On the Lenz-Ising-Onsager Problem in an External Magnetic Field

Abstract

The Lenz-Ising-Onsager (LIO) problem in an external magnetic field in the second quantization representation is the subject of consideration of the paper. It is shown that the operator $V_h$ in the second quantization representation corresponding to Ising spins interaction with the external magnetic field $H$ can be represented in terms of single-subscript creation and anihilation Fermi operators in such a form that the operator $V_h$ commutes with the operator $\hat{P}\equiv(-1)^{\hat{S}}$, where $\hat{S}= \sum_mβ^†_mβ_m$ is the operator of a total number of Fermions. The possible consequences of such representation with it's relation to the LIO is discussed. In particular, the constructive proof of the Lee-Yang theorem on the absence of phase transition for Ising model in nonzero magnetic field $(\Re h\neq 0)$ is demonstrated.