Finite size scaling analysis of the helicity modulus and the inverse dielectric constant in two capacitvely coupled Ultrasmall 2D Josephson Junction Arrays*

We have carried out a finite size scaling analysis of the helicity modulus $\Upsilon_{i}$ and the inverse dielectric constant $\epsilon_{i}$, $(i=1,2)$ of two capacitively coupled Josephson junction arrays with charging energy, $E_c$, and Josephson coupling energy, $E_J$. The arrays are coupled via the capacitance, $C_{\rm inter}$, at each site of the lattices. The parameter that measures the importance of quantum fluctuations in the i-th array is, $\alpha_i\equiv \frac{E_{{c}_i}}{E_{J_i}}$. We have considered the interplay between vortex and charge dominated individual array phases by means of extensive path integral Monte Carlo simulations. It has been found that this system develops a {\it reentrant transition} in $\Upsilon(T,\alpha)$, at low temperatures, when one of the arrays is in the semiclassical limit (i.e. $\alpha_{1}=0.5 $) and the quantum array has $2.0 \leq\alpha_{2} \leq 2.5$, for $C_{{\rm inter}}= 0.26087, 0.52174, 0.78261, 1.04348$ and $1.30435$. Similar behavior was obtained for larger values of $\alpha_{2} =4.0$ with $C_{{\rm inter}}=1.04348$ and 1.30435.