Spin-Field Correspondence

Jakub Mielczarek

doi:10.3390/universe3020029

Abstract

In the recent article Phys. Lett. B 2016, 759, 424–429, a new class of field theories called Nonlinear Field Space Theory was proposed. In this approach, the standard field theories are considered as linear approximations to some more general theories characterized by nonlinear field phase spaces. The case of spherical geometry is especially interesting due to its relation with the spin physics. Here, we explore this possibility, showing that classical scalar field theory with such a field space can be viewed as a perturbation of a continuous spin system. In this picture, the spin precession and the scalar field excitations are dual descriptions of the same physics. The duality is studied in the example of the Heisenberg model. It is shown that the Heisenberg model coupled to a magnetic field leads to a non-relativistic scalar field theory, characterized by quadratic dispersion relation. Finally, on the basis of analysis of the relation between the spin phase space and the scalar field theory, we propose the Spin-Field correspondence between the known types of fields and the corresponding spin systems.

Highlights

The phase space of a classical spin is a two-sphere, S2
Because the phase space is a symplectic manifold, it has to be equipped with the closed symplectic form ω, for which the natural choice is the area two-form
It is worth noticing that for the discrete Heisenberg model in the external magnetic field, the dispersion relation for the spin waves simplifies to the parabolic form (27) in the limit of either small k or small lattice spacing

Summary

Introduction

The phase space of a classical spin is a two-sphere, S2 (see e.g., [1]). Because the phase space is a symplectic manifold, it has to be equipped with the closed symplectic form ω, for which the natural choice is the area two-form. Where q and p are our new phase space variables and R1 and R2 are constants introduced due to dimensional reasons. A magnetic moment couples to an external magnetic field ~B via the vector ~J In such a case, the Hamiltonian of the interaction is μ. One can conclude that the precession of the vector ~J corresponds to a circle in the (q, p) phase space. The picture can be generalized to the area of field theory For this purpose, let us consider a continuous spin distribution, which may be viewed as an approximated description of some discrete spin systems studied experimentally. The continuous spin system is, in correspondence with the scalar field theory with the spherical field phase space. We consider the Heisenberg model of the spin system and find what is the field theoretical counterpart in such a case

The Heisenberg Model

The Spin-Field Correspondence

Conclusions