Geometric phase around multiple exceptional points

Abstract

We study the geometrical phase when multiple exceptional points (EPs) are involved. In an optical microcavity of a stadium shape, we find two EPs, connected through three interacting modes, in a two-dimensional parameter space. It is shown that the geometrical phase imprinted on wave functions becomes zero when the two EPs are encircled three times in the parameter plane. In addition, we examine geometrical phases when three EPs are involved in a $3\ifmmode\times\else\texttimes\fi{}3$ matrix model, and show that single- and double-loop return modes have the geometrical phase of $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}$ or zero, depending on the type of the three EPs encircled.