Symmetry distills the simplicity of natural laws from the complexity of physical phenomena. High symmetries are typically studied in high energy physics, and their applications in condensed matter physics are typically as a mathematical tool rather than accounting for realistic physics. We point out an opportunity, in atomic sys-tems, to explore physics and new states of matter characterized by large symmetries, e.g. Sp(2N) and SU(2N). For example, a generic SO(5), or, isomorphically Sp(4) symmetry is proved in spin-3/2 systems. Moreover, an exact SO(7) symmetry is identified possessing an extraordinarily unifying power: Its η-pairing operator extends Yang's eta-pairing to a high-rank Lie algebra, integrating 21 orders in both particle-hole and particle-particle channels into a unified framework. Such systems also exhibit multi-fermion orderings, including quartetting superuidity (charge 4e) and quartet density wave, which are baryon-like orderings. The resonant quantum plaquette states of SU(4) antiferromagnetism are described by a high-order gauge theory. A quantum phase transition occurs from the Slater region to the Mott region in the SU(6) Hubbard model. A tendency of convergence of itineracy and locality is revealed in 1D SU(N) systems as N goes large. These works also shed new light on the current experimental efforts on exploring novel states with alkaline earth fermions.
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