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arXiv:cond-mat/0010039 · Physical review. B, Condensed matter

Nonperturbative approach to the attractive Hubbard model

S. Allen, A.–M. S. Tremblay

Published Jul 31, 2001• 42 citations
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Abstract

A nonperturbative approach to the single-band attractive Hubbard model is presented in the general context of functional-derivative approaches to many-body theories. As in previous work on the repulsive model, the first step is based on a local-field-type ansatz, on enforcement of the Pauli principle and a number of crucial sumrules. The Mermin-Wagner theorem in two dimensions is automatically satisfied. At this level, two-particle self-consistency has been achieved. In the second step of the approximation, an improved expression for the self-energy is obtained by using the results of the first step in an exact expression for the self-energy, where the high- and low-frequency behaviors appear separately. The result is a cooperon-like formula. The required vertex corrections are included in this self-energy expression, as required by the absence of a Migdal theorem for this problem. Other approaches to the attractive Hubbard model are critically compared. Physical consequences of the present approach and agreement with Monte Carlo simulations are demonstrated in the accompanying paper (following this one).

DOI
10.1103/physrevb.64.075115

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A nonperturbative approach to the single-band attractive Hubbard model is presented in the general context of functional-derivative approaches to many-body theories. As in previous work on the repulsive model, the first step is based on a local-field-type ansatz, on enforcement of the Pauli principle and a number of crucial sumrules. The Mermin-Wagner theorem in two dimensions is automatically satisfied. At this level, two-particle self-consistency has been achieved. In the second step of the approximation, an improved expression for the self-energy is obtained by using the results of the first step in an exact expression for the self-energy, where the high- and low-frequency behaviors appear separately. The result is a cooperon-like formula. The required vertex corrections are included in this self-energy expression, as required by the absence of a Migdal theorem for this problem. Other approaches to the attractive Hubbard model are critically compared. Physical consequences of the present approach and agreement with Monte Carlo simulations are demonstrated in the accompanying paper (following this one).
In this thesis, I present a non-perturbative approach to the single-band attractive Hubard model which is an extension of previous work by Vilk and Tremblay on the repulsive model. Exact results are derived in the general context of functional derivative approaches to many-body theories. The first step of the approximation is based on a local field type ansatz. All physical quantities can be expressed as a function of double-occupancy (in addition to temperature and filling). Double-occupancy is determined without adjustable parameter by imposing the Pauli principle and a crucial sum-rule, making the first step of the approximation Two-Particle Self-Consistent. The final expression for the self-energy is obtained by calculating the low-frequency part of the exact expression with the two-particle correlation, Green function and renormalized vertex obtained in the first step of the approximation. The Mermin-Wagner theorem in two dimensions is automatically satisfied. Application of this non-perturbative many-body approach to the intermediate coupling regime shows quantitative agreement with quantum Monte Carlo calculations. Both approaches predict the existence of a pseudogap in the single-particle spectral weight. I present some physical properties, such as correlation lengths, superfluid density, and characteristic pair fluctuation energy, to highlight the origin of the pseudogap in the weak to intermediate coupling regime. These results suggest that two-dimensional systems that are described by a symmetry group larger than SO(2) could have a larger region of pseudogap behavior. High-temperature superconductors may belong to that category of systems.
A new approach to the single-band Hubbard model is described in the general context of many-body theories. It is based on enforcing conservation laws, the Pauli principle and a number of crucial sum-rules. More specifically, spin and charge susceptibilities are expressed, in a conserving approximation, as a function of two constant irreducible vertices whose values are found self-consistently. The Mermin-Wagner theorem in two dimensions is automatically satisfied. The effect of collective modes on single-particle properties is then obtained by a paramagnon-like formula that is consistent with the two-particle properties in the sense that the potential energy obtained from TrΣGTrΣGTrΣG is identical to that obtained using the fluctuation-dissipation theorem for susceptibilities. The vertex corrections are included through constant irreducible vertices. The theory is in quantitative agreement with Monte Carlo simulations for both single-particle and two-particle properties. In the two-dimensional renormalized classical regime, spin fluctuations lead to precursors of antiferromagnetic bands (shadow bands) and to the destruction of the Fermi-liquid quasiparticles in a wide temperature range above the zero-temperature phase transition. The analogous phenomenon of pairing pseudogap can occur in the attractive model in two dimensions when the pairing fluctuations become critical. Other many-body approaches are critically compared. It is argued that treating the spin fluctuations as if there was a Migdal's theorem can lead to wrong predictions, in particular with regard to the the single-particle pseudogap.
We calculate the one-electron spectral function of the attractive (negative-U)U)U) Hubbard model. We work in the intermediate-coupling and low-density regime and obtain the self-energy in an approximate analytical form. The excitation spectrum is found to consist of three branches. The results are obtained in a framework based on the self-consistent TTT-matrix approximation, which is compatible with the Mermin-Wagner theorem.
In this work, we adapt the formalism of the dynamical vertex approximation ($\mathrm{D}\mathrm{\ensuremath{\Gamma}}\mathrm{A}$), a diagrammatic approach including many-body correlations beyond the dynamical mean-field theory, to the case of attractive on-site interactions. We start by exploiting the ladder approximation of the $\mathrm{D}\mathrm{\ensuremath{\Gamma}}\mathrm{A}$ scheme, in order to derive the corresponding equations for the nonlocal self-energy and vertex functions of the attractive Hubbard model. Second, we prove the validity of our derivation by showing that the results obtained in the particle-hole symmetric case fully preserve the exact mapping between the attractive and the repulsive models. It will be shown how this property can be related to the structure of the ladders, which makes our derivation applicable for any approximation scheme based on ladder diagrams. Finally, we apply our $\mathrm{D}\mathrm{\ensuremath{\Gamma}}\mathrm{A}$ algorithm to the attractive Hubbard model in three dimensions, for different fillings and interaction values. Specifically, we focus on the parameters region in the proximity of the second-order transition to the superconducting and charge-density wave phases, respectively, and calculate (i) their phase-diagrams, (ii) their critical behavior, as well as (iii) the effects of the strong nonlocal correlations on the single-particle properties.
Nonperturbative solutions to the Hubbard model are found by using the functional derivative method. A system of closed equations is obtained for the Green's functions of the Hubbard model. Exact expressions for the self-energy are derived which involve only first order functional derivatives. A scheme is proposed for systematically calculating self-energy corrections. We apply the solution to finite rings of two and four lattice sites and compare to the results of numerical calculations on these systems.
We have investigated the attractive Hubbard model in the low-density limit for the 2D square lattice using the ladder approximation for the vertex function in a self-consistent, conserving formulation. In the parameter region where the on-site attraction is of the order of the bandwidth, we found no evidence of a pseudo-gap. Furthermore, we have observed that the suppression of the Fermi surface known to destroy superconductivity in one and two dimensions, when these systems are treated using a non-self-consistent theory (Schmitt-Rink, Varma C M and Ruckenstein A E 1989 Phys. Rev. Lett. 63 445), does not occur when pair-pair interactions are included. However, we do find a quasi-particle lifetime that varies linearly with temperature, which is similar to the findings from many experiments. Thus, although this system has a Fermi surface, it shows non-Fermi-liquid-type behaviour over a wide temperature range. We stress that our work uses thermal Green's functions along the real-time axis, and thus allows for a more accurate determination of the dynamical properties of a model than theories that require extrapolations from the imaginary-frequency axis.