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

TL;DR

This paper explores the Ising model in an external magnetic field, demonstrating a fermionic operator representation that preserves certain symmetries and provides a constructive proof of the Lee-Yang theorem regarding phase transitions.

Contribution

It introduces a fermionic operator representation of the external magnetic field interaction in the Ising model, facilitating a new proof of the Lee-Yang theorem.

Findings

Operator $V_h$ commutes with $ ext{operator } extbf{P}$ in the fermionic representation.

Constructive proof of the Lee-Yang theorem for nonzero magnetic field.

Discussion of implications for the Lenz-Ising-Onsager problem.

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 $\hat{S}= \sum_m\beta^{\dag}_m\beta_m$ 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.