Implicit renormalization approach to the problem of Cooper instability

In the vast majority of cases, superconducting transition takes place at exponentially low temperature $T_c$ out of the Fermi liquid regime. We discuss the problem of determining $T_c$ from known system properties at temperatures $T\gg T_c$, and stress that this cannot be done reliably by following the standard protocol of solving for the largest eigenvalue of the original gap-function equation. However, within the implicit renormalization approach, the gap-function equation can be used to formulate an alternative eigenvalue problem, where solving leads to an accurate prediction for both $T_c$ and the gap function immediately below $T_c$. With diagrammatic Monte Carlo techniques, this eigenvalue problem can be solved without invoking the matrix inversion or even explicitly calculating the four-point vertex function.

Figure 1

(a), (b) Graphic representation of the equation for the gap function. The two eigenvalue equations are identical to each other up to the substitution $F=GG\mathrm{\Delta }$. (c) Equation for the effective low-frequency vertex function ${\mathrm{\Gamma }}_{\mathrm{eff}}$ for model (1). All incoming and outgoing frequencies satisfy $({\omega }_n,{\omega }_m)<\mathrm{\Omega }$, while all intermediate lines are restricted to high frequencies ${\omega }_k>\mathrm{\Omega }$.

Figure 2

Graphic representation of the pairing interaction $V({\omega }_m,{\omega }_n)=V_{\mathrm{RS}}({\omega }_m,{\omega }_n)$ in the Rietschel-Sham model (left panel) and the model with $V_{\mathrm{eff}}({\omega }_m-{\omega }_n)$ (right panel).

Figure 3

Analytical (blue circles), Eq. (22), vs numerical (black line) solutions of Eq. (13) for $T=T_c$.

Figure 4

The largest eigenvalue flow with logarithm of temperature for model (1) when condition (10) is violated, Eq. (9), (black line). Flow of the modified eigenvalue problem is shown by the red line.

Figure 5

Flow of ${\lambda }_{\epsilon }=\epsilon \lambda$ (black line) and ${\overline{\lambda }}_{\epsilon }$ (red line) for $\epsilon =0.85$, when Eq. (10) is satisfied.

Figure 6

Analytical solution (blue curve), Eq. (30), of the eigenvalue problem Eq. (13) and the corresponding numerical solution (black circles) for the same model parameters as in Fig. 3. Filled squares and open circles show the numerical solutions of the same model within the implicit renormalization approach with the energy scale separation set, respectively, at ${\mathrm{\Omega }}_c=0.1\mathrm{\Omega }$ and ${\mathrm{\Omega }}_c=0.01\mathrm{\Omega }$.

Figure 7

Prediction of $T_c$ (for the same model parameters as in Fig. 3) by linear in $ln1/T$ extrapolation of $\overline{\lambda }$ data from the temperature interval ${10}^{-5}<T<{10}^{-4}$ using various high-frequency cutoffs ${\mathrm{\Omega }}_c$. The flow saturates to the correct value only for ${\mathrm{\Omega }}_c\lesssim \mathrm{\Omega }$.