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arXiv:1910.08931 · Physical Review X

Absence of Superconductivity in the Pure Two-Dimensional Hubbard Model

Mingpu Qin, Chia-Min Chung, Hao Shi, Ettore Vitali, Claudius Hubig, Ulrich Schollwöck, Steven R. White, Shiwei Zhang

Published Jul 21, 2020• 296 citations
arXivHTMLPDF

Abstract

We study the superconducting pairing correlations in the ground state of the doped Hubbard model-in its original form without hopping beyond nearest neighbor or other perturbing parameters-in two dimensions at intermediate to strong coupling and near optimal doping.The nature of such correlations has been a central question ever since the discovery of cuprate high-temperature superconductors.Despite unprecedented effort and tremendous progress in understanding the properties of this fundamental model, a definitive answer to whether the ground state is superconducting in the parameter regime most relevant to cuprates has proved exceedingly difficult to establish.In this work, we employ two complementary, state-of-the-art, many-body computational methods-constrained-path (CP) auxiliary-field quantum Monte Carlo (AFQMC) and density matrix renormalization group (DMRG) methods-deploying the most recent algorithmic advances in each.Systematic and detailed comparisons between the two methods are performed.The DMRG is extremely reliable on small width cylinders, where we use it to validate the AFQMC.The AFQMC is then used to study wide systems as well as fully periodic systems, to establish that we have reached the thermodynamic limit.The ground state is found to be nonsuperconducting in the moderate to strong coupling regime in the vicinity of optimal hole doping.

DOI
10.1103/physrevx.10.031016

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We study the superconducting pairing correlations in the ground state of the doped Hubbard model-in its original form without hopping beyond nearest neighbor or other perturbing parameters-in two dimensions at intermediate to strong coupling and near optimal doping.The nature of such correlations has been a central question ever since the discovery of cuprate high-temperature superconductors.Despite unprecedented effort and tremendous progress in understanding the properties of this fundamental model, a definitive answer to whether the ground state is superconducting in the parameter regime most relevant to cuprates has proved exceedingly difficult to establish.In this work, we employ two complementary, state-of-the-art, many-body computational methods-constrained-path (CP) auxiliary-field quantum Monte Carlo (AFQMC) and density matrix renormalization group (DMRG) methods-deploying the most recent algorithmic advances in each.Systematic and detailed comparisons between the two methods are performed.The DMRG is extremely reliable on small width cylinders, where we use it to validate the AFQMC.The AFQMC is then used to study wide systems as well as fully periodic systems, to establish that we have reached the thermodynamic limit.The ground state is found to be nonsuperconducting in the moderate to strong coupling regime in the vicinity of optimal hole doping.
We study the ground state of the doped Hubbard model on the honeycomb lattice in the small-doping and strongly interacting region. The nature of the ground state after doping holes into antiferromagnetic Mott insulating states on the honeycomb lattice remains a long-standing unsolved issue, even though tremendous effort has been spent investigating this challenging problem. In this work, we employ two complementary, state-of-the-art, many-body computational methods: constrained path (CP) auxiliary-field quantum Monte Carlo (AFQMC) with self-consistent constraint and density matrix renormalization group (DMRG) methods. Systematic and detailed cross validations are performed between these two methods for narrow systems for which DMRG can produce reliable results. AFQMC is then utilized to study wider systems to investigate the thermodynamic limit properties. The ground state is found to be a half-filled stripe state in the small-doping and strongly interacting region. The pairing correlation shows ddd-wave symmetry locally but decays exponentially with the distance between two pairs.
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.
We determine the spin and charge orders in the ground state of the doped two-dimensional (2D) Hubbard model in its simplest form, namely with only nearest-neighbor hopping and on-site repulsion. At half-filling, the ground state is known to be an antiferromagnetic Mott insulator. Doping Mott insulators is believed to be relevant to the superconductivity observed in cuprates. A variety of candidates have been proposed for the ground state of the doped 2D Hubbard model. A recent work employing a combination of several state-of-the-art numerical many-body methods established the stripe order as the ground state near 1/81/81/8 doping at strong interactions. In this paper, we apply one of these methods, the cutting-edge constrained-path auxiliary field quantum Monte Carlo (AFQMC) method with self-consistently optimized gauge constraints, to systematically study the model as a function of doping and interaction strength. With careful finite size scaling based on large-scale computations, we map out the ground state phase diagram in terms of its spin and charge order. We find that modulated antiferromagnetic order persists from near half-filling to about 1/51/51/5 doping. At lower interaction strengths or larger doping, these ordered states are best described as spin-density waves, with essentially delocalized holes and modest oscillations in charge correlations. When the charge correlations are stronger (large interaction or small doping), they are best described as stripe states, with the holes more localized near the node in the antiferromagnetic spin order. In both cases, we find that the wavelength in the charge correlations is consistent with so-called filled stripes.
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 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.
Extensive numerical studies have demonstrated that the two-dimensional single-band Hubbard model contains much of the key physics in cuprate high-temperature superconductors. However, there is no definitive proof that the Hubbard model truly possesses a superconducting ground state or, if it does, of how it depends on model parameters. To answer these longstanding questions, we study an extension of the Hubbard model including an infinite-range ddd-wave pair field term, which precipitates a superconducting state in the ddd-wave channel. Using exact diagonalization on 16-site square clusters, we study the evolution of the ground state as a function of the strength of the pairing term. This is achieved by monitoring the fidelity metric of the ground state, as well as determining the ratio between the two largest eigenvalues of the ddd-wave pair/spin/charge-density matrices. The calculations show a ddd-wave superconducting ground state in doped clusters bracketed by a strong antiferromagnetic state at half filling controlled by the Coulomb repulsion UUU and a weak short-range checkerboard charge ordered state at larger hole doping controlled by the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. We also demonstrate that negative ${t}^{\ensuremath{'}}$ plays an important role in facilitating ddd-wave superconductivity.