In this paper, we present a study of the ground-state phase diagram of a one-dimensional quantum chain, the Penson-Kolb-Hubbard model, H=-${\mathcal{J}}_{\mathrm{i},\mathrm{\eta }=\pm 1,\mathrm{\sigma }}$ (${\mathit{tc}}_{\mathit{i}+\mathrm{\eta }\mathrm{\sigma }}^{\mathrm{°}}$${\mathit{c}}_{\mathit{i}\mathrm{\sigma }}$+${\mathit{Vc}}_{\mathit{i}+\mathrm{\eta }\mathrm{\uparrow }}^{\mathrm{°}}$${\mathit{c}}_{\mathit{i}+}$ ${}_{\mathrm{\eta }\mathrm{\downarrow }}{}^{\mathrm{°}}{}_{}{}^{}$${\mathit{c}}_{\mathit{i}\mathrm{\downarrow }}$${\mathit{c}}_{\mathit{i}\mathrm{\uparrow }}$)+ ${\mathcal{J}}_{\mathit{i}}$ U${\mathit{n}}_{\mathit{i}\mathrm{\uparrow }}$${\mathit{n}}_{\mathit{i}\mathrm{\downarrow }}$ at half filling. We have examined the system using exact diagonalization for samples of up to 12 sites and employed two techniques, eigenprojection decomposition and twisted-boundary conditions, in analyzing the data. These techniques allow us to characterize the ground state in a manner insensitive to changes in sample size and provide us with a clean way to visualize the physics. When used with the ‘‘correct’’ order parameter, qualitative features emerge even for sample sizes as small as six sites. We find that the second-order charge-density-wave–spin-density-wave transition in the weak-coupling limit (t≫U∼2V) turns into a first-order superconducting–antiferromagnetic transition in the strong-coupling regime [t≪U∼(4/π)V]. We also observe evidence of a charge-density-wave–superconducting transition in the parameter range (t∼V≫U). These three transition lines meet together at a tricritical point at (t:U:V)∼(0.04:0.54:0.42). A naive renormalization-group analysis in the intermediate-coupling regime produces results consistent with this conclusion.

• Received 21 January 1993

DOI:https://doi.org/10.1103/PhysRevB.48.2063