Jordan-wigner type transformation and Hamiltonian circuits in a rectangular lattice

Abstract

In the present study a generating function$\bar \Gamma ^{(h)}$ is considered for Hamiltonian circuits in a rectangular lattice of dimensionN×M. A generating function Γ(^e) for closed Euler diagrams (with valence number for the vertices δ=0, 2, 4) with constant step is introduced for this lattice. It is proved that these two generating functions coincide as in the case of the corresponding generating functions relative to a single node (in limit as N, M → ∞). To construct the proof, an auxiliary function that is, in fact, the statistical sum for the two-dimensional Ising model is introduced. The two-dimensional Jordan-Wigner type transformations that were introduced in [7] are also used.