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Physical review. B, Condensed matter

Phase diagram for the attractive Hubbard model in two dimensions in a conserving approximation

J. J. Deisz, D. W. Hess, Joseph W. Serene

Published Jul 25, 2002• 18 citations
Publisher

Abstract

We calculate the superconducting transition temperature as a function of interaction strength and density for a two-dimensional attractive Hubbard model in the fluctuation exchange approximation (FEA). Significant corrections to mean-field theory are apparent, including a maximum in ${T}_{c}$ as a function of U and a fluctuation-broadened specific heat peak. At $n=1,$ competing pair fluctuations and charge-density fluctuations fail to drive ${T}_{c}$ to zero. This failure of the FEA reflects an inadequate account of the coupling among fluctuations. Because ${T}_{c}$ is determined from self-consistent calculations of thermodynamic potentials, the calculations are extremely sensitive to numerical errors, requiring accuracy up to 1 part in ${10}^{6}.$ We outline a procedure for systematically reducing numerical errors associated with the finite-frequency cutoff in numerical many-body calculations such as ours.

DOI
10.1103/physrevb.66.014539

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We calculate the superconducting transition temperature as a function of interaction strength and density for a two-dimensional attractive Hubbard model in the fluctuation exchange approximation (FEA). Significant corrections to mean-field theory are apparent, including a maximum in Tc{T}_{c}Tc​ as a function of U and a fluctuation-broadened specific heat peak. At n=1,n=1,n=1, competing pair fluctuations and charge-density fluctuations fail to drive Tc{T}_{c}Tc​ to zero. This failure of the FEA reflects an inadequate account of the coupling among fluctuations. Because Tc{T}_{c}Tc​ is determined from self-consistent calculations of thermodynamic potentials, the calculations are extremely sensitive to numerical errors, requiring accuracy up to 1 part in 106.{10}^{6}.106. We outline a procedure for systematically reducing numerical errors associated with the finite-frequency cutoff in numerical many-body calculations such as ours.
We report a superconducting transition in the fully conserving fluctuation exchange approximation for a two-dimensional attractive Hubbard model. The transition temperature agrees well with quantum Monte Carlo calculations, but the superfluid density and specific heat near Tc{T}_{c}Tc​ have anomalous temperature dependences for lattice sizes up to $64\ifmmode\times\else\texttimes\fi{}64$.
We study the extension of the superconducting fluctuation-exchange approximation for the attractive Hubbard model into the ordered phase. We show that including all possible diagrams with anomalous propagators in the Luttinger-Ward functional leads to a mismatch between the critical temperature determined by the instability of the normal phase and that associated with the emergence of anomalous long-range order. We then derive a consistent theory by restricting two-particle propagation in the ladder diagrams in the ordered phase. Our main contributions are a derivation of this consistent approximation and the numerical analysis of its thermodynamic and spectral properties.
On the basis of spin-pairing-fluctuation-exchange approximation, we study the superconductivity in the quasi-two-dimensional Hubbard model. The integral equations for the Green's function are self-consistently solved by numerical calculation. Solutions for the order parameter, London penetration depth, density of states, and transition temperature are obtained. Some of the results are compared with the experiments for the cuprate high-temperature superconductors. Numerical techniques are presented in detail. With these techniques, the amount of numerical computations can be greatly reduced.
[[abstract]]The thermodynamic phase diagram of the attractive Hubbard model in the presence of magnetic field is calculated in wide range of temperature, electron concentration and coupling strength within the generalized self-consistent field (GSCF) approach, including the spin density correlations. The theory provides a smooth transition driven by applied magnetic field IL and temperature T from the homogeneous superconducting state into a spatially inhomogeneous state with the order parameter Delta ((-))(Q) not equal 0.
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.
We investigate the superfluid phase transition in a gas of Fermi atoms loaded on a three-dimensional optical lattice. When the lattice potential is strong, this system can be well described by an attractive Hubbard model. In this model, we calculate the superfluid phase transition temperature Tc{T}_{\mathrm{c}}Tc​, including both superfluid and (spin and charge) density fluctuations within the self-consistent ttt-matrix theory and fluctuation exchange approximation, respectively. Since we treat these fluctuations in a consistent manner, our theory satisfies the required particle-hole symmetry over the entire BCS--Bose-Einstein-condensation (BEC) crossover region. We show that charge density fluctuations compete against superfluid fluctuations near the half-filling, leading to the suppression of Tc{T}_{\mathrm{c}}Tc​. As a result, the maximum Tc{T}_{\mathrm{c}}Tc​ is obtained away from the half-filling. Since the strong density fluctuations originate from the nesting property of the Fermi surface at the half-filling (which is absent in a uniform gas with no lattice potential), our results would be useful in considering lattice effects on strong-coupling superfluidity.