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Ground-state Properties of Tonks-Girardeau Gas in One Dimensional Periodic Potential

The relations among the occupation number of the lowest natural orbital (ONLNO), momentum distributions (MD) and off-diagonal long-range element (ODLRE) of the reduced single-particle density matrix (RSPDM) are studied while Tonks-Girardeau gas in one dimensional periodic potential is in the ground state. For $N$-body systems of large enough, RSPDM and its lowest natural orbital do not vary with $N$ in overlapped areas in commensurate and incommensurate cases correspondingly. In commensurate case, the ODLRE is exponential attenuation with $N$, which results in that the ONLNO and MD are invariant with $N$. While in contrast, in incommensurate case, the off-diagonal elements are inversely proportional to $\sqrt{N}$, which results in the different behavior of the ONLNO and MD.