1D action and partition function for the 2D Ising model with boundary magnetic field

TL;DR

This paper derives exact results for the 2D Ising model with boundary magnetic fields using a Grassmann algebra approach, providing new insights into partition functions and correlation functions for finite systems.

Contribution

It introduces an alternative method based on Grassmann algebra to analyze the 2D Ising model with boundary magnetic fields, extending previous work and handling inhomogeneous boundary conditions.

Findings

Exact partition function for boundary magnetic field cases

Correlation functions on the boundary computed

Results agree with previous homogeneous field studies

Abstract

In this article we obtain some exact results for the 2D Ising model with a general boundary magnetic field and for a finite size system, by an alternative method to that developed by B. McCoy and T.T. Wu. This method is a generalization of ideas from V.N. Plechko presented for the 2D Ising model in zero field, based on the representation of the Ising model using a Grassmann algebra. In this way, a Gaussian 1D action is obtained for a general configuration of the boundary magnetic field. In the special case where the magnetic field is homogeneous, we check that our results are in agreement with McCoy and Wu's previous work, and we also compute the two point correlation functions on the boundary. We use this correlation function to obtain the exact partition function and the free energy in the special case of an inhomogeneous boundary magnetic field.