Josephson lattice model for phase fluctuations of local pairs in copper oxide superconductors

We derive an expression for the effective Josephson coupling from the microscopic Hubbard model. It serves as a starting point for the description of phase fluctuations of local Cooper pairs in $d_{x^2-y^2}$-wave superconductors in the framework of an effective $XY$ model of plaquettes, the Josephson lattice. The expression for the effective interaction is derived by means of the local-force theorem, and it depends on local symmetry-broken correlation functions that we obtain using the cluster dynamical mean-field theory. Moreover, we apply the continuum limit to the Josephson lattice to obtain an expression for the gradient term in the Ginzburg-Landau theory and compare predicted London penetration depths and Kosterlitz-Thouless transition temperatures with experimental data for ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-x}$.

Figure 1

Illustration of the Hubbard-plaquette lattice $(t_{ij},U)$ with lattice vector $r$, self-energies ${\mathrm{\Sigma }}_i$ and plaquette sites 0...3. It is mapped to the Josephson lattice model with effective coupling $J_{ij}$ of plaquettes due to phase fluctuations $\delta {\theta }_i$ of the $d$-wave superconducting order parameter ${\mathrm{\Phi }}_i$.

Figure 2

Josephson coupling $J_r$ as a function of doping $\delta$ (left) and interlayer hopping $t_{\perp }$ (right) for different plaquette translations $r$ at $T=1/52$

Figure 3

Josephson coupling $J_r$ (left) and its constituents, $GSGS$ (center) and $FSFS$ (right), as functions of doping $\delta$ and for different plaquette translations $r$ at $T=1/52\sim 0.02,t_{\perp }=0.15$.

Figure 4

Superconducting stiffness $I_{\parallel }$ (top) and order parameter for local Cooper-pair formation ${\mathrm{\Phi }}_{\mathrm{dSC}}^{\mathrm{CDMFT}}$ (bottom) as functions of the temperature $T$ for various dopings $\delta \left(t_{\perp }=0\right)$.

Figure 5

In-plane superconducting stiffness $I_{\parallel }$ (top, left), in-plane penetration depth ${\lambda }_{ab}$ (top, right), perpendicular superconducting stiffness $I_{\perp }$ (center, left), perpendicular penetration depth (center, right), and CDMFT dSC order parameter ${\mathrm{\Phi }}_{\mathrm{dSC}}^{\mathrm{CDMFT}}$ (bottom) as functions of doping $\delta$ at $T=1/52$ for different interlayer hoppings $t_{\perp }$.

Figure 6

Phase diagram of the local dSC order parameter ${\mathrm{\Phi }}_{\mathrm{dSC}}^{\mathrm{CDMFT}}$ with critical temperature $T_c^{\mathrm{CDMFT}}$ depending on the temperature $T$ and doping $\delta \left(t_{\perp }=0\right)$. Circles denote CDMFT calculations. The transition temperature of the Josephson lattice model $T_{\text{KT}}$ has been calculated from the superconducting stiffness at $T=1/52$, at that $I_{\parallel }\left(T\right)$ is (not) saturated for the solid (dotted) part.

Figure 7

Electronic tight-binding band structures of the different hopping lattices 3D and $3{\text{D}}^{*}$ and next-nearest-neighbor hoppings $t^{\prime }$. $k$ is the reciprocal lattice vector in the reduced Brillouin zone. The four bands correspond to four sites within the two-by-two plaquette unit cell. With our choice of cluster the reduced Brillouin zone has the same shape as the original Brillouin zone of the square lattice. Thus, we label the high symmetry points accordingly but with a prime. ${\mathrm{\Gamma }}^{\prime }=(0,0,0),X^{\prime }=(1,0,0),M^{\prime }=(1,1,0),R^{\prime }=(1,1,1)$ in units of half of the reduced Brillouin zone.

Figure 8

The quasiparticle residue $Z$ (top) and energy $\tilde{\epsilon }$ of $k=X$ (bottom) as functions of the hole-doping $\delta$. The noninteraction quasiparticle energy $\left(U=0\right)$ and the anomalous part $S$ are also shown (bottom) $\left(T=1/52,t_{\perp }=0\right)$.

Figure 9

Convergence of the dSC stiffness $I$ with number of Matsubara frequencies ${\omega }_n\left(N_{{\omega }_n}^{\max }=N_k=128\right)$.

Figure 10

Convergence of the dSC stiffness $I$ with number of $k$-points per dimension $\left(N_k^{\max }=N_{{\omega }_n}=128\right)$.

Figure 11

In-plane superconducting stiffness $I_{\parallel }$ (top, left), in-plane penetration depth ${\lambda }_{ab}$ (top, right), perpendicular superconducting stiffness $I_{\perp }$ (center, left), perpendicular penetration depth (center, right), and CDMFT dSC order parameter ${\mathrm{\Phi }}_{\mathrm{dSC}}^{\mathrm{CDMFT}}$ (bottom) as functions of doping $\delta$ at $T=1/52\sim 0.02$. Quantities are shown for different interlayer hoppings $t_{\perp }$, next-nearest-neighbor hoppings $t^{\prime }$, and also tight-binding lattices $t\left(k\right)$.

Figure 12

Superconducting stiffness $I_{\parallel /\perp }$ and penetration depth ${\lambda }_{ab/c}$ as functions of the interlayer hopping $t_{\perp }$ with in-plane next-nearest-neighbor hopping $t^{\prime }=-0.3\left(\beta =52,\delta =0.05\right)$. Results are shown for $\text{3D}s$ and 3D lattice dispersions $t\left(k\right)$. $t_{\perp }$ changes only in the Josephson lattice model. The small numbers are values of $\lambda$.

Figure 13

Superconducting stiffness $I_{\parallel /\perp }$ and penetration depth ${\lambda }_{ab/c}$ as functions of the next-nearest-neighbor hopping $t^{\prime }$ with interlayer hopping $t_{\perp }=0.15\left(\beta =52,\delta =0.075\right)$. Results are shown for the $\text{3D}s$ lattice dispersion $t\left(k\right)$. $t^{\prime }$ changes only in the Josephson lattice model. The small numbers are values of $\lambda$.