On the geometric algebras of the Ising model

TL;DR

This paper reinterprets the classical transfer matrix solution of the Ising model using conformal Clifford algebras, providing a unified geometric framework that clarifies fermionic modes and duality.

Contribution

It introduces a geometric algebra perspective to the Ising model, unifying transfer matrix elements, eigenvectors, and excitations within a conformal Clifford algebra framework.

Findings

Transfer matrix viewed as a dilation generated by a conformal bivector

Eigenvectors as null combinations of Clifford generators

Connection between Ising model and free Majorana fermions

Abstract

We revisit the classical transfer matrix solution of the one- and two-dimensional Ising model from the perspective of Clifford and conformal geometric algebras. Building on Kaufman's spinor formulation, we show that all elements entering the solution, including the transfer matrix, its eigenvectors, and the quasiparticle excitations, admit a natural and unified interpretation as elements of an appropriate conformal Clifford algebra. In particular, the transfer matrix can be viewed as a dilation generated by a conformal bivector, while its eigenvectors correspond to null combinations of Clifford generators, closely paralleling the emergence of Majorana fermionic degrees of freedom. In the two-dimensional case, the standard eigenvalue equation for the row-to-row transfer matrix is reinterpreted as a dispersion relation for quasiparticle excitations, exposing the connection between the Ising model and a theory of free Majorana fermions. While all the explicit exact results recovered are well known, this geometric reformulation provides a unified algebraic framework which is compact and physically interpretable. Specifically, this clarifies the role of scale transformations, fermionic modes, and duality in the Ising model. We believe this approach offers a useful pedagogical complement to more conventional fermionic, Grassmann, or field theoretic treatments.