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.
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 ground state of electron-doped t−t′−J model on cylinders
We perform a comprehensive study of the electron-doped t−t′−J model on cylinders with Density Matrix Renormalization Group (DMRG). We adopt both periodic and anti-periodic boundary conditions along the circumference direction to explore the finite size effect. 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 6 systems, the ground state for fixed doping switches between anti-ferromagnetic Neel state and stripe state under different boundary conditions and with 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 changes 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-ranged 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 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 in 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 ground state on boundary conditions and size observed in this work is similar to that in the t′- Hubbard model.
We report results of large-scale ground-state density matrix renormalization group (DMRG) calculations on t-[Formula: see text]-J cylinders with circumferences 6 and 8. We determine a rough phase diagram that appears to approximate the two-dimensional (2D) system. While for many properties, positive and negative [Formula: see text] values ([Formula: see text]) appear to correspond to electron- and hole-doped cuprate systems, respectively, the behavior of superconductivity itself shows an inconsistency between the model and the materials. The [Formula: see text] (hole-doped) region shows antiferromagnetism limited to very low doping, stripes more generally, and the familiar Fermi surface of the hole-doped cuprates. However, we find [Formula: see text] strongly suppresses superconductivity. The [Formula: see text] (electron-doped) region shows the expected circular Fermi pocket of holes around the [Formula: see text] point and a broad low-doped region of coexisting antiferromagnetism and d-wave pairing with a triplet p component at wavevector [Formula: see text] induced by the antiferromagnetism and d-wave pairing. The pairing for the electron low-doped system with [Formula: see text] is strong and unambiguous in the DMRG simulations. At larger doping another broad region with stripes in addition to weaker d-wave pairing and striped p-wave pairing appears. In a small doping region near [Formula: see text] for [Formula: see text], we find an unconventional type of stripe involving unpaired holes located predominantly on chains spaced three lattice spacings apart. The undoped two-leg ladder regions in between mimic the short-ranged spin correlations seen in two-leg Heisenberg ladders.
Uniform-Charge-Density d-Wave Superconductivity in the Pure t-J Model at 1/8 Doping on an Infinite Cylinder
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 Ly=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 Ly=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.
We revisit the ground state of the Hubbard model on two-legged ladders in this paper. We perform density matrix renormalization group (DMRG) calculations on large system sizes with large numbers of kept states and perform extrapolation of DMRG results with truncation errors in the converged region. We find that the superconducting correlation exponent Ksc extracted from the pair-pair correlation is very sensitive to the position of the reference bond, reflecting a huge boundary effect on it. By systematically removing the effects from boundary conditions, finite sizes, and truncation errors in DMRG, we obtain more accurate values of Ksc and ${K}_{\ensuremath{\rho}}$ with DMRG. With these exponents, we confirm that the two-legged Hubbard model is in the Luther-Emery liquid phase with ${K}_{sc}\ifmmode\cdot\else\textperiodcentered\fi{}{K}_{\ensuremath{\rho}}=1$ from tiny doping near half filling to 1/8 hole doping. When the doping is increased to $\ensuremath{\delta}⪆1/6$, the behaviors of charge, pairing, and spin correlations do not change qualitatively, but the relationship ${K}_{sc}\ifmmode\cdot\else\textperiodcentered\fi{}{K}_{\ensuremath{\rho}}=1$ is likely to be violated. With the further increase of the doping to $\ensuremath{\delta}=1/3$, the quasi-long-ranged charge correlation turns into a true long-ranged charge order, and the spin gap is closed, while the pair-pair correlation still decays algebraically. Our work provides a standard way to analyze the correlation functions when studying systems with open boundary conditions.
We study the ground state of the doped Hubbard model on honeycomb lattice with both nearest (t) and next-nearest-neighbor hoppings $({t}^{\ensuremath{'}})$ in the small doping and strongly interacting region. Previous study on the model without ${t}^{\ensuremath{'}}$ showed the ground state is a half-filled stripe. We employ density matrix renormalization group and extrapolate the results with truncation errors in the converged region. In the ${t}^{\ensuremath{'}}<0$ side, we find the half-filled stripe phase at ${t}^{\ensuremath{'}}=0$ is stable against the frustration of ${t}^{\ensuremath{'}}$ until a critical point $\ensuremath{-}0.4<{t}_{c}^{\ensuremath{'}}<\ensuremath{-}0.3$, beyond which the ground state switches to antiferromagnetic N\'eel phase with charge modulation. With further increase of ${t}^{\ensuremath{'}}$ to $\ensuremath{-}0.7$, the ground state becomes paramagnetic. In the ${t}^{\ensuremath{'}}>0$ side, the half-filled stripe stretches to ${t}^{\ensuremath{'}}\ensuremath{\approx}0.7$. We do not find obvious enhancement of pairing for the range of ${t}^{\ensuremath{'}}$ studied. We study width-4 cylinders in this paper but the results for spin, charge, and pairing correlation agree qualitatively for periodic and antiperiodic boundary conditions in the half-filled stripe and antiferromagnetic N\'eel phases, suggesting the results are likely to be representative for true two-dimensional systems. The half-filled stripe to antiferromagnetic N\'eel phase transition can be realized on real materials or ultracold atom platform.
The square-lattice Hubbard and closely related t-J models are considered as basic paradigms for understanding strong correlation effects and unconventional superconductivity (SC). Recent large-scale density matrix renormalization group simulations on the extended t-J model have identified d-wave SC on the electron-doped side (with the next-nearest-neighbor hopping t_{2}>0) but a dominant charge density wave (CDW) order on the hole-doped side (t_{2}<0), which is inconsistent with the SC of hole-doped cuprate compounds. We re-examine the ground-state phase diagram of the extended t-J model by employing the state-of-the-art density matrix renormalization group calculations with much enhanced bond dimensions, allowing more accurate determination of the ground state. On six-leg cylinders, while different CDW phases are identified on the hole-doped side for the doping range δ=1/16-1/8, a SC phase emerges at a lower doping regime, with algebraically decaying pairing correlations and d-wave symmetry. On the wider eight-leg systems, the d-wave SC also emerges on the hole-doped side at the optimal 1/8 doping, demonstrating the winning of SC over CDW by increasing the system width. Our results not only suggest a new path to SC in general t-J model through weakening the competing charge orders, but also provide a unified understanding on the SC of both hole- and electron-doped cuprate superconductors.