Resonating-valence-bond liquid in low dimensions

The Hubbard model in $D$ dimensions, with the on-site repulsion $U$ and the transfer integral between nearest neighbors $-t/\sqrt{D}$, is studied on the basis of the Kondo-lattice theory. If $U/|t| \gg 1$, $|n - 1| \lesssim |t|/(DU)$, where $n$ is the number of electrons per unit cell, and $D$ is so small that $|J|/D \gg k_{\rm B}T_c$, where $J = -4t^2/U$ and $T_c$ is $0 {\rm K}$ for $D = 1$ and is the highest critical temperature among possible ones for $D \ge 2$, a low-$T$ phase where $T_c < T \ll |J|/(k_{\rm B}D)$ is a frustrated electron liquid. Since the liquid is stabilized by the Kondo effect in conjunction with the resonating-valence-bond (RVB) mechanism, it is simply the RVB electron liquid; in one dimension, it is also the Tomonaga-Luttinger liquid. The Kondo energy of the RVB liquid is $k_{\rm B}T_{\rm K} = O(|J|/D)$; its effective Fermi energy is $O(k_{\rm B}T_{\rm K})$. A midband appears on the chemical potential between the upper and lower Hubbard bands; the Hubbard gap is a pseudogap. As regards the density of states per unit cell of the midband, its bandwidth is $O(k_{\rm B}T_{\rm K})$ or $O(|J|/D)$, its peak height is $O(1/U)$, and its spectral weight is $O[t^2/(DU^2)]$. Since the midband almost disappears in the Heisenberg limit, the RVB electron liquid in the Heisenberg limit is simply the RVB spin liquid. The RVB electron and spin liquids adiabatically continue to each other. Since local moments form in a high-$T$ phase where $T \gtrsim T_{\rm K}$, the high-$T$ phase is simply the Mott insulator.