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

Abstract

We use the Bethe ansatz equations to calculate the charge stiffness ${\mathit{D}}_{\mathit{c}}$=(L/2)${\mathit{d}}^{2}$${\mathit{E}}_{0}$/d${\mathrm{\ensuremath{\Phi}}}_{\mathit{c}}^{2}$${\mathrm{\ensuremath{\Vert}}}_{\mathrm{\ensuremath{\Phi}}\mathit{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 ${\mathit{E}}_{0}$ is the ground-state energy, L is the circumference of the system (assumed to have periodic boundary conditions), and (\ensuremath{\Elzxh}c/e)${\mathrm{\ensuremath{\Phi}}}_{\mathit{c}}$ is the magnetic flux enclosed. We obtain an exact result for the asymptotic form of ${\mathit{D}}_{\mathit{c}}$(L) as L\ensuremath{\rightarrow}\ensuremath{\infty} at n=1, which defines and yields an analytic expression for the correlation length \ensuremath{\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 ${\mathit{D}}_{\mathit{c}}$(n,L,U)=${\mathit{Y}}_{+}$(\ensuremath{\xi}\ensuremath{\delta},\ensuremath{\xi}/L), where \ensuremath{\delta}=\ensuremath{\Vert}1-n\ensuremath{\Vert} and ${\mathit{Y}}_{+}$ is a universal scaling function which we calculate. The physical significance of \ensuremath{\xi} in the metallic phase of the model is that it defines the characteristic size of the charge-carrying solitons, or holons. We construct an explicit mapping for arbitrary U and \ensuremath{\xi}\ensuremath{\delta}\ensuremath{\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 holelike transport for 01-n\ensuremath{\ll}${\ensuremath{\xi}}^{\mathrm{\ensuremath{-}}1}$.