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arXiv:cond-mat/9802239 · Physical Review Letters

Density-Induced Breaking of Pairs in the Attractive Hubbard Model

Bumsoo Kyung, Elyakum G. Klepfish, P. E. Kornilovitch

Published Apr 6, 1998• 30 citations
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Abstract

A conserving $T$-matrix approximation is applied to the two-dimensional attractive Hubbard model in the low-density regime. A set of self-consistent equations is solved in the real-frequency domain to avoid the analytic continuation procedure. By tuning the chemical potential the particle density was varied in the limits $0.01<n<0.18$. For the value of the attractive potential $U\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}8t$ the binding energy of pairs monotonically decreases with increasing $n$, from its zero-density limit $2.3t$, and vanishes at a critical density ${n}_{\mathrm{cr}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.19$. A pairing-induced pseudogap in the single-particle density of states is found at low densities and temperatures.

DOI
10.1103/physrevlett.80.3109

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A conserving TTT-matrix approximation is applied to the two-dimensional attractive Hubbard model in the low-density regime. A set of self-consistent equations is solved in the real-frequency domain to avoid the analytic continuation procedure. By tuning the chemical potential the particle density was varied in the limits 0.01<n<0.180.01<n<0.180.01<n<0.18. For the value of the attractive potential U=8tU\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}8tU=8t the binding energy of pairs monotonically decreases with increasing nnn, from its zero-density limit 2.3t2.3t2.3t, and vanishes at a critical density ncr=0.19{n}_{\mathrm{cr}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.19ncr​=0.19. A pairing-induced pseudogap in the single-particle density of states is found at low densities and temperatures.
The two-dimensional attractive Hubbard model is studied in the weak-to-intermediate-coupling regime by employing a nonperturbative approach. It is shown that this approach is in quantitative agreement with Monte Carlo calculations for both single-particle and two-particle quantities. Both the density of states and the single-particle spectral weight show a pseudogap at the Fermi energy below some characteristic temperature T∗,{T}^{*},T∗, also in good agreement with quantum Monte Carlo calculations. The pseudogap is caused by critical pairing fluctuations in the low-temperature renormalized classical regime $(\ensuremath{\Elzxh}\ensuremath{\omega}<{k}_{B}T)$ of the two-dimensional system. With increasing temperature the spectral weight fills in the pseudogap instead of closing it, and the pseudogap appears earlier in the density of states than in the spectral function. Small temperature changes around T∗{T}^{*}T∗ can modify the spectral weight over frequency scales much larger than temperature. Several qualitative results for the s-wave case should remain true for d-wave superconductors.
One-particle spectral properties in the normal phase of the two-dimensional attractive Hubbard model are investigated in the weak-coupling regime using the non-self-consistent T-matrix approximation. The corresponding equations are evaluated numerically directly on the real-frequency axis. For temperatures sufficiently close to the superconducting transition temperature, a pseudogap in the one-particle spectral function is observed, which can be assigned to the increasing importance of pair fluctuations.
Angle-resolved photoemission spectra are calculated microscopically for the two-dimensional attractive Hubbard model. A system of self-consistent T -matrix equations are solved numerically in the real-time domain. The single-particle spectral function has a two-peak structure resulting from the presence of bound states. The spectral function is suppressed at the chemical potential, leading to a pseudogap-like behaviour. At high temperatures and densities the pseudogap diminishes and finally disappears; these findings are similar to the experimental observations for the cuprates.
We explore the effect of fluctuations for T≥TcT \geq T_cT≥Tc​ in the 2-D negative Hubbard model within the framework of the selfconsistent T-matrix formalism, which goes beyond the BCS approximation and includes pair fluctuations. We enter the regime where correlations are important, namely, U/t=−4.0U/t = -4.0U/t=−4.0, where ttt denotes the hopping matrix element between nearest neighboors and UUU is the strength of the on-site interaction. Our results include: 1) In comparison to the free case, the distribution function, n(k)n({\bf k})n(k), reveals considerable renormalization. This fact makes us conclude that the Fermi surface is blurred for correlated electron systems; 2) There is a pseudogap in the density of states, N(ω)N(ω)N(ω), around the chemical potential, μμμ; and 3) The real part of the T-matrix, at zero frequency and zero momentum, vs T/tT/tT/t shows a divergence at a particular temperature. This last result shows that more approximate T-matrix calculations, extended to T = 0T~=~0T = 0 are meaningless.
The atomic limit of the Hubbard model is a simple single-site problem which can be solved exactly, and all one- and two-particle Green's functions can be obtained analytically. These solutions can thus serve as a means of critiquing the success of various approximate theories which might be applied to the full Hubbard model. In particular, we have examined the TTT-matrix approximation for the attractive Hubbard model in the atomic limit, which should give reasonable results at low electronic densities, if one can avoid the spurious phase transition that results when a fully non-self-consistent TTT-matrix approximation is employed---previously we have shown that any level of self-consistency guarantees that this phase transition is correctly suppressed to zero temperature in two dimensions or less. Here, a minimally self-consistent TTT-matrix approximation is shown to be successful in reproducing the exact results for the atomic limit, while fully self-consistent TTT-matrix results do not agree with the known solutions. Of particular note is that the minimally self-consistent TTT-matrix approximation reproduces not only one- and two-particle (static) thermodynamic quantities, but it also exactly reproduces the one-particle spectral function at low but nonzero temperatures. We also make a comparison to the two-particle self-consistent approach of Vilk and Tremblay, and find that the minimally self-consistent TTT-matrix theory can give better results over a broader temperature range.
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.