Thermodynamic local-occupation averages and local-number-occupation fluctuations are studied in small-cluster many-body systems by introducing a single-site chemical potential at a particular site. This procedure allows the study of differential properties of thermodynamic functions by providing continuous variation of local occupations. The method, which starts from the quantum-mechanical grand canonical ensemble, gives a criterion to distinguish particular features of the small cluster that are likely to survive in the thermodynamic limit from those discontinuities that are characteristic of the finiteness of the cluster and the resulting discreteness of the energy spectrum. In particular the Mott-insulating state (a discontinuity in the chemical potential at a particular occupation) can be clearly tested this way. Similar indications are obtained for spin-polarized states and for particle-pairing conditions. Two four-site Hubbard-model clusters--a ring and a tetrahedron--are used as examples.
We study the Drude weight, optical conductivity, and flux-periodicity properties of one-dimensional Hubbard chains using Bethe-ansatz, Lanczos, and exact-diagonalization techniques. We find that the Drude weight D is unexpectedly negative for half-filled Hubbard rings with N=4n sites reflecting a strong paramagnetic response, while it is positive (diamagnetic) for half-filled rings of N=4n+2 sites. In both cases at half filling, D vanishes exponentially with N on a length scale set by the inverse of the gap for small U/t. Near half filling, we find that D approaches a constant, positive, diamagnetic value as N increases, indicating metallic behavior. Combining Bethe-ansatz results near half filling with the known U=0 and U=\ensuremath{\infty} limits and Lanczos finite-size extrapolations, we obtain a general picture of D in the thermodynamic limit as a function of band filling and the ratio U/t. We note similarities with the low-frequency integrated spectral weight of certain hole-doped and electron-doped high-
Tc compounds. We investigate the finite-frequency optical conductivity, finding structure at \ensuremath{\omega}\ensuremath{\approxeq}U and, for one hole off half filling, at \ensuremath{\omega}\ensuremath{\propto}
t2/U. We find that minima in the energy of a Hubbard ring enclosing a flux \ensuremath{\Phi} occur with changes of half a flux quantum rather than a full flux quantum when the Coulomb repulsion is turned on. Lastly, we discuss the relationship of the Drude weight computed from open chains to that obtained from rings, and also the effect of arbitrary phases at the ring boundary.