Modeling superconducting states of arbitrary pairing symmetry with the fluctuation exchange approximation applied to Hubbard-like models

We present an algorithm for microscopic modeling of superconducting states of arbitrary pairing symmetry via an implementation of self-consistent perturbation theory, the fluctuation exchange approximation (FLEX), applied to Hubbard-like models. Like the original formulation of FLEX, all ring diagrams that are generated with the bare interaction vertex are included in the generating functional for the self-energy. Thus, this algorithm is suitable for FLEX-based Hubbard model calculations in both the attractive and repulsive cases.

Figure 1

Bare vertex function in the particle-hole representation, ${\Gamma }_{{\alpha }_1,{\alpha }_3;{\alpha }_4{\alpha }_2}^{\left(0\right)ph}$ and particle-particle representation ${\Gamma }_{{\alpha }_1,{\alpha }_2;{\alpha }_3{\alpha }_4}^{\left(0\right)}$.

Figure 2

Diagrammatic representation of third and fourth order particle-hole ring diagram contributions to $\Phi$, the generating functional for the self-energy, $\Sigma$.

Figure 3

(Top) Pairing amplitude, $m_p$, vs scaled temperature, $T/t$, for the repulsive Hubbard model with $U=4t$, a density of 0.85 electrons per site and $t$ is the nearest-neighbor-hopping amplitude in the limit that $h_p\to 0$. Calculation was performed with a $16\times 16$ dynamical cluster embedded in a $64\times 64$ lattice using the dynamical cluster approximation. A fit (dashed line) to the FLEX data (circles) is used to estimate the transition temperature $T_c$. (Bottom) Scaled superconducting transition temperature, $T_c$, versus electron density per atomic site, $n$. All calculations were performed with $U=4t$ and an underlying $64\times 64$ lattice. The results are consistent with those produced by Pao and Bickers by analyzing pairing instabilities in the normal state. Some variation is observed with respect to cluster size in going from $4\times 4$ to $16\times 16$ clusters. Results for large clusters and densities near $n=1.0$ are difficult to obtain on account of singular features in FLEX calculations that make convergence slow at low temperatures.

Figure 4

Scaled superconducting transition temperature, $T_c/t^{\ast }$, versus scaled interaction strength, $\left|U\right|/\left(t^{\ast }+\left|U\right|\right)$, for the three-dimensional attractive Hubbard model with a density of one-electron per site where $t^{\ast }=2\sqrt{d}t\simeq 2.8t$ is a measure of the electron bandwidth as a function of spatial dimension $d$. Calculations were performed using an $N_L={16}^3$ lattice, but with correlations evaluated using $N_c=1^3$ (open circles) and $N_c=4^3$ (filled diamonds) clusters as prescribed by the dynamical cluster approximation. The weak dependence of $T_c/t^{\ast }$ results as a function of $N_c$ demonstrate the smallness of corrections to dynamical mean-field theory, which becomes exact in the infinite dimensional limit, at least for the parameters and approximation scheme explored here. These results are in excellent agreement with those obtained by Freericks in the infinite dimensional limit. (Ref. 25)

Figure 5

Pairing susceptibility, ${\chi }_p$, in the $p$-wave channel for the one-band repulsive Hubbard model at $n=0.3$ electrons per site, $U=4t$ and a nearest-neighbor-hopping term $t^{\prime }=-0.5t$. Although ferromagnetic correlations are strong, no transition to a $p$-wave superconducting state is observed as the temperature is lowered, in concurrence with previous results based on an instability analysis.