Reentrant quantum phase transitions in two capacitively coupled Josephson arrays in perpendicular magnetic fields

Abstract

We have studied the phase diagram structure of two capacitively coupled Josephson junction arrays as a function of their charging energy ${E}_{c}$, Josephson coupling energy ${E}_{J}$, and a homogeneous perpendicular magnetic field. The arrays are coupled via a site interaction capacitance, ${C}_{\mathrm{int}}={C}_{\mathrm{inter}}∕{C}_{m}$, with ${C}_{\mathrm{inter}}$ as the interlayer mutual capacitance and ${C}_{m}$ as the intralayer mutual capacitance defined as the nearest neighbor grain mutual capacitance. The parameter that measures the competition between thermal and quantum fluctuations in the $i\text{th}$ array $(i=1,2)$ is ${\ensuremath{\alpha}}_{i}\ensuremath{\equiv}{E}_{{c}_{i}}∕{E}_{{J}_{i}}$. The phase structure of the system is dominated by the thermally induced and magnetically induced vortices as well as intergrain charge induced excitations. We have studied the capacitively coupled array behavior when one of them is in the vortex dominated regime, and the other in the quantum charge dominated regime. We determined the different possible phase boundaries by carrying out extensive quantum path integral Monte Carlo calculations of the helicity modulus ${\ensuremath{\Upsilon}}_{1,2}(\ensuremath{\alpha},f)$ and the inverse dielectric constant ${ϵ}_{1,2}^{\ensuremath{-}1}(\ensuremath{\alpha},f)$ for each array as a function of temperature, interlayer capacitance ${C}_{\mathrm{int}}$, quantum parameter $\ensuremath{\alpha}$, and frustration values $f\ensuremath{\equiv}\frac{\ensuremath{\Phi}}{{\ensuremath{\Phi}}_{0}}=1∕2$ and $f=1∕3$. Here, $\ensuremath{\Phi}$ is the total flux in a plaquette and ${\ensuremath{\Phi}}_{0}$ is the quantum of flux. We found an intermediate temperature range when array 1 is in the semiclassical regime $({\ensuremath{\alpha}}_{1}=0.5)$ and array 2 is in the quantum regime with $1.25\ensuremath{\leqslant}{\ensuremath{\alpha}}_{2}<2$, in which ${\ensuremath{\Upsilon}}_{2}(T,\ensuremath{\alpha},f=1∕2)>0$ and then goes down to zero while ${\ensuremath{\epsilon}}_{2}^{\ensuremath{-}1}(T,\ensuremath{\alpha},f=1∕2)$ increases from zero up to a finite value. This behavior is similar to the one previously found for unfrustrated capacitively coupled arrays. However, for ${\ensuremath{\alpha}}_{2}=2.0$, a reentrant transition in ${\ensuremath{\Upsilon}}_{2}(T,\ensuremath{\alpha},f=1∕2)$ occurs at intermediate temperatures for ${C}_{\mathrm{int}}=0.782\phantom{\rule{0.2em}{0ex}}61$, 1.043 48, and 1.304 35. For smaller values of the interlayer capacitance no phase coherence was found in array 2. This suggest that the increase between the array capacitive coupling induces a normal-superconducting-normal (N-SC-N) reentrant phase transition. For values of ${\ensuremath{\alpha}}_{2}>2.0$, the quantum array only exhibits an insulating phase, while the semiclassical array shows a superconducting behavior. In contrast, for phase frustration, $f=1∕3$, we found that when array 2 is in the full quantum regime, $2\ensuremath{\leqslant}{\ensuremath{\alpha}}_{2}\ensuremath{\leqslant}4$, the semiclassical array is the one that shows a reentrant N-SC-N behavior at relatively low temperatures. This reentrance in the coupled array behavior is a manifestation of the gauge invariant capacitive interaction and the duality relation between vortices, in the semiclassical array, and charges in the quantum-fluctuation dominated array. We find that the phase diagrams for $f=1∕2$ and $f=1∕3$ are very different in nature.