Superfluid stiffness in cuprates: Effect of Mott transition and phase competition

Superfluid stiffness ${\rho }_s$ is a defining characteristic of the superconducting state, allowing phase coherence and supercurrent. It is accessible experimentally through the penetration depth. Coexistence of $d$-wave superconductivity with other phases in underdoped cuprates, such as antiferromagnetism or charge-density waves, may drastically alter ${\rho }_s$. To shed light on this physics, the zero-temperature value of ${\rho }_s={\rho }_{zz}$ along the $c$ axis was computed for different values of Hubbard interaction $U$ and different sets of tight-binding parameters describing the high-temperature superconductors YBCO and NCCO. We used cellular dynamical mean-field theory for the one-band Hubbard model with exact diagonalization as impurity solver and state-of-the-art bath parametrization. We conclude that Mott physics plays a dominant role in determining the superfluid stiffness on the hole-doped side of the phase diagram. On the electron-doped side, antiferromagnetism wins over superconductivity near half-filling. But, upon approaching optimal electron-doping, homogeneous coexistence between superconductivity and antiferromagnetism causes the superfluid stiffness to drop sharply. Hence, on the electron-doped side, it is competition between antiferromagnetism and $d$-wave superconductivity that plays a dominant role in determining the value of ${\rho }_{zz}$ near half-filling. At large overdoping, ${\rho }_{zz}$ behaves in a more BCS-type manner in both the electron- and hole-doped cases. We comment on some qualitative implications of these results for the superconducting transition temperature.

Figure 1

The original Brillouin zone (BZ) is enclosed by the yellow square. The AF Brillouin zone (AF-BZ) is enclosed by the green diamond figure and the supercluster reduced Brillouin (rBZ) zone by the black square. $\mathcal{G}(\tilde{\mathit{k}},i{\omega }_n)$ is defined on the rBZ and has to be periodized to map onto the AF-BZ for the full Green's function $\mathcal{G}(\mathit{k},i{\omega }_n)$ to have dimension $4\times 4$. In the case where there is only superconductivity, the wave vectors ${\mathit{K}}_i$, with $i\in \{1,2,3,4\}$, are the reciprocal-superlattice wave vectors: ${\mathit{K}}_1=(0,0)$, ${\mathit{K}}_2=(\pi ,0)$, ${\mathit{K}}_3=(0,\pi )$, and ${\mathit{K}}_4=(\pi ,\pi )$.

Figure 2

(a) ${\rho }_{zz}$ as a function of band filling $n$ in both pure $d\mathrm{SC}$ and coexisting $\mathrm{AF}+d\mathrm{SC}$ states for $U=12t$, $t^{\prime }=-0.3t$, and $t^{\prime }=0.2t$. The green squares show ${\rho }_{zz}^{\text{AF}+\text{SC}}$ in coexistence regime, and the black squares show ${\rho }_{zz}^{\text{SC}}$ computed in the pure regime (with coexistence forbidden). The calculated values of ${\lambda }_c$ in physical units are of the same order of magnitude as experimental measurements of the $c$-axis superfluid penetration depth in hole-doped compounds (Ref. [38]). The bottom subfigure illustrates the $d\mathrm{SC}$ order parameter in the pure regime $\langle D\rangle \left(\text{pure}\right)$ (orange), and the $d\mathrm{SC}\langle D\rangle$ (blue) and AF $\langle M\rangle$ (red) order parameters as a function of $n$ in the coexistence regime. (b) ${\rho }_{zz}$ as a function of the electron density $n$ in both pure $d\mathrm{SC}$ and microscopic $\mathrm{AF}+d\mathrm{SC}$ states for $U=8t$, $t^{\prime }=-0.3t$, and $t^{\prime }=0.2t$. The symbols are the same as in (a).

Figure 3

(a) ${\rho }_{zz}$ as a function of band filling $n$ in both pure $d\mathrm{SC}$ and $\mathrm{AF}+d\mathrm{SC}$ states for $U=6.55t$, $t^{\prime }=-0.17t$, and $t^{\prime }=0.03t$. The symbols are defined in Fig. 2. (b) ${\rho }_{zz}$ as a function of band filling $n$ in both pure $d\mathrm{SC}$ and $\mathrm{AF}+d\mathrm{SC}$ states for $U=5t$, $t^{\prime }=-0.17t$, and $t^{\prime }=0.03t$. The symbols are defined in Fig. 2.

Figure 4

(a) Effect of the ${\mathit{k}}_{\parallel }$ dependence of $t_{\perp }$ on ${\rho }_{zz}^{\text{AF}+\text{SC}}$ in the coexistence regime for $U=12t$, $t^{\prime }=-0.3t$, and $t^{\prime }=0.2t$. Only the electron-doped side is shown. The green and black squares are for ${\rho }_{zz}$ as a function of $n$ with the ${\mathit{k}}_{\parallel }$ dependence of $t_{\perp }$ included: these results are taken from Fig. 2. The magenta (gray) diamonds on the other hand show ${\rho }_{zz}^{\text{AF}+\text{SC}}$ with (without) coexistence as a function of $n$ replacing the ${\mathit{k}}_{\parallel }$ modulation of $t_{\perp }$ in Eq. (23) by its reduced-Brillouin-zone average. Thus, the gray diamonds show the effect of a missing $t_{\perp }$ when $\langle M\rangle$ vanishes and $\langle D\rangle$ dominates in the case where both order parameters are allowed in the calculations. The bottom subfigure illustrates the AF and $d\mathrm{SC}$ order parameter amplitudes $\langle M\rangle$ and $\langle D\rangle$, respectively, as a function of $n$. (b) ${\rho }_{zz}$ for $U=6.55t$, $t^{\prime }=-0.17t$, and $t^{\prime }=0.03t$. The symbols have the same meaning as in (a). The green and black squares are taken from Fig. 3.

Figure 5

Schematic representation of the $2\times 2$ cluster cut out of the original lattice expressed in the orbital basis. The labels $A$ and $B$ on the sites account for the two sublattices resulting from AF order. We also illustrate the Fourier transforms of the nearest-neighbor ${\epsilon }_{\tilde{\mathit{k}}}$, second-nearest-neighbor ${\zeta }_{\tilde{\mathit{k}}}$, and third-nearest-neighbor hoppings ${\mathrm{\Omega }}_{\tilde{\mathit{k}}}$. These Fourier transforms take the same form as for the infinite, translationally invariant lattice. For clarity, there is no repetition of the various hoppings on the figure.

Figure 6

Example of stacked ${\mathrm{CuO}}_2$ planes along the $c$ axis ($z$ axis). The different AF sublattices $A$ and $B$ are shown in orange and yellow, respectively (cf. Fig. 5). The red arrow illustrates a nearest-neighbor hopping, the cyan arrow a second-neighbor hopping, and the blue arrows third-neighbor hoppings between two stacked ${\mathrm{CuO}}_2$ planes. To lighten the figure, only half of the overall second- and third-neighbor hoppings are shown and a wide range of hoppings are colored gray. All the hopping terms contained in $t_{\perp }$ [Eq. (19)] shift electrons from one AF sublattice to another when hopping from one plane to another.