In this work we study one family of liquid states of $k$-orbital $\text{SU}\left(N\right)$ spin systems, focusing on the case of $k=2$ which can be realized by ultracold alkaline-earth atoms trapped in optical lattices, with $N$ as large as 10. Five different algebraic liquid states with selectively coupled charge, spin, and orbital quantum fluctuations are considered. The algebraic liquid states can be stabilized with large enough $N$ and the scaling dimension of physical order parameters is calculated using a systematic $1/N$ expansion. The phase transitions between these liquid states are also studied and all the algebraic liquid states discussed in this work can be obtained from one “mother” state with $\text{SU}\left(2\right)\times \text{U}\left(1\right)$ gauge symmetry.

• Received 2 November 2009

DOI:https://doi.org/10.1103/PhysRevB.81.144431