Dynamics of superconducting pairs in the two-dimensional Hubbard model

Abstract

The frequency structure of the superconducting correlations in cuprates gives insights on the pairing mechanism. Here we present an exhaustive study of this problem in the two-dimensional Hubbard model with cellular dynamical mean-field theory. To this end, we systematically quantify the dependence on doping and interaction strength of the superconducting gap, of the frequency scales where pairing occurs, and of their relative contribution to pairing. We find pair-forming alternating with pair-breaking processes as a function of frequency, providing new evidence that pairing can arise in principle from both low- and high-frequency processes. However, we find that the high-frequency pair-forming processes are outweighed by pair-breaking ones. Hence, the net contribution to pairing comes only from the low-frequency pair-forming processes, a result that underscores their key role in the pairing mechanism.

Figures (11)

• Figure 1: (a) Superconducting transition temperature $T_c^{\rm CDMFT}$ versus $\delta$ for different values of $U$ (data taken from Ref. CaitlinPNAS2021). (b) Density of states in the superconducting state $A_{\rm nor}(\omega)$. The half distance between the position of the coherence peaks (marked by open circles) gives the superconducting gap $\Delta_{\rm sc}$ and is analysed in Fig. \ref{['fig:SCgap']}. (c) Anomalous spectral function $A_{\rm an}(\omega)$. The frequency regions where $A_{\rm an}(\omega)$ is positive are those that contribute to pairing: they are shaded with blue vertical bands and are examined in Fig. \ref{['fig:freq-pairing']}. The areas where $A_{\rm an}(\omega)$ is positive are colored and analysed in Fig. \ref{['fig:areas']}. (d) Cumulative spectral weight of the order parameter $I_F(\omega)$. Horizontal line denotes the superconducting order parameter $|\Phi|$ computed independently. Data in panels (b), (c) and (d) are for $U=12$, $T=1/50$ and $\delta \approx 0.02$. Frequencies are converted into physical units with $t=350$ meV.
• Figure 2: (a): Superconducting gap $\Delta_{\rm sc}$ versus $\delta$ for different values of $U$, as determined by the half of the frequency difference of the coherence peaks in $A_{\rm nor}(\omega)$, see e.g. Fig. \ref{['fig:intro']}(b). Error bars indicate the uncertainty in determining the peaks. For each value of $U$, the filled symbol denotes the doping at which the superconducting order parameter $|\Phi(\delta)|$ has a maximum. (b) $\Delta_{\rm sc}$ at fixed doping $\delta \approx 0.02$ versus $U$. The dashed vertical line marks the critical threshold $U_{\rm MIT}\approx 5.95$CaitlinSb for opening the Mott gap at half-filling. Filled up triangles denote the position of the low-frequency peak in $A_{\rm an}(\omega)$, which tracks $\Delta_{\rm sc}$. Data are converted into physical units with $t=350$ meV.
• Figure 3: Vertical bars indicate the frequency regions where $A_{\rm an}(\omega)$ is positive, for several values of doping. Up (down) triangles denote the maxima (minima) in each frequency region. Each panel shows data for a given interaction strength $U$.
• Figure 4: The largest areas, divided by $2\pi$, under the positive regions of $A_{\rm an}(\omega)$: $\mathcal{C}^{+}_{\rm low}$ (low frequencies, yellow filled circles), $\mathcal{C}^{+}_{\rm int}$ (intermediate frequencies, blue filled diamonds), $\mathcal{C}^{+}_{\rm high}$ (high frequencies, red filled squares). They are calculated as follows. With the composite trapezoidal method, we find the four largest areas under the positive regions of $A_{\rm an}(\omega)$ and their support $[\omega_{\rm in}^{i}, \omega_{\rm fin}^{i}]$, with ${i=1, \ldots, 4}$. If $\omega_{\rm in}^{i}$ is in the $(0, 0.4]$ range, the corresponding area is $\mathcal{C}^{+}_{\rm low}$. Similarly, if $\omega_{\rm in}^{i} \in (0.4, 2.5]$, the area is denoted by $\mathcal{C}^{+}_{\rm int}$, and if $\omega_{\rm in}^{i} \in (2.5, 8.0]$, the area is denoted by $\mathcal{C}^{+}_{\rm high}$. Yellow crosses indicate the value of the low-frequency peak of the cumulative spectral weight of the order parameter, $I_F(\omega_p)$. They follow $\mathcal{C}^{+}_{\rm low}$. Open squares denote the superconducting order parameter $|\Phi|$. Panels (a)-(e) show the areas versus doping $\delta$, for different values of $U$. Panel (f) shows the areas versus $U$ for fixed doping $\delta \approx 0.02$. The dashed vertical line marks $U_{\rm MIT}$.
• Figure 5: Low-frequency zoom of Fig. \ref{['fig:freq-pairing']}. For each value of $\delta$ and $U$, the frequency position of the low-frequency maximum of $A_{\rm an}(\omega)$ (filled up triangles) tracks the size of the superconducting gap $\Delta_{\rm sc}$ (open circles), suggesting that $\Delta_{\rm sc}$ is the energy scale where pairing is maximum reymbautPRB2016.