Uniform-Charge-Density $d$-Wave Superconductivity in the Pure $t$-$J$ Model at $1/8$ Doping on an Infinite Cylinder

Abstract

The ground state of the square-lattice $t$-$J$ model at $1/8$ hole doping has been the subject of a long-standing controversy, with conflicting numerical reports of stripe order versus superconductivity. Here, we address this debate using the variational uniform matrix product state (VUMPS) method, which optimizes the wave function directly in the thermodynamic limit along the cylinder axis and eliminates the boundary pinning effects that can arise in finite-cylinder calculations. On an infinite cylinder of circumference $L_y = 8$, we find a uniform-charge-density $d$-wave superconducting ground state with quasi-long-range pairing correlations. The dominant pairing occurs at zero center-of-mass momentum, while no sizable finite-momentum pairing component is detected. Twisted-boundary-condition calculations reveal a finite superconducting phase stiffness, with the order-parameter phase winding smoothly to follow the applied flux, providing an independent probe of the robustness of the superconductivity. Moreover, both charge and spin correlations decay rapidly, with no signatures of long-range stripe or magnetic order, in contrast to an $L_y=6$ cylinder at $1/6$ doping where the same method readily identifies stripe order. Our results provide methodologically distinct evidence in the long-standing stripe--superconductivity debate in the pure $t$-$J$ model and reveal uniform $d$-wave superconductivity in a minimal model of doped Mott insulators.