Mean-Field Pairing Theory for the Charge-Stripe Phase of High-Temperature Cuprate Superconductors

Striped high-$T_c$ superconductors such as ${\mathrm{La}}_{2-y-x}{\mathrm{Nd}}_y{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ and ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$ near $x=1/8$ show a fascinating competition between spin and charge order and superconductivity. A theory for these systems therefore has to capture both the spin correlations of an antiferromagnet and the pair correlations of a superconductor. For this purpose we present here an effective Hartree-Fock theory incorporating both electron pairing with finite center-of-mass momentum and antiferromagnetism. We show that this theory reproduces the key experimental features such as the formation of the antiferromagnetic stripe patterns at $7/8$ band filling or the quasi-one-dimensional electronic structure observed by photoemission spectroscopy.

Figure 1

Real-space characterization of the striped SC state. (a) The magnetization $m_i=n_{i\uparrow }-n_{i\downarrow }$ exhibits AFM stripes separated by nonmagnetic antiphase domain walls. (b) The charge density $n_i=n_{i\uparrow }+n_{i\downarrow }$ reaches nearly one electron per site inside the AFM stripes which are separated by single lines near quarter filling, resulting in an overall mean charge density $7/8$, i.e., hole doping $1/8$. (c) The SC bond order parameter on the horizontal (purple) and on the vertical bonds (blue). It is largest on the vertical bonds along the line of nonmagnetic sites. Results were obtained for $V=2t$ on a $16\times 12$ lattice.

Figure 2

Momentum-space characterization of the striped superconductor. (a) Momentum distribution $n(\mathbf{k})$. (b) Integrated spectral weight [see Eq. (3)]. (c) The pair density $P(\mathbf{k})$ shows the distribution of the SC pairs in momentum space. (d) Fourier transformed spin density ${\rho }_S(\mathbf{k})$. Results were obtained for $V=2t$ on a $16\times 12$ lattice using $9\times 9$ supercells.

Figure 3

Local density of states (LDOS). The colors correspond to the three distinct sites of the striped superconductor: sites with minimum charge density (dark blue), sites in the center (light blue), and on the edge (medium blue) of the AFM stripes.

Figure 4

Doping dependence of the SC order parameter. The purple and blue curves for the maximum SC order parameter $\underset{ij}{max}{\Delta }_{ij}$ correspond to $V=2t$ for isotropic $t_x=t_y$ and anisotropic hopping $t_x/t_y=0.83$, respectively. The gray line is the result for $t_x=t_y$ with a reduced $V=1.8t$. The dashed lines indicate an extrapolation to smaller values of $x$ where the numerical procedure does not converge to the discussed solution. Inset: Measured doping dependence of $T_c$ in ${\mathrm{La}}_{2-y-x}{\mathrm{Nd}}_y{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ for $y=0.2$ (Ref. 9) (red squares), $y=0.4$ (blue dots), and $y=0$ (gray) (Ref. 10).