Deconfinement in $d=1$: Asymptotic and half-asymptotic particles

The notion of deconfinement in two $d=1$ models, the Schwinger model and the Heisenberg chain, is reexamined. Both have half-asymptotic excitations (where particles and antiparticles must alternate) and also truly asymptotic particles which are half as many in number. The two kinds of particles are related by a complicated transformation. The main purpose of this paper is to highlight the relationship between asymptotic and half-asymptotic particles. The relevance of our findings to higher dimensions is briefly discussed.

Figure 1

The scalar potential $v\left(\varphi \right)$ for $\gamma =\theta -\pi =0$ for two cases, with single and double minima. When $\gamma \ne 0$, there is a single asymmetric minimum.

Figure 2

Phase diagram in the $\lambda =e^2∕2\pi m\Lambda c$ and $\gamma =\pi -\theta$ plane. The dark line denotes a first order transition terminating at $C$, a second order critical point. In the Ising model mapping, $\lambda -{\lambda }_C$ plays the role of $t$, the temperature, and $\gamma =\pi -\theta$ plays the role of the magnetic field $h$.