Quantum critical point and scaling in a layered array of ultrasmall Josephson junctions

TL;DR

This paper models a layered array of ultrasmall Josephson junctions to analyze quantum phase transitions, deriving a new effective model, phase diagram, and universal conductivity scaling near the quantum critical point.

Contribution

It introduces a novel quantum spherical model for Josephson junction arrays and provides improved phase diagrams and universal conductivity scaling relations.

Findings

Derived a new effective quantum spherical model.

Calculated the superconductor-insulator phase diagram.

Established universal scaling forms for conductivity near the quantum critical point.

Abstract

We have studied a quantum Hamiltonian that models an array of ultrasmall Josephson junctions with short range Josephson couplings, $E_J$, and charging energies, $E_C$, due to the small capacitance of the junctions. We derive a new effective quantum spherical model for the array Hamiltonian. As an application we start by approximating the capacitance matrix by its self-capacitive limit and in the presence of an external uniform background of charges, $q_x$. In this limit we obtain the zero-temperature superconductor-insulator phase diagram, $E_J^{\rm crit}(E_C,q_x)$, that improves upon previous theoretical results that used a mean field theory approximation. Next we obtain a closed-form expression for the conductivity of a square array, and derive a universal scaling relation valid about the zero--temperature quantum critical point. In the latter regime the energy scale is determined by temperature and we establish universal scaling forms for the frequency dependence of the conductivity.