Spin-charge stripes belong to the most prominent low-temperature orders besides superconductivity in high-temperature superconductors. This phase is particularly challenging to study numerically due to finite-size effects. By investigating the formation of long, isolated stripes, we offer a perspective complementary to typical finite-doping phase diagrams. We use the density-matrix renormalization group algorithm to extract the ground states of an 18-leg cylindrical strip geometry, making the diameter significantly wider than in previous works. This approach allows us to map out the range of possible stripe filling fractions on the electron versus hole-doped side. We find good agreement with established results, suggesting that the spread of filling fractions observed in the literature is governed by the physics of a single stripe. Taking a microscopic look at stripe formation, we reveal two separate regimes - a high-filling regime captured by a simplified squeezed-space model and a low-filling regime characterized by the structure of individual pairs of dopants. Thereby, we trace back the phenomenology of the striped phase to its microscopic constituents and highlight the different challenges for observing the two regimes in quantum simulation experiments.
Large Isolated Stripes on Short 18-leg t-J Cylinders
Spin-charge stripes belong to the most prominent low-temperature orders besides superconductivity in high-temperature superconductors. This phase is particularly challenging to study numerically due to finite-size effects. By investigating the formation of long, isolated stripes, we offer a perspective complementary to typical finite-doping phase diagrams. We use the density-matrix renormalization group algorithm to extract the ground states of an 18-leg cylindrical strip geometry, making the diameter significantly wider than in previous works. This approach allows us to map out the range of possible stripe filling fractions on the electron versus hole-doped side. We find good agreement with established results, suggesting that the spread of filling fractions observed in the literature is governed by the physics of a single stripe. Taking a microscopic look at stripe formation, we reveal two separate regimes - a high-filling regime captured by a simplified squeezed-space model and a low-filling regime characterized by the structure of individual pairs of dopants. Thereby, we trace back the phenomenology of the striped phase to its microscopic constituents and highlight the different challenges for observing the two regimes in quantum simulation experiments.
Understanding the physics of the two-dimensional Hubbard model is widely believed to be a key step in achieving a full understanding of high- T c cuprate superconductors. In recent years, progress has been made by large-scale numerical simulations at finite doping and, on the other hand, by microscopic theories able to capture the physics of individual charge carriers. In this work, we study single pairs of dopants in a cylindrical system using the density-matrix renormalization group algorithm. We identify two coexisting charge configurations that couple to the spin environment in different ways: a tightly bound configuration featuring (next-)nearest-neighbor pairs and a stripelike configuration of dopants on opposite sides of the cylinder, accompanied by a spin domain wall. Thus, we establish that the interplay between stripe order and uniform pairing, central to the models’ phases at finite doping, has its origin at the single-pair level. By interpolating between the Hubbard and the related t − J model, we are able to quantitatively understand discrepancies in the pairing properties of the two models through the three-site hopping term usually omitted from the t − J Hamiltonian. This term is closely related to a next-nearest-neighbor tunneling t ′ , which we observe to upset the balance between the competing stripe and pair states on the two-dopant level.
The authors employ the density-matrix renormalization group to investigate the doped Hubbard model on six-leg square cylinders. It uncovers a rich quantum phase diagram, intricately sensitive to next-nearest-neighbor electron hopping ($t\ensuremath{'}$). The positive-t\ensuremath{'} region shows a robust d-wave superconducting phase with intertwined superconducting and charge-density-wave orders. In contrast, the negative-t\ensuremath{'} side remains insulating, where doped holes form either long-range charge stripe order at small t\ensuremath{'} or a holon Wigner crystal with one doped hole per emergent unit cell and short-range spin correlations at larger t'.
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.
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 A microscopic understanding of the strongly correlated physics of the cuprates must account for the translational and rotational symmetry breaking that is present across all cuprate families, commonly in the form of stripes. Here we investigate emergence of stripes in the Hubbard model, a minimal model believed to be relevant to the cuprate superconductors, using determinant quantum Monte Carlo (DQMC) simulations at finite temperatures and density matrix renormalization group (DMRG) ground state calculations. By varying temperature, doping, and model parameters, we characterize the extent of stripes throughout the phase diagram of the Hubbard model. Our results show that including the often neglected next-nearest-neighbor hopping leads to the absence of spin incommensurability upon electron-doping and nearly half-filled stripes upon hole-doping. The similarities of these findings to experimental results on both electron and hole-doped cuprate families support a unified description across a large portion of the cuprate phase diagram.
The dualism between superconductivity and charge/spin modulations (the so-called stripes) dominates the phase diagram of many strongly-correlated systems. A prominent example is given by the Hubbard model, where these phases compete and possibly coexist in a wide regime of electron dopings for both weak and strong couplings. Here, we investigate this antagonism within a variational approach that is based upon Jastrow-Slater wave functions, including backflow correlations, which can be treated within a quantum Monte Carlo procedure. We focus on clusters having a ladder geometry with M M legs (with M M ranging from 2 2 to 10 10 ) and a relatively large number of rungs, thus allowing us a detailed analysis in terms of the stripe length. We find that stripe order with periodicity \lambda=8 λ = 8 in the charge and 2\lambda=16 2 λ = 16 in the spin can be stabilized at doping \delta=1/8 δ = 1 / 8 . Here, there are no sizable superconducting correlations and the ground state has an insulating character. A similar situation, with \lambda=6 λ = 6 , appears at \delta=1/6 δ = 1 / 6 . Instead, for smaller values of dopings, stripes can be still stabilized, but they are weakly metallic at \delta=1/12 δ = 1 / 12 and metallic with strong superconducting correlations at \delta=1/10 δ = 1 / 10 , as well as for intermediate (incommensurate) dopings. Remarkably, we observe that spin modulation plays a major role in stripe formation, since it is crucial to obtain a stable striped state upon optimization. The relevance of our calculations for previous density-matrix renormalization group results and for the two-dimensional case is also discussed.