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Journal of Physics Condensed Matter

The ground state of the Penson-Kolb-Hubbard model

B Bhattacharyya, G K Roy

Published Jul 10, 1995• 19 citations
Publisher

Abstract

The ground state of the one-dimensional half-filled Penson-Kolb-Hubbard model (with competing single-particle and pair hoppings in the presence of on-site Coulomb repulsion) has been investigated by a real space renormalization group method. The ground state energy, energy gap, local moment and several correlation functions are studied to obtain the phase diagram. The phase diagram consists of a spin density wave (SDW) state, a superconducting (SC) phase and a quasi-metallic phase dominated by short range SC correlations for positive values of the pair hopping amplitude; when this amplitude is negative, the phase diagram contains a near-metallic commensurate charge density wave and an SC phase of pairs with centre-of-mass momentum q= pi apart from an SDW phase.

DOI
10.1088/0953-8984/7/28/011

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The phase diagrams and superconducting properties of the extended Hubbard model with pair hopping interaction, i.e., the Penson-Kolb-Hubbard model are studied. The analysis of the model is performed for d-dimensional hypercubic lattices, including $d=1$ and $d=\ensuremath{\infty},$ by means of the (broken symmetry) Hartree-Fock approximations and, for $d=\ensuremath{\infty},$ by the slave-boson mean-field method. For $d=1,$ at half filling the phase diagram is shown to consist of nine different phases including two superconducting states with center-of-mass momentum $q=0$ and $q=Q$ $(\ensuremath{\eta}$ pairing), site and bond-located antiferromagnetic and charge-density wave states as well as three mixed phases with coexisting site and bond orderings. The stability range of the bond-type orderings shrinks with increasing lattice dimensionality d and for $d=\ensuremath{\infty}$ the corresponding diagram consists of four phases only, involving exclusively site-located orderings. Comparing the pair hopping model with the attractive Hubbard model we found in both cases gradual evolution from the BCS-like limit to the tightly bound pairs regime and a monotonic increase of the gap in the excitation spectrum with increasing coupling. However, the dynamics of electron pairs in both models is qualitatively different, which results in different dependences of condensation energies and critical temperatures on interaction parameters as well as in different electrodynamic properties, especially in a strong coupling regime.

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The ground state of the one-dimensional half-filled Penson-Kolb-Hubbard model (with competing single-particle and pair hoppings in the presence of on-site Coulomb repulsion) has been investigated by a real space renormalization group method. The ground state energy, energy gap, local moment and several correlation functions are studied to obtain the phase diagram. The phase diagram consists of a spin density wave (SDW) state, a superconducting (SC) phase and a quasi-metallic phase dominated by short range SC correlations for positive values of the pair hopping amplitude; when this amplitude is negative, the phase diagram contains a near-metallic commensurate charge density wave and an SC phase of pairs with centre-of-mass momentum q= pi apart from an SDW phase.
Different kinds of instabilities (charge-density wave, spin-density wave, singlet superconductivity) in the one-dimensional Hubbard model with pair-hopping interaction are investigated using an approximate Bethe-Salpeter equation. The study is performed at any density of electrons and for arbitrary values of the model parameters. In the absence of the on-site interaction, no transition occurs at half-filling for any finite negative pair-hopping parameter, in agreement with recent results; our calculation suggests that the Penson-Kolb model (t,W) with \ensuremath{\Vert}W/t\ensuremath{\Vert}/sinkF{\mathit{k}}_{\mathit{F}}kF​ behaves qualitatively like the Hubbard model (t,U). Phase diagrams of the Penson-Kolb-Hubbard model (t,U,W) at various densities are presented. \textcopyright{} 1996 The American Physical Society.
The ground-state properties of a 1D modified Penson–Kolb–Hubbard model characterized by alternating Hubbard interaction at odd and even sites are discussed by the bosonization scheme and the renormalization group analysis. At quarter filling, the quantum phase diagram consists of the charge-density-wave, spin-density-wave and singlet-superconductivity instabilities. The insulating phases come from the relevant umklapp process induced by the unequal on-site repulsion. The results provide a further understanding of occurrence of the umklapp process in 1D correlated electron systems.
In this paper we study the ground-state phase diagram of the one-dimensional half-filled repulsive (U > 0) Hubbard model supplemented with the pair-hopping interaction (W) (the Penson - Kolb - Hubbard model) using the continuum-limit field theory approach. We compare the low-energy properties of the U > 0 Hubbard model and W > 0 Penson - Kolb model. We show that, despite similar excitation spectra, the character of instabilities in these models is completely different. In contrast to the Hubbard model, in the case of the Penson - Kolb model, the dynamical generation of a charge gap leads to the suppression of spin-density-wave (SDW) fluctuations. The charge-density-wave fluctuations survive and coexist with bond-located SDW (Bd-SDW) instabilities. The Bd-SDW corresponds to a magnetically ordered state with staggered magnetization located on bonds between sites. The possibility of bond-located ordering is connected with the site-off-diagonal nature of the pair-hopping interaction. In the case of PKH the bond-ordered states exist at . For W > U/2 the Bd-SDW is realized while for the dimerized phase is realized.
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{}}}$${\mathit{c}}_{\mathit{i}\mathrm{\ensuremath{\sigma}}}$+${\mathit{Vc}}_{\mathit{i}+\mathrm{\ensuremath{\eta}}\mathrm{\ensuremath{\uparrow}}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$ci+{\mathit{c}}_{\mathit{i}+}ci+​ $_{\mathrm{\ensuremath{\eta}}\mathrm{\ensuremath{\downarrow}}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${\mathit{c}}_{\mathit{i}\mathrm{\ensuremath{\downarrow}}}$${\mathit{c}}_{\mathit{i}\mathrm{\ensuremath{\uparrow}}}$)+ Ji{\mathcal{J}}_{\mathit{i}}Ji​ U${\mathit{n}}_{\mathit{i}\mathrm{\ensuremath{\uparrow}}}$${\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.
By relaxing the conventional assumption of the equal one-electron hopping amplitude, we modify the one-dimensional half-filled Penson-Kolb (PK) model to obey the density-dependent hopping (DDH). For positive small modulation parameter [Formula: see text] and pair hopping [Formula: see text], we use bosonization and renormalization-group analysis to determine the ground state phase diagram. In the attractive case([Formula: see text]), the existence of the DDH leads to a gap transition in the spin sector and produces a fully gapless phase where a Luttinger liquid metal phase with dominant triplet superconducting correlations is realized for [Formula: see text], with the charge transition point [Formula: see text] and the spin transition point [Formula: see text]. Besides, there is a Luther-Emery matal phase with dominant singlet-superconducting correlations for [Formula: see text] and a Haldane insulator phase characterized by dominant bond-spin-density-wave correlations for [Formula: see text] in the repulsive regime ([Formula: see text]).
The phase diagrams and superconducting properties of the extended Hubbard model with pair hopping interaction, i.e., the Penson-Kolb-Hubbard model are studied. The analysis of the model is performed for d-dimensional hypercubic lattices, including d=1d=1d=1 and $d=\ensuremath{\infty},$ by means of the (broken symmetry) Hartree-Fock approximations and, for $d=\ensuremath{\infty},$ by the slave-boson mean-field method. For d=1,d=1,d=1, at half filling the phase diagram is shown to consist of nine different phases including two superconducting states with center-of-mass momentum q=0q=0q=0 and q=Qq=Qq=Q $(\ensuremath{\eta}$ pairing), site and bond-located antiferromagnetic and charge-density wave states as well as three mixed phases with coexisting site and bond orderings. The stability range of the bond-type orderings shrinks with increasing lattice dimensionality d and for $d=\ensuremath{\infty}$ the corresponding diagram consists of four phases only, involving exclusively site-located orderings. Comparing the pair hopping model with the attractive Hubbard model we found in both cases gradual evolution from the BCS-like limit to the tightly bound pairs regime and a monotonic increase of the gap in the excitation spectrum with increasing coupling. However, the dynamics of electron pairs in both models is qualitatively different, which results in different dependences of condensation energies and critical temperatures on interaction parameters as well as in different electrodynamic properties, especially in a strong coupling regime.