Exactly solvable model of strongly correlated $d$-wave superconductivity

We present an infinite-dimensional lattice of two-by-two plaquettes, the quadruple Bethe lattice, with Hubbard interaction and solve it exactly by means of the cluster dynamical mean-field theory. It exhibits a $d$-wave superconducting phase that is related to a highly degenerate point in the phase diagram of the isolated plaquette at that the ground states of the particle number sectors $N=2,3,\text{and}4$ cross. The superconducting gap is formed by the renormalized lower Slater peak of the correlated, hole-doped Mott insulator. We engineer parts of the interaction and find that pair hoppings between $X/Y$ momenta are the main two-particle correlations of the superconducting phase. The suppression of the superconductivity is caused by the diminishing of pair hopping correlations in the overdoped regime and by the charge blocking in the underdoped one. The optimal doping is $\sim 0.15$ at which the underlying normal state shows a Lifshitz transition. Tuning intra- and interplaquette hoppings allows to disentangle superconductivity from antiferromagnetism as the latter requires larger interplaquette hoppings. We expect our results to provide additional insight into the nature of the superconductivity in cuprates.

Figure 1

Quadruple Bethe lattice, four Bethe lattices (dashed lines) interconnected via plaquettes (solid lines). The coordination number for each Bethe lattice of this figure is set to $z=3$, and four sites of each Bethe lattice are depicted. An entire Bethe lattice exhibits an infinite number of sites with self-similar structure. Next-nearest-neighbor hoppings of the plaquette are omitted for convenience.

Figure 2

Illustration of the plaquette orbitals/momenta $\mathrm{\Gamma },X,\text{and}M$ ($Y$ omitted), i.e., the basis that diagonalizes the hopping in plaquette-site space ($0,1,2,\text{and}3$). Colors denote symmetries of the orbitals.

Figure 3

Retarded pairing susceptibility ${\chi }^{\mathrm{pair}}\left(\omega \right)$ of pairs with plaquette momentum $X$ in the isolated plaquette dependent on the screened Coulomb repulsion $U$ and chemical potential $\mu$ for $\beta =30$. The ground-state sectors $N=2,3,4$ (solid lines) cross at a plaquette degenerate point (square) with $U_c=2.78$ and ${\mu }_c=0.24$. The maximum of ${\chi }^{\mathrm{pair}}\left(\omega \right)$ (black circle) lies at the $N=2,4$ crossover that becomes a nonground state crossover at $U_c<U$ (dotted line).

Figure 4

Plaquette eigenenergies $E$ as functions of the chemical potential $\mu$ around $\mu =0.24,U=2.78$ at that $N=2,3,\text{and}4$ cross. All energies are plotted relative to the ground state energy of the normal state $E_0^n$. The kets label the normal states particle number $N$, spin number $S$ and plaquette momentum $K$. The superconducting (SC) fields are ${\mathrm{\Delta }}_x=0.1$ for the different orders $x$: $d$-wave $d\mathrm{SC}$, $s$-wave $s\mathrm{SC}$, and spin-triplet $\pi \mathrm{SC}$.

Figure 5

Plaquette momentum $K$-resolved spectral function $A^K\left(\omega \right)$ of the isolated plaquette for different chemical potentials $\mu$. The peaks are identified with single-particle transitions of the plaquette eigenstates. $U=2.78$, $t^{\prime }=0.3$, $\beta =30,$ and Lorentzian broadening $\epsilon =\pi /\beta$.

Figure 6

Semicircular densities of states of the noninteracting ($U=0$) quadruple Bethe lattice for the different orbitals/momenta and Bethe lattice hoppings $t_b$ ($t^{\prime }=0.3$). The marks on the $x$ axis correspond to the levels of the isolated noninteracting plaquette. Those levels are broadened by the Bethe hopping to semicirculars with the width $W=4t_b$, which is the same for scalar $t_b$.

Figure 7

Filling $\langle n\rangle$ dependence on the chemical potential $\mu$ and the Bethe hopping $t_b$ for the noninteracting ($U=0$) quadruple Bethe lattice ($t^{\prime }=0.3$). The positions of fillings 0.5, 0.75, 1.0 (half-filling), and 1.25 are shown by color lines, the color correspondence being shown on the legend. Two white lines corresponding to $〈n〉=0.5$ in the low-$t_b$ range correspond to the gap between the $M$ and the $X/Y$ bands. The point where the two lines meet corresponds to the overlap of the two bands. At $t_b=0.7,$ the upper edge of the $M$ band reaches the center of the $X/Y$ band, and the half-filling $\mu$ moves away from its low $t_b$ value of $-t^{\prime }=-0.3$.

Figure 8

$\mu \text{−}U$ phase diagrams of the quadruple Bethe lattice for several next-nearest-neighbor hoppings $t^{\prime }$ and Bethe hoppings $t_b$. Considered spontaneously broken symmetries are the $d$-wave superconductivity (dSC), antiferromagnetism (AFM), and plaquette antiferromagnetism (PAFM). The black lines denote the ground state crossovers of the regions $N=2,3,\text{and}4$ (bottom to top) of the isolated plaquette. The dotted black line marks the chemical potential ${\mu }_{24}\left(U\right)$ at which the $N=2,4$ sector-ground states of the isolated plaquette cross. The maximum dSC order value per diagram is marked by “$+$”. The dashed and solid colored lines correspond to ${\mu }_{\mathrm{opt}}(U,t_b)$ and $U_{\mathrm{opt}}\left(t_b\right)$ fits corresponding to ${\mathrm{\Psi }}_{d\mathrm{SC}}^{\max }$, respectively.

Figure 9

Chemical potential ${\mu }_{\mathrm{opt}}$, that maximizes the $d$-wave superconducting order parameter ${\mathrm{\Psi }}_{d\mathrm{SC}}$ as a function of the Bethe hopping $t_b$, $U=U_c$. Linear fits (dashed) are performed for the next-nearest-neighbor hoppings $t^{\prime }=0$ (top) and 0.3 (bottom) separately and are extrapolated to $t_b=0$. The plaquette degenerate point (PDP) is also shown. ${\mathrm{\Psi }}_{d\mathrm{SC}}$ is depicted as a function of the hole doping $\delta$ (insets) for different $t_b$ (color coded).

Figure 10

(Top) Linear fit (dashed) of the optimal Hubbard interaction $U_{\mathrm{opt}}$ as a function of the Bethe hopping $t_b$ at optimal doping (${\mu }_{\mathrm{opt}}$). The fit is performed for different next-nearest-neighbor hoppings $t^{\prime }$ separately. (Bottom) $d$-wave superconducting order parameter along the optimal $\mu \text{−}U$ line as a function of $t_b$.

Figure 11

Chemical potential $\mu$, Hubbard interaction $U$-phase diagram of the isolated plaquette (black, solid) with the ground states $|N,S,K\rangle$ ($t^{\prime }=0.3$). $W_{\text{plaquette}}=4t$ is the energy range of the plaquette hopping. The crossover of $|2,0,\mathrm{\Gamma }\rangle$ and $|4,0,\mathrm{\Gamma }\rangle$ is also shown for $U>U_{\text{PDP}}$, where it is not a ground state crossover (black, dash dotted). The plaquette degenerate point (PDP) and the maximum of the retarded pairing susceptibility ${\chi }_{\max }^{\text{pair}}(\omega =0)$ of the plaquette are marked. Linear fits of ${\mu }_{\mathrm{opt}}(U,t_b)$ (red, dashed) and $U_{\mathrm{opt}}\left(t_b\right)$ (red, solid) of the quadruple Bethe lattice are shown. The maximum ${\mathrm{\Psi }}_{d\mathrm{SC}}^{\max }$ corresponding to the parameter set $({\mu }_{\mathrm{opt}},U_{\mathrm{opt}},t_b^{\mathrm{opt}})$ is marked by $+$.

Figure 12

Spectral function $A\left(\omega \right)$ for different chemical potentials $\mu$ at approximately half-filling $\delta \approx 0$. For $\mu =0.5,$ we label the one-particle excitations Hubbard (H) and Slater (S) peaks ($t^{\prime }=0.3$, $t_b=0.2$, $U=2.78$). The analytic continuation is obtained by the stochastic optimization method [58, 59, 60].

Figure 13

Quasiparticle residue $Z$ (top) and renormalized quasiparticle energy $\tilde{\epsilon }$ and bandwidth $\tilde{W}$ (shown by shading) of the normal state as functions of the chemical potential $\mu$ (bottom), that at a certain value (dashed vertical line) hole dopes $\delta$ the system. The $K$ differentiation of $Z$ is absent in the Fermi liquid (FL) ($t^{\prime }=0.3$, $t_b=0.2$, and $U=2.78$).

Figure 14

Momentum resolved spectral function $A^K\left(\omega \right)$ for different dopings $\delta$ (corresponding to chemical potentials $\mu$) showing a four-peak structure at half-filling ($\delta =0$) of Hubbard (H) and Slater (S) peaks and for hole-dopings $\delta$ the $d$-wave superconducting gap ($t^{\prime }=0.3$, $t_b=0.2$, and $U=2.78$). The analytic continuation is obtained by the stochastic optimization method [58, 59, 60].

Figure 15

$d$-wave superconducting order parameter ${\mathrm{\Psi }}_{d\mathrm{SC}}$ of the symmetry-broken state (a), renormalized quasiparticle band(width) ${\tilde{\epsilon }}_X$ of the normal state (b), static two-particle observable $\langle \cdots \rangle$ of the normal state (c), and reduced density matrix of the normal state $\rho$ with plaquette many-body state indices $\gamma$ (d) as functions of hole-doping $\delta$. Points of certain features in the doping dependence are marked by circles. $t^{\prime }=0.3$, $t_b=0.2$, and $U=2.78$.

Figure 16

Local density of states for different hole-dopings $\delta$ in the dSC state ($t^{\prime }=0.3$, $t_b=0.2$, and $U=2.78$). The color code in the zoom-in (left) is the same as in the overview (right). The analytic continuation is obtained by the stochastic optimization method [58, 59, 60].

Figure 17

Order parameters $\mathrm{\Psi }$ of antiferromagnetism (AFM), $d$-wave superconductivity (dSC) and spin-triplet superconductivity ($\pi \mathrm{SC}$) dependent on hole doping $\delta$ for Bethe hopping $t_b=0.5$ (left) and dependent on $t_b$ for half-filling $\delta =0$ (right) ($U=2.78$, $t^{\prime }=0.3$).

Figure 18

$d$-wave superconducting order parameter ${\mathrm{\Psi }}_{d\mathrm{SC}}$ as a function of the change in different entries of the Hubbard-interaction $\mathrm{\Delta }{U}_{ijkl}$ normalized by its initial value $U_{ijkl}$ ($t^{\prime }=0.3$, $t_b=0.2$, $\delta =0.15$, and $U=2.78$). Further shown are slopes of linear fits (right) and an illustration (bottom, right) of the two-particle fluctuations, i.e., pair hopping and spin flip.

Figure 19

Superconducting order parameter ${\mathrm{\Psi }}_{d\mathrm{SC}}$ (top) and renormalized quasiparticle energy ${\tilde{\epsilon }}_X$ (bottom) as functions of the extended Bethe lattice hopping at $\delta =0.15$ ($U=2.78$, $t^{\prime }=0.3$, and $t_b=0.2$). The colored area marks the renormalized quasiparticle bandwidth $\tilde{W}$. Noninteracting semicircular density of states for different next-nearest-neighbor Bethe lattice hoppings (inset).