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arXiv:cond-mat/9601065 · Physical review. B, Condensed matter

Strong-coupling expansion for the Hubbard model in arbitrary dimension using slave bosons

P. J. H. Denteneer

Published Apr 15, 1996• 3 citations
arXivHTMLPDF

Abstract

A strong-coupling expansion for the antiferromagnetic phase of the Hubbard model is derived in the framework of the slave-boson mean-field approximation. The expansion can be obtained in terms of moments of the density of states of freely hopping electrons on a lattice, which in turn are obtained for hypercubic lattices in arbitrary dimension. The expansion is given for the case of half-filling and for the energy up to fifth order in the ratio of hopping integral t over on-site interaction U, but can straightforwardly be generalized to the non-half-filled case and be extended to higher orders in t/U. For the energy the expansion is found to have an accuracy of better than 1% for U/t\ensuremath{\ge}8. A comparison is given with an earlier perturbation expansion based on the linear-spin-wave approximation and with a similar expansion based on the Hartree-Fock approximation. The case of an infinite number of spatial dimensions is discussed. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.9764

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A strong-coupling expansion for the antiferromagnetic phase of the Hubbard model is derived in the framework of the slave-boson mean-field approximation. The expansion can be obtained in terms of moments of the density of states of freely hopping electrons on a lattice, which in turn are obtained for hypercubic lattices in arbitrary dimension. The expansion is given for the case of half-filling and for the energy up to fifth order in the ratio of hopping integral t over on-site interaction U, but can straightforwardly be generalized to the non-half-filled case and be extended to higher orders in t/U. For the energy the expansion is found to have an accuracy of better than 1% for U/t\ensuremath{\ge}8. A comparison is given with an earlier perturbation expansion based on the linear-spin-wave approximation and with a similar expansion based on the Hartree-Fock approximation. The case of an infinite number of spatial dimensions is discussed. \textcopyright{} 1996 The American Physical Society.
In this study, we perform a strong-coupling expansion up to third order of the hopping parameter ttt for the spin-1 Bose-Hubbard model with antiferromagnetic interaction. As expected from previous studies, the Mott-insulator phase is considerably more stable against the superfluid phase when filling with an even number of bosons than when filling with an odd number of bosons. The phase-boundary curves are consistent with the perturbative mean-field theory in the limit of infinite dimensions. The critical value of the hopping parameter tC{t}_{\mathrm{C}}tC​ at the peak of the Mott lobe depends on the antiferromagnetic interaction. This result indicates the reliability of the strong-coupling expansion when U2{U}_{2}U2​ possesses large (intermediate) values for Mott lobe with an even (odd) number of bosons. Moreover, in order to improve our results, we apply a few extrapolation methods up to infinite order in ttt. The fitting results of the phase-boundary curves agree better with those of the perturbative mean-field approximation. In addition, the linear fit error of tC{t}_{\mathrm{C}}tC​ is very small for the strong antiferromagnetic interaction.
The attractive Hubbard model for normal and superconducting ground states is examined for arbitrary electron concentration by the use of the slave-boson mean-field approximation (SBMFA) technique. This approach, at a saddle-point level, is equivalent to the Gutzwiller approximation. Several superfluid characteristics of the model are shown for hypercubic lattices of all dimensions, including , and a comparison with the Bardeen-Cooper-Schrieffer Hartree-Fock approximation (BCS-HFA) calculations and exact results is made. Our results show quantitative and qualitative corrections of the SBMFA to the HFA. The improvement of the SBMFA over the HFA diminishes with increasing lattice dimensionality. We have also evaluated the energy difference between the superconducting and the normal states; this is compared with available results for the superconducting critical temperatures in various dimensions and Uemura-type plots are obtained. The results confirm and substantiate the assertion of a continuous evolution of the superfluid properties of the model from the weak-coupling (BCS-like) to the strong-coupling (composite-boson superconductivity) limit.
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.
The superfluid characteristics of the attractive Hubbard model are analyzed for any coupling |U| and arbitrary electron concentration (0n2) by means of the slave-boson mean-field method and also by the perturbative treatment of the strong-coupling limit. The slave boson method takes into account correlations of electrons and yields a reliable description of the crossover from BCS-type superconductivity to local pair (composite bosons) superconductivity with increasing |U|. The results for the ground state (the free energy, the gap in the excitation spectrum) and the electromagnetic characteristics (the critical magnetic field, the London penetration depth, the coherence length) are compared with those obtained by the Hartree-Fock approximation and by the self-consistent second-order perturbation theory in the weak-coupling limit as well as with those obtained using perturbational approaches in the strong-coupling limit. We show that the slave-boson method, in contrast to the Hartree-Fock approximation, gives credible results for all investigated quantities in the whole interaction range, interpolating smoothly between the BCS and local pair regimes. A comparison of theoretical predictions for our simple model with experimental data for various families of short-coherence-length superconductors suggests that the best agreement can be obtained for intermediate values of the local attraction. \textcopyright{} 1996 The American Physical Society.
The slave-boson mean-field method is used to study the two-dimensional Hubbard model. A magnetic phase diagram allowing for paramagnetism, weak and strong ferromagnetism and antiferromagnetism is constructed and compared to the corresponding phase diagram using the Hartree-Fock approximation (HFA). Magnetically ordered regions are reduced by a factor of about three along both the t/U and density axes compared to the HFA. Using the spin-rotation-invariant formulation of the slave-boson method the helicity modulus is computed and for half filling is found practically to coincide with that found using variational Monte Carlo calculations with the Gutzwiller wave function. Off half filling the results can be used to compare with quantum Monte Carlo calculations of the effective hopping parameter. Contrary to the case of half filling, the slave-boson approach is seen to greatly improve the results of the HFA when off half filling.
A strong-coupling expansion for the Green's functions, self-energies, and correlation functions of the Bose-Hubbard model is developed. We illustrate the general formalism, which includes all possible (normal-phase) inhomogeneous effects in the formalism, such as disorder or a trap potential, as well as effects of thermal excitations. The expansion is then employed to calculate the momentum distribution of the bosons in the Mott phase for an infinite homogeneous periodic system at zero temperature through third order in the hopping. By using scaling theory for the critical behavior at zero momentum and at the critical value of the hopping for the Mott insulator--to--superfluid transition along with a generalization of the random-phase-approximation-like form for the momentum distribution, we are able to extrapolate the series to infinite order and produce very accurate quantitative results for the momentum distribution in a simple functional form for one, two, and three dimensions. The accuracy is better in higher dimensions and is on the order of a few percent relative error everywhere except close to the critical value of the hopping divided by the on-site repulsion. In addition, we find simple phenomenological expressions for the Mott-phase lobes in two and three dimensions which are much more accurate than the truncated strong-coupling expansions and any other analytic approximation we are aware of. The strong-coupling expansions and scaling-theory results are benchmarked against numerically exact quantum Monte Carlo simulations in two and three dimensions and against density-matrix renormalization-group calculations in one dimension. These analytic expressions will be useful for quick comparison of experimental results to theory and in many cases can bypass the need for expensive numerical simulations.