Experimental observation of energy-band Riemann surface Cheng, Dali Wang, Heming Zhong, Janet Lustig, Eran Roques-Carmes, Charles Fan, Shanhui Non-Hermiticity naturally arises in physical systems that exchange energy with their environment. The presence of non-Hermiticity leads to many topological physics phenomena and device applications. In the non-Hermitian energy band theory, the foundation of these physics and applications, both energies and wave vectors take complex values. The energy bands thus become a Riemann surface, and such an energy-band Riemann surface underlies all important signatures of non-Hermitian topology. Despite a long history and recent theoretical interests, the energy-band Riemann surface has not been experimentally studied. Here, we provide a photonic observation of the energy-band Riemann surface of a non-Hermitian system. This is achieved by a tunable imaginary gauge transformation in photonic synthetic frequency dimensions. From measured topologies of the Riemann surface, we reveal the complex-energy winding, the open-boundary-condition spectrum, the generalized Brillouin zone, and the branch points. Our findings demonstrate a unified framework in the studies of diverse effects in non-Hermitian topological physics through an experimental observation of energy-band Riemann surfaces. American Association for the Advancement of Science 2026 info:eu-repo/semantics/article doc-type:article text http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/21583 Cheng D, Wang H, Zhong J, Lustig E, Roques-Carmes C, Fan S. Experimental observation of energy-band Riemann surface. <i>Science Advances</i>. 2026;12(12). doi:<a href="https://doi.org/10.1126/sciadv.aec8239">10.1126/sciadv.aec8239</a> eng info:eu-repo/semantics/altIdentifier/doi/10.1126/sciadv.aec8239 info:eu-repo/semantics/altIdentifier/issn/2375-2548 info:eu-repo/semantics/altIdentifier/arxiv/2510.08819 info:eu-repo/semantics/openAccess