Penson-Kolb-Hubbard model: A study of the competition between single-particle and pair hopping in one dimension

Abstract

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=-${\mathcal{J}}_{\mathrm{i},\mathrm{\ensuremath{\eta}}=\ifmmode\pm\else\textpm\fi{}1,\mathrm{\ensuremath{\sigma}}}$ (${\mathit{tc}}_{\mathit{i}+\mathrm{\ensuremath{\eta}}\mathrm{\ensuremath{\sigma}}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${\mathit{c}}_{\mathit{i}\mathrm{\ensuremath{\sigma}}}$+${\mathit{Vc}}_{\mathit{i}+\mathrm{\ensuremath{\eta}}\mathrm{\ensuremath{\uparrow}}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${\mathit{c}}_{\mathit{i}+}$ $_{\mathrm{\ensuremath{\eta}}\mathrm{\ensuremath{\downarrow}}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${\mathit{c}}_{\mathit{i}\mathrm{\ensuremath{\downarrow}}}$${\mathit{c}}_{\mathit{i}\mathrm{\ensuremath{\uparrow}}}$)+ ${\mathcal{J}}_{\mathit{i}}$ U${\mathit{n}}_{\mathit{i}\mathrm{\ensuremath{\uparrow}}}$${\mathit{n}}_{\mathit{i}\mathrm{\ensuremath{\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\ensuremath{\gg}U\ensuremath{\sim}2V) turns into a first-order superconducting--antiferromagnetic transition in the strong-coupling regime [t\ensuremath{\ll}U\ensuremath{\sim}(4/\ensuremath{\pi})V]. We also observe evidence of a charge-density-wave--superconducting transition in the parameter range (t\ensuremath{\sim}V\ensuremath{\gg}U). These three transition lines meet together at a tricritical point at (t:U:V)\ensuremath{\sim}(0.04:0.54:0.42). A naive renormalization-group analysis in the intermediate-coupling regime produces results consistent with this conclusion.