Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0408074 · Physical Review B

Minimally self-consistent T-matrix approximation to describe the low-temperature properties of the Hubbard model in the atomic limit

Simona Verga, Robert J Gooding, F. Marsiglio

Published Apr 18, 2005• 8 citations
arXivHTMLPDF

Abstract

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 $T$-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 $T$-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 $T$-matrix approximation is shown to be successful in reproducing the exact results for the atomic limit, while fully self-consistent $T$-matrix results do not agree with the known solutions. Of particular note is that the minimally self-consistent $T$-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 $T$-matrix theory can give better results over a broader temperature range.

DOI
10.1103/physrevb.71.155111

Related papers

Analyzing the success ofT-matrix diagrammatic theories in representing a modified Hubbard model

P. Pisarski, Robert J Gooding

We present a systematic study of various forms of renormalization that can be applied in the calculation of the self-energy of the Hubbard model within the T-matrix approximation. We compare the exact solutions of the attractive and repulsive Hubbard models, for linear chains of lengths up to eight sites, with all possible taxonomies of the T-matrix approximation. For the attractive Hubbard model, the success of a minimally self-consistent theory found earlier in the atomic limit (Verga et al 2005 Phys. Rev. B 71 155111) is not maintained for finite clusters unless one is in the very strong correlation limit. For the repulsive model, in the weak correlation limit at low electronic densities-that is, where one would expect a self-consistent T-matrix theory to be adequate-we find the fully renormalized theory to be most successful. In our studies we employ a modified Hubbard interaction that eliminates all Hartree diagrams, an idea which was proposed earlier (Zlatić et al 2000 Phys. Rev. B 63 035104).

arXiv:1010.3383•Journal of Physics Condensed Matter•May 4, 2011•3 citations

SELF-CONSISTENT T-MATRIX APPROXIMATION TO THE HUBBARD MODEL FOR A FINITE NUMBER OF SITES

Mohsen Jemaï

Within the 1D Hubbard model, linear closed chains with various numbers of sites are considered in particle–particle self-consistent random phase approximation (SCRPA) for the T-matrix. Encouraging results with a minimal numerical effort are obtained, confirming earlier results with this theory for other models. SCRPA solves the two-site problem exactly. It therefore contains the two electrons and high density Fermi gas limits correctly.

Modern Physics Letters B•Jun 30, 2009•0 citations

A self-consistent, conserving theory of the attractive Hubbard model in two dimensions

Martin Letz, R J Gooding

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.

arXiv:cond-mat/9802107•Journal of Physics Condensed Matter•Aug 10, 1998•17 citations

From the atomic limit to a metal-insulator transition in the Hubbard model

L. Craco, M. A. Gusmão

Starting from the exact solution of the Hubbard model in the atomic limit, and treating the hopping term as a perturbation by means of a diagram technique, we discuss the electronic structure of the model in some simple approximations. We show that a metal-insulator transition is obtained above a critical value of the on-site Coulomb interaction in a self-consistent evaluation of the one-particle Green's function to one-loop order. A relationship with results in infinite dimensions is briefly discussed.

Physical review. B, Condensed matter•Dec 15, 1995•27 citations

Density-Induced Breaking of Pairs in the Attractive Hubbard Model

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

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.

arXiv:cond-mat/9802239•Physical Review Letters•Apr 6, 1998•30 citations

Some global properties of the attractive Hubbard model in the superconducting phase: theT-matrix approximation in two dimensions

Sandra Schafroth, José Jesús Rodríguez-Núñez, H Beck

We have applied the fast Fourier transform, which allows one to compute efficiently convolution sums, to solve the set of self-consistent T-matrix equations to get the Green function of the two-dimensional attractive-U Hubbard model below , extending previous calculations by the same authors. Using a constant order parameter , we calculated as a function of the electron density and interaction strength U. These global results deviate from the BCS behaviour remarkably.

arXiv:cond-mat/9612246•Journal of Physics Condensed Matter•Mar 3, 1997•8 citations
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 present a systematic study of various forms of renormalization that can be applied in the calculation of the self-energy of the Hubbard model within the T-matrix approximation. We compare the exact solutions of the attractive and repulsive Hubbard models, for linear chains of lengths up to eight sites, with all possible taxonomies of the T-matrix approximation. For the attractive Hubbard model, the success of a minimally self-consistent theory found earlier in the atomic limit (Verga et al 2005 Phys. Rev. B 71 155111) is not maintained for finite clusters unless one is in the very strong correlation limit. For the repulsive model, in the weak correlation limit at low electronic densities-that is, where one would expect a self-consistent T-matrix theory to be adequate-we find the fully renormalized theory to be most successful. In our studies we employ a modified Hubbard interaction that eliminates all Hartree diagrams, an idea which was proposed earlier (Zlatić et al 2000 Phys. Rev. B 63 035104).
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.
Starting from the exact solution of the Hubbard model in the atomic limit, and treating the hopping term as a perturbation by means of a diagram technique, we discuss the electronic structure of the model in some simple approximations. We show that a metal-insulator transition is obtained above a critical value of the on-site Coulomb interaction in a self-consistent evaluation of the one-particle Green's function to one-loop order. A relationship with results in infinite dimensions is briefly discussed.
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.