GEOMETRICAL STRING and DUAL SPIN SYSTEMS

TL;DR

This paper develops a duality transformation for a lattice spin system related to strings, resulting in a new gauge-invariant spin model with complex local interactions and potential spin glass behavior.

Contribution

It introduces a geometrical and algebraic duality transformation for a string-related spin system in higher dimensions, leading to a novel gauge-invariant spin model with complex interactions.

Findings

Dual Hamiltonian describes a new gauge-invariant spin system.

Special cases reduce to known Ising ferromagnets and antiferromagnets.

Generalization to p-branes provided.

Abstract

We are able to perform the duality transformation of the spin system which was found before as a lattice realization of the string with linear action. In four and higher dimensions this spin system can be described in terms of a two-plaquette gauge Hamiltonian. The duality transformation is constructed in geometrical and algebraic language. The dual Hamiltonian represents a new type of spin system with local gauge invariance. At each vertex $\xi$ there are $d(d-1)/2$ Ising spins $\Lambda_{\mu,\nu}= \Lambda_{\nu,\mu}$, $\mu \neq \nu = 1,..,d$ and one Ising spin $\Gamma$ on every link $(\xi,\xi +e_{\mu})$. For the frozen spin $\Gamma \equiv 1$ the dual Hamiltonian factorizes into $d(d-1)/2$ two-dimensional Ising ferromagnets and into antiferromagnets in the case $\Gamma \equiv -1$. For fluctuating $\Gamma$ it is a sort of spin glass system with local gauge invariance. The generalization to $p$-branes is given.