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Physical review. B, Condensed matter

Onsager reaction terms for quantum many-body systems: Application to antiferromagnetic and superconducting order in the Hubbard model

Antoine Georges, Jonathan S. Yedidia

Published Feb 1, 1991• 40 citations

Abstract

We demonstrate that an expansion in powers of the strength of the interaction, of the free energy at fixed order parameter, can be used to generate and correct mean-field theories for interacting quantum many-body systems. The first two terms in the expansion generally yield ordinary Hartree-Fock mean-field theory and the next term gives an ``Onsager reaction field'' correction to Hartree-Fock theory. This method can be used to directly generate expansions for inverse susceptibilities. We illustrate the method for the one- and two-dimensional Hubbard model, for which we consider corrections to mean-field theories of antiferromagnetism for the repulsive-U half-filled case and superconductivity in the attractive-U case. These corrections give a quantitative account superior to that of the random-phase approximation (RPA) for the correlation energy at small and intermediate values of U. For susceptibilities, we recover from the first two terms in the expansion the usual RPA results, while higher-order terms give systematic corrections to the RPA susceptibilities. For the case of superconductivity in the repulsive-U Hubbard model, we show that the higher-order terms in the expansion must be considered to determine whether or not an instability exists. We find that there is no superconducting instability in the repulsive-U case, at least towards ordinary singlet or triplet pairing. We also find no evidence for a superconducting instability driven by a coexisting antiferromagnetic order.

DOI: 10.1103/physrevb.43.3475

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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.

Physical review. B, Condensed matter•Nov 1, 1996•27 citations

We demonstrate that an expansion in powers of the strength of the interaction, of the free energy at fixed order parameter, can be used to generate and correct mean-field theories for interacting quantum many-body systems. The first two terms in the expansion generally yield ordinary Hartree-Fock mean-field theory and the next term gives an ``Onsager reaction field'' correction to Hartree-Fock theory. This method can be used to directly generate expansions for inverse susceptibilities. We illustrate the method for the one- and two-dimensional Hubbard model, for which we consider corrections to mean-field theories of antiferromagnetism for the repulsive-U half-filled case and superconductivity in the attractive-U case. These corrections give a quantitative account superior to that of the random-phase approximation (RPA) for the correlation energy at small and intermediate values of U. For susceptibilities, we recover from the first two terms in the expansion the usual RPA results, while higher-order terms give systematic corrections to the RPA susceptibilities. For the case of superconductivity in the repulsive-U Hubbard model, we show that the higher-order terms in the expansion must be considered to determine whether or not an instability exists. We find that there is no superconducting instability in the repulsive-U case, at least towards ordinary singlet or triplet pairing. We also find no evidence for a superconducting instability driven by a coexisting antiferromagnetic order.

The Onsager-reaction-field correction to Hartree-Fock theory recently derived by Georges and Yedidia [Phys. Rev. B 43, 3475 (1991)] is evaluated analytically for the half-filled positive-U Hubbard model on a d-dimensional hypercubic lattice. It is shown that the Onsager correction is for all d\ensuremath{\ge}3 irrelevant in the weak-coupling limit, while for d=1 and d=2 it strongly renormalizes the Hartree-Fock result, although it does note completely destabilize the spin-density wave. We examine the perturbation theory for the energy change induced by a small staggered magnetization m to all orders in U, and show that one obtains an infinite series in powers of (U/t)ln[${\mathit{m}}^{\mathrm{\ensuremath{-}}1}$ln(${\mathit{m}}^{\mathrm{\ensuremath{-}}1}$)] for d=1, and in powers of (U/t)

{\mathrm{ln}}^{2}

[${\mathit{m}}^{\mathrm{\ensuremath{-}}1}$

{\mathrm{ln}}^{2}

(${\mathit{m}}^{\mathrm{\ensuremath{-}}1}$)] for d=1. The close similarity between d=1 and d=2 is interpreted as evidence that antiferromagnetism in the weak coupling regime is not stable in d=2.

We analyze the competition between antiferromagnetism and superconductivity in the two-dimensional Hubbard model by combining a functional renormalization group flow with a mean-field theory for spontaneous symmetry breaking. Effective interactions are computed by integrating out states above a scale ${\ensuremath{\Lambda}}_{\mathrm{MF}}$ in one-loop approximation, which captures in particular the generation of an attraction in the

{\mathrm{ln}}^{2}

-wave Cooper channel from fluctuations in the particle-hole channel. These effective interactions are then used as an input for a mean-field treatment of the remaining low-energy states, with antiferromagnetism, singlet superconductivity, and triplet $\ensuremath{\pi}$ pairing as the possible order parameters. Antiferromagnetism and superconductivity suppress each other, leaving only a small region in parameter space where both orders can coexist with a sizable order parameter for each. Triplet $\ensuremath{\pi}$ pairing appears generically in the coexistence region, but its feedback on the other order parameters is very small.

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.

Perturbative corrections to the mean field theory for particle-hole instabilities of interacting electron systems are computed within a scheme which is equivalent to the recently developed variational approach to the Kohn-Luttinger superconductivity. This enables an unbiased comparison of particle-particle and particle-hole instabilities within the same approximation scheme. A spin-rotation invariant formulation for the particle-hole instabilities in the triplet channel is developed. The method is applied to the phase diagram of the t-t' Hubbard model on the square lattice. At the Van Hove density, antiferromagnetic and d-wave Pomeranchuk phases are found to be stable close to half filling. However, the latter phase is confined to an extremely narrow interval of densities and away from the singular filling, d-wave superconducting instability dominates.

The auxiliary boson method and 1/N expansion are applied to the solution of the Hubbard model in the U =

\infty

limit, in functional integral form. The zero order term in the expansion gives the mean-field result of non-interacting enhanced mass quasiparticles, and the next order gives Gaussian fluctuations around the mean-field value, representing effective interactions between the quasiparticles. From the density-density correlation function, the collective mode indicates a very different behavior away from and close to half filling. The superconducting instability is examined to 1/$N\sp2$ order from the dominant double collective mode exchange. For a parabolic band, there is a p-wave triplet instability. But for a 2D square lattice, there is no superconducting pairing to this order. The effects of disorder on this system are also investigated. For moderate filling, the specific heat and scattering rate have the same results as predicted in the weakly interacting theories. Close to half-filling, the specific heat and scattering rate show a much sharper temperature and energy dependence.

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.