Length Scale-Dependent Superconductor–Insulator Quantum Phase Transitions in One Dimension: Renormalization Group Theory of Mesoscopic SQUIDs Array

Abstract

The transition at zero temperature or at very low temperature belongs to the class of phase transition that is driven by quantum fluctuations of the system [1]. The quantum fluctuations are controlled by system parameters such as the charging energies and Josephson couplings of the Josephson junction array [2]. Here we present a field-theoretical renormalization group study to find the quantum dissipative phases of a lumped superconducting quantum interference device (SQUID) . The quantum fluctuations of the system are controlled by the externally applied magnetic field and α, which is the ratio of quantum resistance to tunnel junction resistance. In the SQUID the total current of device is modulated by the applied magnetic flux. Therefore the total current in the SQUID is \( I= 2 {I_{\mathrm{c}}} \sin( \theta ) | \cos ( \frac{\pi {\Phi} }{{\Phi}_0} ) |\), where \(I_{\mathrm{c}}\) is the critical current, Φ is the magnetic flux and \({{\Phi}_0} (= \frac{h}{2 e}) \) is the flux quantum, and θ is the phase of superconducting order parameter. Similar relation holds for Josephson coupling: \( E_J = 2 E_{J0} \cos( \theta ) | \cos ( \frac{\pi {\Phi} }{{\Phi}_0} )|\), where \(E_{J0}\) is the bare Josephson coupling. So one may consider that a lumped SQUID system (Fig. 13.1a) can be described in terms of an array of superconducting quantum dots (SQD) but with a modulated Josephson coupling (Fig. 13.1b) and critical current [3], [4]. This effective mapping will help us to analyze (analytically) the system in detail. The experimentalists of [3], [4] have also considered the mesoscopic SQUID system as a modulated Josephson junction array, and we have been motivated by these well-accepted experimental findings [3], [4]. The plan of this manuscript is as follows. Section 13.2 contains the analytical derivations and the physical explanation for the occurrence of quantum dissipative phase in the lumped SQUID system; conclusions are presented in Sect. 13.4.