On Araki’s extension of the Jordan–Wigner transformation
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Published:2023-01-05
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ISSN:0129-055X
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Container-title:Reviews in Mathematical Physics
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language:en
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Short-container-title:Rev. Math. Phys.
Affiliation:
1. Université de Toulon, Aix Marseille Université, CNRS, CPT, Toulon, France
Abstract
In his seminal paper [On the XY-model on two-sided infinite chain, Publ. RIMS Kyoto Univ. 20 (1984) 277–296], Araki introduced an elegant extension of the Jordan–Wigner transformation which establishes a precise connection between quantum spin systems and Fermi lattice gases in one dimension in the so-called infinite system idealization of quantum statistical mechanics. His extension allows in particular for the rigorous study of numerous aspects of the prominent XY chain over the two-sided infinite discrete line without having to resort to a thermodynamic limit procedure at an intermediate or at the final stage. We rigorously review and elaborate this extension from scratch which makes the paper rather self-contained. In the course of the construction, we also present a simple and concrete realization of Araki’s crossed product extension.
Publisher
World Scientific Pub Co Pte Ltd
Subject
Mathematical Physics,Statistical and Nonlinear Physics