Why the critical temperature of high-$T_c$ cuprate superconductors is so low: The importance of the dynamical vertex structure

To fathom the mechanism of high-temperature ($T_c$) superconductivity, the dynamical vertex approximation is evoked for the two-dimensional repulsive Hubbard model. After showing that our results reproduce well the cuprate phase diagram with a reasonable $T_c$ and dome structure, we keep track of the scattering processes that primarily affect $T_c$. We find that local particle-particle diagrams significantly screen the bare interaction at low frequencies, which in turn suppresses antiferromagnetic spin fluctuations and hence the pairing interaction. Thus we identify dynamical vertex corrections as one of the main oppressors of $T_c$, which may provide a hint toward higher $T_c$'s.

Figure 1

(a) Antiferromagnetic spin fluctuations captured for weak interaction $U$ (wiggled line) in terms of particle-hole ladder diagrams (solid line: Green's function). (b) $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ diagrams describe similar spin fluctuations but now for strong correlation, with ladders in terms of the local $\mathrm{\Gamma }$, which is nonperturbative and frequency-dependent [33] instead of $U$. (c) The spin fluctuations can act, in turn, as a pairing glue for superconductivity in the particle-particle channel (an exemplary diagram is shown).

Figure 2

$d$-wave eigenvalue $\lambda$ against the band filling $n$ for $U=6t,T/t=0.010,0.013,0.020$ with (a) $t^{\prime }=t^{\prime \prime }=0$ and (b) $t^{\prime }/t=-0.20,t^{\prime \prime }/t=0.16$. (c,d) Momentum dependence of the Green's function $\left|G\right(\pi /\beta ,\mathit{k}\left)\right|$ (c) and the pairing interaction vertex ${\mathrm{\Gamma }}_{\mathrm{pp},\mathit{Q}}$ (d) for $n=0.775$ (overdoped), 0.825 (optimally doped), and 0.85 (underdoped), at $T/t=0.02$ with other parameters as in (a). In (d) we specifically display the $\mathit{Q}$ dependence of the pairing vertex $[{\mathrm{\Gamma }}_{\mathrm{pp},\mathit{Q}}(\pi /\beta ,\pi /\beta )+{\mathrm{\Gamma }}_{\mathrm{pp},\mathit{Q}}(-\pi /\beta ,\pi /\beta )]/2$, which is symmetrized for $d$-wave (singlet, even-frequency) pairing.

Figure 3

(a) Eigenvalue $\lambda$ against the frequency range $n_{\mathrm{vertex}}$ [exemplified in (b) as a dashed line], over which the local vertex correction $\delta {\mathrm{\Gamma }}_{\mathrm{m}}$ is considered, for $U=6t,t^{\prime }=t^{\prime \prime }=0,n=0.825$, and $T/t=0.040,0.067$. (b,c,d) Dynamical vertex structure of (b) the local vertex correction $\delta {\mathrm{\Gamma }}_{\mathrm{m}}({\nu }_n,{\nu }_{n^{\prime }},\omega =0)$ in the magnetic channel relative to $-U$, (c) the nonlocal vertex in the magnetic channel, $F_{\mathrm{m},\mathit{Q}=(\pi ,\pi )}({\nu }_n,{\nu }_{n^{\prime }},\omega =0)$, and (d) the pairing interaction ${\mathrm{\Gamma }}_{\mathrm{pp},\mathit{Q}=(\pi ,\pi )}({\nu }_n,{\nu }_{n^{\prime }},\omega =0)$, for the same $U,t^{\prime },t^{\prime \prime },n$ with $T/t=0.067$ here. The inset in (d) shows a typical structure of ${\mathrm{\Gamma }}_{\mathrm{pp}}$ in mean-field-like approaches.

Figure 4

Frequency structure of the local vertex correction, $\delta {\mathrm{\Gamma }}_{\mathrm{m}}({\nu }_n,{\nu }_{n^{\prime }},\omega =0)\equiv {\mathrm{\Gamma }}_{\mathrm{m}}({\nu }_n,{\nu }_{n^{\prime }},\omega =0)-(-U)$, for (a) second-order or (b) third-order perturbation theory for $U=6t,t^{\prime }=t^{\prime \prime }=0,n=0.825$, and $T/t=0.067$. In each panel, the inset shows a typical diagram taken into account, and the corresponding eigenvalue $\lambda$ is indicated.

Figure 5

Typical diagrams that contribute to the magnetic vertex $F_{\mathrm{m}}$ in the Bethe-Salpeter ladder. Left: a typical diagram as in RPA with $U$ (red wavy line) as an irreducible building block, connected by ${\chi }_0$ (with two Green's functions having fermionic frequencies ${\nu }_1$ and ${\nu }_1+\omega$). Right: local (second-order) vertex correction $\delta {\mathrm{\Gamma }}_{\mathrm{m}}$ with a particle-particle bubble. Such terms are particle-hole irreducible, hence they need to be inserted in the Bethe-Salpeter ladder for spin fluctuations. They lead to the suppression of ${\mathrm{\Gamma }}_{\mathrm{m}}$ in Fig. 4, and through the whole ladder suppress $F_{\mathrm{m}}$ and ${\mathrm{\Gamma }}_{\mathrm{pp}}$ in Figs. 3 and 3.