The superconductor-insulator transition in a Josephson junction chain with quantum fluctuation

The superconductor-insulator transition in a Josephson junction chain with quantum fluctuation is studied by a density matrix renormalization group. The critical Josephson coupling is found to be J(c) similar to 1.85, while the Villain approximation gives J(c) similar to 1.1. When extending the phase space from (0, 2 pi) to (0, 4 pi), corresponding to a very small quasi-particle tunnelling effect, the superconductor-insulator transition still exists but at a larger critical coupling J(c) similar to 2.3.