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In open, driven systems where parity-time symmetry is preserved, phenomena that defy conventional wisdom emerge near exceptional points, promising advances in photonics. While most studies focus on two-level systems of a conventional exceptional point, unconventional exceptional points as well as reentrant phases have been discovered in separate studies of higher-dimensional phase spaces. In this Letter, we present a minimal, analytical model that encompasses several key phenomena in higher-dimensional phase spaces, including reentrant PT phases, higher-order exceptional points, and anisotropic exceptional points. Using the exact analytical solution, we identify a new topological index as the unifying origin of these different phenomena. The simplicity of the model may furthermore facilitate experimental implementations for enhanced sensing and efficient polariton devices., Comment: 12 pages, 7 figures

GeSn alloys have been regarded as a potential lasing material for a complementary metal-oxide-semiconductor (CMOS)-compatible light source. Despite their remarkable progress, all GeSn lasers reported to date have large device footprints and active areas, which prevent the realization of densely integrated on-chip lasers operating at low power consumption. Here, we present a 1D photonic crystal (PC) nanobeam with a very small device footprint of 7 ${\mu}m^2$ and a compact active area of ~1.2 ${\mu}m^2$ on a high-quality GeSn-on-insulator (GeSnOI) substrate. We also report that the improved directness in our strain-free nanobeam lasers leads to a lower threshold density and a higher operating temperature compared to the compressive strained counterparts. The threshold density of the strain-free nanobeam laser is ~18.2 kW cm$^{ -2}$ at 4 K, which is significantly lower than that of the unreleased nanobeam laser (~38.4 kW cm$^{ -2}$ at 4 K). Lasing in the strain-free nanobeam device persists up to 90 K, whereas the unreleased nanobeam shows a quenching of the lasing at a temperature of 70 K. Our demonstration offers a new avenue towards developing practical group-IV light sources with high-density integration and low power consumption., Comment: 14 pages, 6 figures