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arXiv:2409.18131 · Physical Review X

Two-Dopant Origin of Competing Stripe and Pair Formation in Hubbard and t − J Models

Tizian Blatz, Ulrich Schollwöck, Fabian Grusdt, Annabelle Bohrdt

Published Jul 25, 2025• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/dpfl-12st

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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.
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 Hubbard model is an iconic model in quantum many-body physics and has been intensely studied, especially since the discovery of high-temperature cuprate superconductors. Combining the complementary capabilities of two computational methods, we found superconductivity in both the electron- and hole-doped regimes of the two-dimensional Hubbard model with next-nearest-neighbor hopping. In the electron-doped regime, superconductivity was weaker and was accompanied by antiferromagnetic Néel correlations at low doping. The strong superconductivity on the hole-doped side coexisted with stripe order, which persisted into the overdoped region with weaker hole-density modulation. These stripe orders varied in fillings between 0.6 and 0.8. Our results suggest the applicability of the Hubbard model with next-nearest hopping for describing cuprate high–transition temperature ( T c ) superconductivity.
') neighbor hoppings. Our work investigates the consequences of increasing hole doping on the instauration of stripes and the behavior of the superconducting order parameter, with a discussion on how the two phenomena affect each other. We consider two different values of the next-nearest neighbor hopping parameter, that are appropriate for describing cuprate superconductors. We observe that stripes are the optimal state in a wide doping range; the stripe wavelength reduces at increasing doping, until stripes melt into a uniform state for large values of doping. Superconducting pair-pair correlations, indicating the presence of superconductivity, are always suppressed in the presence of stripes. Our results suggest that the phase diagram for the single-band Hubbard model is dominated by stripes, with superconductivity being possible only in a narrow doping range between striped states and a nonsuperconducting metal.
Unravelling competing orders emergent in doped Mott insulators and their interplay with unconventional superconductivity is one of the major challenges in condensed matter physics. To explore the possible superconducting state in a doped Mott insulator, we study the square-lattice t-J model with both the nearest-neighbor and next-nearest-neighbor electron hoppings and spin interactions. By using the state-of-the-art density matrix renormalization group calculation with imposing charge U(1) and spin SU(2) symmetries on the six-leg cylinders, we establish a quantum phase diagram including three phases: a stripe charge density wave phase, a superconducting phase without static charge order, and a superconducting phase coexistent with a weak charge stripe order. Crucially, we demonstrate that the superconducting phase has a power-law pairing correlation that decays much slower than the charge density and spin correlations, which is a quasi-1D descendant of the uniform d-wave superconductor in two dimensions. These findings reveal that enhanced charge and spin fluctuations with optimal doping is able to produce robust d-wave superconductivity in doped Mott insulators, providing a foundation for connecting theories of superconductivity to models of strongly correlated systems.
The two-dimensional Hubbard model was originally proposed as the minimal model of uniform ddd-wave superconductivity in the cuprate high-temperature superconductors, but recent calculations with relevant parameters in a realistic doping regime have often found other ground states. Further, uncertainty remains over such results due to the intrinsic bias of different computational methods. Here we address this uncertainty through state-of-the-art neural tensor network and neural backflow state simulations, enabled by large-scale GPU computation, that together reach an unprecedented variational accuracy on lattices with up to 1024 sites. At 1/81/81/8 doping in the pure Hubbard model, as well as with next-nearest-neighbor hopping relevant to the hole-doped cuprates, we reveal a near degeneracy among a low-energy uniform ddd-wave state, superconducting stripes, and non-superconducting stripe states on the milli-ttt energy scale. Our work rebalances the narrative that the Hubbard model supports a single dominant order in relevant parameter regimes towards a picture of extreme competition between stripes and superconductivity.
Abstract Several state-of-the-art numerical methods have observed static or fluctuating spin and charge stripes in doped two-dimensional Hubbard models, suggesting that these orders play a significant role in shaping the cuprate phase diagram. Many experiments, however, also indicate that the cuprates have strong electron-phonon (e-ph) coupling, and it is unclear how this interaction influences stripe correlations. We study static and fluctuating stripe orders in the doped single-band Hubbard-Holstein model using zero temperature variational Monte Carlo and finite temperature determinant quantum Monte Carlo. We find that the lattice couples more strongly with the charge component of the stripes, leading to an enhancement or suppression of stripe correlations, depending on model parameters like the next-nearest-neighbor hopping t′{t}^{{\prime} }t′ t′ or phonon energy Ω. Our results help elucidate how thee-ph interaction can tip the delicate balance between stripe and superconducting correlations in the Hubbard-Holstein model with implications for our understanding of the high-Tccuprates.