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.
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.
The relevance of the single-band two-dimensional Hubbard model to superconductivity in the doped cuprates has recently been questioned, based on density matrix renormalization group (DMRG) computations that found superconductivity over an unrealistically broad doping region upon electron-doping, yet a complete absence of superconductivity for hole-doping. We report very similar results from DMRG calculations on a Cu2O3 two-leg ladder within the parent three-band correlated-electron Hamiltonian. The strong asymmetry found in our calculations are in contradiction to the deep and profound symmetry in the experimental phase diagrams of electron- and hole-doped cuprate superconductors, as seen from the occurrence of quantum critical points within the superconducting domes in both cases that are characterized by Fermi surface reconstruction, large jumps in carrier density, and strange metal behavior.
The Hubbard and closely related t-J models are exciting platforms for unconventional superconductivity (SC). Through state-of-the-art density matrix renormalization group calculations using the grand canonical ensemble, we address open issues regarding the ground-state phase diagram of the extended t-J model on a square lattice in the parameter regime relevant to cuprate superconductors. On large 8-leg cylinders, we demonstrate that the pure t-J model with only nearest-neighbor hoppings and superexchange interactions, for a wide range of doping ( δ = 0.1 to 0.2 ), hosts robust d-wave superconductivity possibly coexisting with weak unidirectional pair density wave. Furthermore, a small next nearest neighbor hopping t 2 suppresses pair and charge density waves, resulting in a uniform d-wave SC phase in both electron- and hole-doped cuprate model systems. Our work validates the t-J model as a proper minimum model for the emergence of superconductivity in cuprate superconductors.
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.
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.
Recently, robust d-wave superconductive (SC) order has been unveiled in the ground state of the 2D t-t^{'}-J model-with both nearest-neighbor (t) and next-nearest-neighbor (t^{'}) hoppings-by density matrix renormalization group studies. However, there is currently a debate on whether the d-wave SC holds up strong on both t^{'}/t>0 and t^{'}/t<0 cases for the t-t^{'}-J model, which correspond to the electron- and hole-doped sides of the cuprate phase diagram, respectively. Here, we exploit state-of-the-art thermal tensor network approach to accurately obtain the phase diagram of the t-t^{'}-J model on cylinders with widths up to W=6 and down to low temperature as T/J≃0.06, pushing the boundaries of contemporary finite-T calculations. For t^{'}/t>0, we find a domelike SC regime with a diverging d-wave pairing susceptibility, χ_{SC}∝1/T^{α} below a characteristic temperature T_{c}^{*}. Near optimal doping, T_{c}^{*} reaches its highest value of about 0.15J. Above T_{c}^{*} yet below a higher crossover temperature T^{*}, the magnetic susceptibility becomes suppressed, which can be related to the onset of pseudogap (PG) behaviors. On the other hand, for t^{'}/t<0, we find the pairing correlations are much weaker, although there exhibits a node-antinode structure in the PG regime as observed in the hole-doped cuprates. The thermal tensor network calculations of the t-t^{'}-J model underscore both the similarities and differences in the finite-temperature phase diagram between the fundamental model and cuprates, yielding unique insights into their intricate behaviors.