  Higher-order topological band theory has transformed the landscape of
topological phases in quantum and classical systems. Here, we experimentally
demonstrate a two-dimensional (2D) higher-order topological phase (HOTP),
referred to as the multiple chiral topological phase (MCTP), which is protected
by a multipole chiral number (MCN). Our realization differs from previous HOTPs
in that it possesses a larger-than-unity MCN, which arises when the
nearest-neighbor couplings (NNCs) are weaker than long-range couplings (LRCs).
Our phase has an MCN of 4, protecting the existence of 4 mid-gap topological
corner modes (TCMs) at each corner. The multiple TCMs demonstrated here could
lead to enhanced quantum-inspired devices for sensing and computing. Our study
also highlights the rich and untapped potential of LRC manipulation for future
research in topological phases.
