Induced current by a magnetic flux in $(1+2)-$dimensional conical spacetime in a Ho{ř}ava-Lifshitz Lorentz-violating scenario

Abstract

We investigate the vacuum expectation value of bosonic current induced by a magnetic flux in a $(2+1)-$dimensional conical spacetime in the presence of a circular boundary, in a Ho{ř}ava-Lifshitz Lorentz violation symmetry scenario. We assume that the circular boundary is concentric with magnetic flux, and the massive scalar quantum field obeys the Robin boundary condition on the boundary. In order to develop this analysis, we calculate the positive frequency Wightman functions for both regions, inside and outside the boundary. Using these functions, we obtain analytical expressions for the vacuum expectation bosonic currents. As we will see, these functions are presented in the form of the sum of boundary-free and boundary-induced parts. As to the boundary-induced currents, some asymptotic behaviors are investigated for specific limiting cases; moreover, in order to provide a better understanding about the behavior of the currents, some plots are given.

Figures (3)

• Figure 1: These plots exhibit the behavior of boundary free induced currents in units of $me$ as function of $mr$, considering different values of the critical exponent $\xi=1$ (upper), $\xi=2$ (middle) and $\xi=3$ (bottom). For each plot we assumed $q=1.0, \ 1.5, \ 2.0$. Moreover, we adopted $\alpha_0=1/4$, and $ml=10^{-3}$. Note that the horizontal axis values are on a logarithmic scale.
• Figure 2: These plots exhibit the boundary induced of $\frac{a\langle {\hat{j}}_{\varphi}(x) \rangle_{b}^{(in)}}{e}$ as function of $(r/a)$ in the region inside the circle, considering Dirichlet boundary condition for the left plots and Neumann boundary condition for the right plots. The upper plots are for $\xi=3$ and the lower ones for $\xi=5$. Different values of $q$ (the numbers near the curves) are adopted; moreover we assume $\alpha_0=1/4$, $(a/l)=10^2$ and $ml=10^{-2}$. Note that the horizontal axis values are on a logarithmic scale.
• Figure 3: These plots exhibit the boundary induced of $\frac{a\langle {\hat{j}}_{\varphi}(x) \rangle_{b}^{(in)}}{e}$ as function of $(r/a)$ in the region outside the circle. Wr consider Dirichlet boundary condition for the left plots and Neumann boundary condition for the right plots. Moreover, the upper plots are for $\xi=3$ and the lower ones for $\xi=5$. Different values of $q$ (the numbers near the curves) are adopted; moreover we assume $\alpha_0=1/4$, $(a/l)=10^2$ and $ml=10^{-2}$. Note that the horizontal axis values are on a logarithmic scale.