Jordan-Wigner transformations and their generalizations for multidimensional systems

TL;DR

This paper introduces new nonlinear Jordan-Wigner transformations for multidimensional systems, enabling the expression of multi-index Pauli operators in terms of Fermi operators, with applications in statistical mechanics and graph theory.

Contribution

The paper presents novel nonlinear Jordan-Wigner transformations for multidimensional systems, expanding the algebraic framework and applications beyond previous linear forms.

Findings

Developed two- and three-dimensional JW-type transformations

Analyzed algebraic properties and transposition relations

Applied transformations to lattice models and graph theory problems

Abstract

In the paper nonlinear transformations of the Jordan-Wigner (JW) type are introduced in the form different from the ones known previously, for the purpose of expressing multi-index Pauli operators in terms of multi-index Fermi creation and annihilation operators. These JW transformations in the general case being a subject of a rather complicated algebra of transposition relations between various sets of Fermi creation and annihilation operators, depending on the common multiindex of the latter, is shown. As an example, the two- and three- dimensional transformations of the JW type are investigated, their properties and possible applications in analysis of a couple of lattice models of statistical mechanics and also an example of application of these transformations to problems of self-avoiding walks in graph theory, are discussed. The relation of the obtained transformations to the previously known transformations of the JW type for higher dimensions is shown.