Two-particle self-consistent approach to unconventional superconductivity

A nonperturbative approach to unconventional superconductivity is developed based on the idea of two-particle self-consistent (TPSC) theory by Vilk and Tremblay. A sum rule, which the momentum-dependent pairing susceptibility satisfies, is derived. An effective pairing interaction between quasiparticles is determined so that the susceptibility should fulfill this exact sum rule, in which fluctuations belonging to different symmetries couple at finite momentum. It is demonstrated that the mode coupling between $d$-wave and $s$-wave pairing fluctuations leads to suppression of the $d$-wave fluctuation near the Mott insulator.

Figure 1

Diagrammatic representations of $Q_{\alpha \alpha }$ for $\alpha =$ $A_{1g}$, $B_{1g}$, and $B_{2g}$.

Figure 2

$Q_{\alpha \alpha }$ evaluated in TPSC [Eq. (24)] as a function of $n$ for $t^{\prime }=0$, $U=8$, and $L=128$. The dots are results computed by ED with $L=4$.

Figure 3

The effective pairing interaction $V_{\alpha }$ at $T=0.1$ for $t^{\prime }=0$, $U=8$, and $L=128$. The arrow indicates a minimum of $V_{B_{1g}}$ in approximation (ii).

Figure 4

The inverse of the static susceptibility $P_{\alpha \alpha }^{\mathrm{eff}}(\mathbf{0},0)$ with $\alpha =B_{1g}$ as a function $n$ for $t^{\prime }=0$, $U=8$, and $L=128$.

Figure 5

A phase diagram of the “quasi-two-dimensional” system calculated in the present theory ($B_{1g}$ superconductivity) and in TPSC (magnetism) for $U=8$. AF-C indicates the antiferromagnetism with $\mathit{q}=(\pi ,\pi )$ and AF-IC indicates the ordering with $\mathit{q}\ne (\pi ,\pi )$ The horizontal dotted line shows the lowest temperature $T=0.01$ in the calculation. The system size is $L=128$ (closed symbol) and $L=64$ (open symbol). (a) $t^{\prime }=0$ and (b) $t^{\prime }=-0.1$.