Geometric perspective on the one-dimensional Lenz–Ising model in a non-zero magnetic field

Abstract

Having been linked to numerous applications, the one-dimensional Lenz–Ising model in a non-zero magnetic field is well-known in graduate- and senior-level statistical physics. The transfer matrix method is the standard way to solve the model. In this line, we reformulate the transfer matrix by subjecting the usual Pauli matrices to rotation. The transfer matrix is then shown as a ‘spin’-representative of a two-dimensional rotation just like the special case when the magnetic field is zero. Consequently, eigenvalues in exponential form are obtained, and the zeroes of the partition function lie on a unit circle in agreement with the Lee–Yang Circle Theorem. The zeroes pinching the positive side of the real axis give the known trivial transition temperature, K. Finally, we discuss the significance of the results in teaching the model.