u‰‰ŽาF Sung G. Chung ŽiPhysics & Nanotechnology Research and Computation Center, Western Michigan University, USAj “๚ŽžF ‚QŒŽ@‚Q‚W“๚i‰ฮj@@Œ฿Œใ‚SŽž‚R‚O•ช‚ฉ‚็ ๊ŠF ยŽRŠw‰@‘ๅŠw@—HŠw•”i‘Š–อŒดƒLƒƒƒ“ƒpƒXj‚k“‚UŠK@‚k‚U‚O‚RŽบ ‘่–ฺF uSpontaneous symmetry breaking in Josephson junction arraysv —vŽ|F The superconductor-insulator transition in the two dimensional XY model is identified as a spontaneous symmetry breaking. The strong size dependence in the spontaneous symmetry breaking predicts an order of magnitude increase in the critical coupling constant Ej/T with the array size, where Ej is the Josephson energy and T the temperature. The order of magnitude change experimentally observed in the critical coupling constant Ej/Ec, where Ec is the Coulomb energy, from a single Josephson junction to the 48x40 arrays with ultra-small island sizes may be accounted for likewise with an opposite tendency in the quantum phase transition in contrast to the classical phase transition in the XY model; the critical Ej/Ec decreases with the array size. Our spontaneous-symmetry-breaking theory bridges the two limiting cases; the Likharev-Zorin band theory for the single Josephson junction and the Berezinskii-Kosterlitz-Thouless transition in the infinite-size limit. --------------------------------- ‹ครF@ยŽRŠw‰@‘ๅŠw@—HŠw‰๏