasi-product state associated with the one-dimensional hard-core lattice gas. If κ>0 denotes the activity (fugacity), the transfer-matrix construction gives the stationary transition matrix P=p&q 1&0, p+q=1, κ=q/p^2. The stationary Markov measure on the hard-core path space induces a faithful diagonal state on the corresponding path AF-algebra. We give a state-preserving identification of its GNS von Neumann algebra with the Feldman--Moore algebra of the measured tail equivalence relation. The relation is hyperfinite and ergodic, and its nonzero ratio set is κ^ℤ. Consequently, the generated algebra is the hyperfinite factor of type II_1 when κ=1, and of type III_λ, where λ=min{κ,κ^−1}, when κ≠1. The point κ=1 is not a thermodynamic phase transition: the pressure and particle density are analytic for all κ>0. We also describe the precise support-corner relation with the GNS algebra of the diagonal Gibbs state on the quasi-local spin algebra, and we determine the centralizer and the flow of weights.
ER -