Superconducting phase diagram in a model for tetragonal and cubic systems with strong antiferromagnetic correlations

We calculate the superconducting phase diagram as a function of temperature and $z$-axis anisotropy in a model for tetragonal and cubic systems having strong antiferromagnetic fluctuations. The formal basis for our calculations is the fluctuation exchange approximation applied to the single-band Hubbard model near half-filling. For nearly cubic lattices, two superconducting phase transitions are observed as a function of temperature with the low-temperature state having the time-reversal symmetry-breaking form $d_{x^2-y^2}\pm i{d}_{3z^2-r^2}$. With increasing tetragonal distortion, the time-reversal symmetry-breaking phase is suppressed, giving way to only $d_{x^2-y^2}$ or $d_{3z^2-r^2}$ single-component phases. Based on these results, we propose that a time-reversal symmetry-breaking superconducting state may be found in cubic systems with pairing driven by antiferromagnetic fluctuations.

Figure 1

Numerical systematics in the evaluation of pairing amplitude $m_p$ vs the scaled temperature $k_{B}T∕W_0$, where $W_0$ is the noninteracting electron bandwidth, for two-dimensional lattices with 0.85 electrons/site and $U∕W_0=0.5$. Top: Variation as a function of number of Matsubara frequencies $m$. Middle: Dependence on the number of lattice points $N_L$. Bottom graph: Variation as a function of DCA cluster size $N_c$.

Figure 2

Fluctuation exchange approximation results for the mean-field superconducting phase diagram of the Hubbard model as a function of the ratio of the interplanar to intraplanar hopping $t_z∕t_{xy}$ and reduced temperature $k_{B}T∕W_0$, for 0.85 electrons/site and $U∕W_0=0.5$. The $d_{x^2-y^2}$ and $d_{3z^2-r^2}$ pairing states are most stable for $t_z∕t_{xy}<1$ and $t_z∕t_{xy}>1$, respectively. For $t_z∕t_{xy}\sim 1$, a second transition occurs as a function of decreasing temperature into the time-reversal symmetry-breaking state $d_{x^2-y^2}+i{d}_{3z^2-r^2}$. The phase diagram was calculated using $N_c=4^3$ DCA clusters (see text for the definition of this numerical parameter). Our aim is to extract $N_c\to \infty$ results using as small of $N_c$ values as possible. Boundary points recalculated using $N_c=8^3$ clusters (closed circles) show good qualitative agreement and reasonable quantitative agreement. A recalculation of $T_c$ at $t_{xy}∕t_z=0$ and 1 using an $N_c={16}^3$ confirms the overall stability of these results