Jackiw and Rossi had a classic paper Zero modes of the vortex-fermion system (1981). In that nice-written paper, they found fermionic zero modes of Dirac operator under nontrivial Higgs vortex in 2D space, which is a 2+1D spacetime problem. The winding number n of Higgs vortex corresponds to the number n of fermionic zero modes.

May I please ask: **is there known result in the literature where fermion zero mode is formed in 1+1D spacetime under 1+1D spacetime Higgs vortex (i.e. 1D space +1D Euclidean-time vortex formed by a complex scalar Higgs field)?**

To clarify, I assume this problem is similar to their 1981's analysis, but the Higgs vortex now I just ask is not a 2D space vortex, but a 1+1D spacetime vortex. (I assume a major difference between these cases should be the number of components of spinor, Jackiw and Rossi had 4-component spinor, here I have 2-component spinor.) Thank you for your time of thoughts and reply.

[Below for the details:]

Here the complex scalar Higgs $\Phi(x,t) = \Phi_{Re}(x,t)+I \Phi_{Im}(x,t)$, with $\Phi_{Re}, \Phi_{Im} \in \mathbb{R}$
which couple to the fermions by Yukawa coupling $\bar{\Psi} \Phi \Psi$:

The full 1+1D action is:
$$
S=\int dt dx \;\bar{\Psi} (i \not{\partial}+ \Phi_{Re}(x,t)+I \gamma^5 \Phi_{Im}(x,t))\Psi +L_{\text{Higgs}}
$$
with $\Psi=(\Psi_L,\Psi_R)$ a 2-component spinor. $L_{\text{Higgs}}=a |\Phi|^2+b |\Phi|^4 \dots$.

The 1+1D spacetime vortex of Higgs can be, for example, written in Euclidean time $t_E=-it$:
$$
\Phi(z) \equiv \Phi(x,t_E) \simeq \frac{t_E+ix}{|t_E+ix|}=\frac{z}{|z|}
$$
with $t_E+ix \equiv z$ as a complex coordinate, which gives 1 winding mode from the homotopy mapping:
$$
S^1 \text{of} \;z \to S^1 \text{of}\; \Phi(z)
$$

We do not consider the gauge field profile here. We only consider the 1+1D spacetime vortex of Higgs (1D space +1D Euclidean-time Higgs vortex).

This post imported from StackExchange Physics at 2014-06-04 11:36 (UCT), posted by SE-user Idear