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Universal temperature dependence of electron number in one-dimensional Hubbard model

We investigate the temperature region in which a Tomonaga-Luttinger liquid (TLL) description of the charge sector of the one-dimensional Hubbard model is valid. By using the thermodynamic Bethe ansatz method, electron number is calculated at finite temperatures and fixed chemical potential. We observe maximum electron number as a function of temperature close to the chemical potential of the upper critical value that corresponds to half filling. As the chemical potential approaches the upper critical value from below, the temperature $(T_{\rm M})$ at which the electron number shows its maximum asymptotically approaches a universal relation. We show that, below the energy corresponding to $T_{\rm M}$, the charge excitation spectrum nearly obeys a linear dispersion relation. The results demonstrate that $T_{\rm M}$ marks the important temperature below which TLL is realized.