Superconductivity in the two-dimensional Hubbard model with cellular dynamical mean-field theory: A quantum impurity model analysis

Doping a Mott insulator gives rise to unconventional superconducting correlations. Here we address the interplay between $d$-wave superconductivity and Mott physics using the two-dimensional Hubbard model with cellular dynamical mean-field theory on a $2\times 2$ plaquette. Our approach is to study superconducting correlations from the perspective of a cluster quantum impurity model embedded in a self-consistent bath. At the level of the cluster, we calculate the probabilities of the possible cluster electrons configurations. Upon condensation, we find an increased probability that cluster electrons occupy a four-electron singlet configuration, enabling us to identify this type of short-range spin correlation as key to superconducting pairing. The increased probability of this four-electron singlet comes at the expense of a reduced probability of a four-electron triplet with no significant probability redistribution of fluctuations of charges. This allows us to establish that superconductivity at the level of the cluster primarily involves a reorganization of short-range spin correlations rather than charge correlations. We gain information about the bath by studying the spectral weight of the hybridization function. Upon condensation, we find a transfer of spectral weight leading to the opening of a superconducting gap. We use these insights to interpret the signatures of superconducting correlations in the density of states of the system and in the zero-frequency spin susceptibility.

Figure 1

Real-space sketch of the cluster DMFT framework. (a) Sketch of the 2D Hubbard model on a square lattice; up and down spins are represented by spheres with arrows. (b) Cluster DMFT maps the lattice model onto a cluster impurity ($2\times 2$ shaded plaquette) embedded in a self-consistent bath. As a function of time, the cluster exchanges electrons with the bath. Our work shows how superconducting correlations emerge both at the level of the cluster and of the bath. At the level of the cluster, we can extract the relative time the cluster spends in a given electronic configuration; at the level of the bath, we can extract the bath hybridization function, which encodes the dynamics of the processes between cluster and bath.

Figure 2

(a)–(d) Temperature vs doping phase diagram of the plaquette CDMFT solution of 2D Hubbard model for several values of the interaction strength $U$. Only normal and superconducting phases are shown. Data are taken from Ref. [42]. Filled hexagons indicate the data set analyzed in Figs. 3, 4, 5, 6, 7, 8, 9, 10. Open red squares denote the CDMFT superconducting transition temperature $T_c^d$, below which the superconducting order parameter $\mathrm{\Phi }$ is nonzero. Open blue circles indicate the crossover temperature $T^{*}$ corresponding to the opening of the pseudogap, as determined by the drop in the spin susceptibility vs $T$ at fixed doping. Red and blue shaded areas are a guide to the eye for the superconducting and pseudogap phases. In (b), the superconducting dome hides a pseudogap to metal transition in the underlying normal state. This transition is first order and bounded by spinodal lines (filled blue triangles), and it terminates in a critical end point (filled cyan circle), from which a supercritical crossover emerges (Widom line [60, 61, 62]), as determined here by the loci of the maximum of the charge compressibility vs $\delta$. The shaded gray area corresponds to the low-temperature region that is not accessible to our calculations because of the sign problem. (e)–(h) Superconducting order parameter $\left|\mathrm{\Phi }\right|$ vs $\delta$ for several values of the interaction strength $U$ and at temperature $T=1/50$. Data are taken from Ref. [42].

Figure 3

Histograms of the probability of the plaquette sectors $\left\{m\right\}$. Data are shown at $T=1/50$ for several values of interaction strength $U$ and doping $\delta$, corresponding to the filled hexagons in the $T\text{−}\delta$ phase diagrams of Fig. 2. The index on the $x$ axis refers to the label of the plaquette sectors in the last column of Table 1. Vertical dotted lines bound the sectors with same quantum number $N$. Empty bars correspond to the normal state CDMFT solution, whereas filled bars correspond to the superconducting state solution. The sectors with the highest probabilities for a given $N$ and $S_z$ are shown in color, and they correspond to the sets plotted in Fig. 4. The “singlets” with two and four electrons with $S_z=0$ in the cluster momentum $\mathbf{K}=(0,0)$ (indexes 16 and 36 in Table 1) are shown in yellow and red, respectively. The “triplet” with four electrons with $S_z=0,\pm 1$ in the cluster momentum $\mathbf{K}=(\pi ,\pi )$ (indices 39, 46, and 47 in Table 1) is shown in blue. The “doublet” with three electrons with $S_z=\pm 1/2$ in the cluster momentum $\mathbf{K}=(\pi ,0)$ and its degenerate counterpart $\mathbf{K}=(0,\pi )$ (indexes 30–33 in Table 1) are shown in green.

Figure 4

Histogram showing superconducting and normal state probabilities for selected key plaquette sets of sectors [see Eq. (23)]. They are the four-electron singlet ${\mathcal{S}}_4$ (red), the four-electron triplet ${\mathcal{T}}_4$ (blue), the three-electron doublet ${\mathcal{D}}_3$ (green), the two-electron singlet ${\mathcal{S}}_2$ (yellow), and the remnant eigenstates $\mathcal{R}$ (gray). Data are shown for the same values of $U$ and $\delta$ as Fig. 3 corresponding to the filled hexagons of Fig. 2, at $T=1/50$. Filled and unfilled bars indicate the superconducting and normal phases, respectively. Error bars indicate the root-mean-square error over the last 20 CDMFT iterations.

Figure 5

Probability of selected superconducting and normal state plaquette sets of sectors as a function of doping for several values of the interaction strength $U$, at $T=1/50$. Selected plaquette sets are those shown in Fig. 4, using the same color code and listed in Eq. (23): ${\mathcal{S}}_4, {\mathcal{S}}_2, {\mathcal{T}}_4$, and ${\mathcal{D}}_3$. Filled symbols correspond to the superconducting state solution. Empty symbols correspond to the normal state solution. Insets contain the difference between the superconducting and normal state probabilities of the singlet ${\mathcal{S}}_4$ (red) and triplet ${\mathcal{T}}_4$ (blue), respectively. The shaded area around each curve shows the root-mean-square error. For each $U$ above $U_{\mathrm{MIT}}$, a gray filled circle indicates the estimate of the doping level of the pseudogap to metal critical end point in the normal state (see Fig. 2).

Figure 6

Imaginary part of the hybridization function $-\text{Im}{\mathrm{\Delta }}_{\mathbf{K}}\left(\omega \right)$ for several values of $U$ and $\delta$, corresponding to the filled hexagons in Fig. 2. The dashed line shows the normal state solution, whereas the filled line with a shaded region denotes the normal component of the superconducting state solution. Each panel shows the spectral function corresponding to cluster momenta $\mathbf{K}=(\pi ,0),(0,0),(\pi ,\pi )$ (shifted by 0.5 each for a better visualization). Analytical continuation has been done using the method of Ref. [72]. Each spectral function is normalized to 1.

Figure 7

Local density of states of the bath $N_{\text{bath}}\left(\omega \right)=-\text{Im}{\mathrm{\Delta }}_{\mathbf{R}=(0,0)}\left(\omega \right)$ for the same values of $U$ and $\delta$ as in Fig. 6, which correspond to the filled hexagons in Fig. 2. The dashed line shows the normal state solution, whereas the filled line with a shaded region denotes the superconducting state solution. Insets in each panel show $N_{\mathrm{bath}}\left(\omega \right)$ in the superconducting state on the full spectrum of the frequency. As in Fig. 6, the normalization of each density of states is equal to 1.

Figure 8

Imaginary part of the Green's function $-\text{Im}{\overline{G}}_{\mathbf{K}}\left(\omega \right)$ for different values of $U$ and $\delta$ corresponding to the filled hexagons in Fig. 2. The dashed line shows the normal state solution, whereas the filled line with a shaded region denotes the normal component of the superconducting state solution. Each panel shows the spectral function corresponding to cluster momenta $\mathbf{K}=(\pi ,0),(0,0),(\pi ,\pi )$ (shifted by 0.5 each for a better visualization). Each spectral function is normalized to 1.

Figure 9

Low-frequency part of the local density of states $N\left(\omega \right)$ of the system for the values of $U, \delta$, and $T$ indicated by filled hexagons in Fig. 2. The solid line with a shaded region indicates the superconducting state, while the dashed line indicates the normal state. Insets show the density of states of the system $N\left(\omega \right)$ in the superconducting state on the full spectrum of the frequency. Each density of states is normalized to 1.

Figure 10

(a)–(d) Zero-frequency spin susceptibility ${\chi }_0$ vs temperature $T$. Data are shown for several values of $U$ and $\delta$, corresponding to the filled hexagons in the $T\text{−}\delta$ phase diagram of Fig. 2. Filled squares correspond to the superconducting state solution. Empty circles correspond to the normal state solution. (e)–(h) Temperature dependence of the superconducting and normal state probabilities of the four-electron singlet ${\mathcal{S}}_4$. Data are shown for the same values of top panels.