Scaling theory of the Mott-Hubbard metal-insulator transition in one dimension

Abstract

We use the Bethe ansatz equations to calculate the charge stiffness $D_{\mathrm{c}}=(L/2)d^2{E}_0/d{\Phi }_{\mathrm{c}}^2{|}_{{\Phi }_{\mathrm{c}}=0}$ of the one-dimensional repulsive-interaction Hubbard model for electron densities close to the Mott insulating value of one electron per site ($n=1$), where $E_0$ is the ground state energy, $L$ is the circumference of the system (assumed to have periodic boundary conditions), and $(\hslash c/e){\Phi }_{\mathrm{c}}$ is the magnetic flux enclosed. We obtain an exact result for the asymptotic form of $D_{\mathrm{c}}(L)$ as $L\to \infty$ at $n=1$, which defines and yields an analytic expression for the correlation length $\xi$ in the Mott insulating phase of the model as a function of the on-site repulsion $U$. In the vicinity of the zero temperature critical point U=0, $n=1$, we show that the charge stiffness has the hyperscaling form $D_{\mathrm{c}}(n,L,U)=Y_{+}(\xi \delta ,\xi /L)$, where $\delta =|1-n|$ and $Y_{+}$ is a universal scaling function which we calculate. The physical significance of $\xi$ in the metallic phase of the model is that it defines the characteristic size of the charge-carrying solitons, or {\em holons}. We construct an explicit mapping for arbitrary $U$ and $\xi \delta \ll 1$ of the holons onto weakly interacting spinless fermions, and use this mapping to obtain an asymptotically exact expression for the low temperature thermopower near the metal-insulator transition, which is a generalization to arbitrary $U$ of a result previously obtained using a weak- coupling approximation, and implies hole-like transport for $0<1-n\ll {\xi }^{-1}$.

arXiv comment34 pages, REVTEX (5 figures by request)

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