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arXiv:2609.37191

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

G. Yu, Zheng Zhu

Published Sep 29, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2609.37191

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The ground state of electron-doped t−t′−Jt-t'-Jt−t′−J model on cylinders
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Abstract We perform a comprehensive study of the electron-doped t–t′–J model on cylinders with density matrix renormalization group (DMRG). We conduct a systematic study on the finite-size and boundary condition effects on t–t′–J model on cylinders. Periodic and anti-periodic boundary conditions are implemented along the circumference direction, with the system’s width extending up to as large as 8 lattice units. We study doping levels of 1/6, 1/8, and 1/12, which represent the most interesting region in the phase diagram of electron-doped cuprates. We find that for width-4 and width-6 systems, the ground state for fixed doping switches between anti-ferromagnetic Neel state and stripe state under different boundary conditions and system widths, indicating the presence of large finite size effect in the t–t′–J model. We also have a careful analysis of the d-wave pairing correlations which also change quantitatively with boundary conditions and widths of the system. However, the pairing correlations are enhanced when the system becomes wider for all dopings, suggesting the existence of possible long-range superconducting order in the thermodynamic limit. The width-8 results are found to be dependent on the starting state in the DMRG calculation for the kept states we can reach. For the width-8 system, only Neel (stripe) state can be stabilized in DMRG calculation for 1/12 (1/6) doping, while both stripe and Neel states are stable in the DMRG sweep for 1/8 doping, regardless of the boundary conditions. These results indicate that 1/8 doping is likely to lie on the boundary of a phase transition between the Neel phase with lower doping and the stripe phase with higher doping, consistent with the previous study. The sensitivity of the ground state on boundary conditions and size observed for narrow systems is similar to that found in the t′-Hubbard model, where the t′ term introduces frustration and makes the stripe state fragile. The study of different boundary conditions provides a useful tool to check the finite size effect in the future DMRG calculations.
The long-studied Hubbard model is one of the simplest models of copper-oxide superconductors. However, the connection between the model and the experimental phase diagram is still under debate, in particular regarding the existence and extent of the ddd-wave superconducting phase. Recent rapid progress in improving the accuracy of numerical solvers has opened a way to answer this question reliably. Here, we study the hole-doping concentration ($\ensuremath{\delta}$) dependence of the Hubbard model in the ground states on a square lattice at strong coupling U/t=10U/t=10U/t=10, for the on-site interaction UUU and the transfer ttt, using a variational Monte Carlo method. The method, which combines tensor network and Lanczos methods on top of Pfaffian wave functions, reveals a rich phase diagram, in which many orders compete severely and degenerate within the energy range of 0.01t0.01t0.01t. We have identified distinct phases including a uniform ddd-wave superconducting phase for $0.17\ensuremath{\lesssim}\ensuremath{\delta}\ensuremath{\lesssim}0.22$ and a stripe charge/spin ordered phase for $\ensuremath{\delta}\ensuremath{\lesssim}0.17$ with the stripe period depending on $\ensuremath{\delta}$, together with presumable spatially coexisting antiferromagnetic and stripe order for $\ensuremath{\delta}\ensuremath{\lesssim}0.07$ and coexisting stripe and ddd-wave superconductivity for $0.07\ensuremath{\lesssim}\ensuremath{\delta}\ensuremath{\lesssim}0.17$. The present, improved method revealed a wider region of a charge uniform superconducting phase than the previous studies and shows a qualitative similarity to the phase diagram of the cuprate superconductors. The superconducting order parameter is largest at doping of around $\ensuremath{\delta}=0.17$ in the ground state, which undergoes phase transitions from an inhomogeneous to a uniform state.