Geometrical String and Spin Systems

TL;DR

This paper introduces a new geometrical string formulation on Euclidean lattices, linking it to specific spin systems like the Ising model with additional interactions, and extends the concept to higher-dimensional random surfaces.

Contribution

It presents a novel geometrical string model on lattices and connects it to known spin systems with specific interactions, extending to high-dimensional membranes.

Findings

Spin systems reproduce surface dynamics with local interactions.

In 3D, the model is an Ising ferromagnet with diagonal antiferromagnetic interactions.

In 4D, it matches a gauge Ising system with double-plaquette interactions.

Abstract

We formulate a new geometrical string on the euclidean lattice. It is possible to find such spin systems with local interaction which reproduce the same surface dynamics.In the three-dimensional case this spin system is a usual Ising ferromagnet with additional diagonal antiferromagnetic interaction and with specially adjusted coupling constants. In the four-dimensional case the spin system coincides with the gauge Ising system with an additional double-plaquette interaction and also with specially tuned coupling constants. We extend this construction to random walks and random hypersurfaces (membrane and p-branes) of high dimensionality. We compare these spin systems with the eight-vertex model and BNNNI models.