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

We use the Bethe ansatz equations to calculate the charge stiffness D_c=(L/2)d²E₀/dΦ²_c||_Φc=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 E₀ is the ground-state energy, L is the circumference of the system (assumed to have periodic boundary conditions), and (ħc/e)Φ_c is the magnetic flux enclosed. We obtain an exact result for the asymptotic form of D_c(L) as L-->∞ at n=1, which defines and yields an analytic expression for the correlation length ξ 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_c(n,L,U)=Y₊(ξδ,ξ/L), where δ=||1-n|| and Y₊ is a universal scaling function which we calculate. The physical significance of ξ 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 ξδ<<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 0<1-n<<ξ^-1.