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arXiv:2307.14835 · Physical Review Letters

Frustration-Induced Superconductivity in the t − t ′ Hubbard Model

Changkai Zhang, Jheng-Wei Li, Dimitra Nikolaidou, Jan von Delft

Published Mar 18, 2025• 9 citations
arXivHTMLPDF

Abstract

The two-dimensional Hubbard model is widely believed to capture key ingredients of high-T_{c} superconductivity in cuprate materials. However, compelling evidence remains elusive. In particular, various magnetic orders may emerge as strong competitors of superconducting orders. Here, we study the ground state properties of the doped two-dimensional t-t^{'} Hubbard model on a square lattice via the infinite projected entangled-pair state method with U(1) or SU(2) spin symmetry. The former is compatible with antiferromagnetic orders, while the latter forbids them. Therefore, we obtain by comparison a detailed understanding of the magnetic impact on superconductivity. Moreover, an additional t^{'} term accommodates the particle-hole asymmetry, which facilitates studies on the discrepancies between electron- and hole-doped systems. We demonstrate that (i) a positive t^{'}/t significantly amplifies the strength of superconducting orders; (ii) at sufficiently large doping levels, the t-t^{'} Hubbard model favors a uniform state with superconducting orders instead of stripe states with charge and spin modulations; and (iii) the enhancement of magnetic frustration, by increasing either the strength of next-nearest neighbor interactions or the charge doping, impairs stripe orders and helps stabilize superconductivity.

DOI
10.1103/physrevlett.134.116502

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The two-dimensional Hubbard model is widely believed to capture key ingredients of high-T_{c} superconductivity in cuprate materials. However, compelling evidence remains elusive. In particular, various magnetic orders may emerge as strong competitors of superconducting orders. Here, we study the ground state properties of the doped two-dimensional t-t^{'} Hubbard model on a square lattice via the infinite projected entangled-pair state method with U(1) or SU(2) spin symmetry. The former is compatible with antiferromagnetic orders, while the latter forbids them. Therefore, we obtain by comparison a detailed understanding of the magnetic impact on superconductivity. Moreover, an additional t^{'} term accommodates the particle-hole asymmetry, which facilitates studies on the discrepancies between electron- and hole-doped systems. We demonstrate that (i) a positive t^{'}/t significantly amplifies the strength of superconducting orders; (ii) at sufficiently large doping levels, the t-t^{'} Hubbard model favors a uniform state with superconducting orders instead of stripe states with charge and spin modulations; and (iii) the enhancement of magnetic frustration, by increasing either the strength of next-nearest neighbor interactions or the charge doping, impairs stripe orders and helps stabilize superconductivity.
We study the two-dimensional $t\text{\ensuremath{-}}J$ model on a square lattice using infinite projected entangled pair states (iPEPS). At small doping, multiple orders, such as antiferromagnetic order, stripe order and superconducting order, are intertwined or compete with each other. We demonstrate the role of spin symmetry at small doping by either imposing SU(2) spin symmetry or its U(1) subgroup in the iPEPS ansatz, thereby excluding or allowing spontaneous spin-symmetry breaking, respectively, in the thermodynamic limit. From a detailed comparison of our simulations, we provide evidence that stripe order is pinned by long-range antiferromagnetic order. We also find SU(2) iPEPS, enforcing a spin-singlet state, yields a uniform charge distribution and favors d-wave singlet pairing.
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 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 study the competition between stripe states with different periods and a uniform ddd-wave superconducting state in the extended two-dimensional Hubbard model at 1/8 hole doping using infinite projected entangled-pair states (iPEPS). With increasing strength of negative next-nearest neighbor hopping ${t}^{\ensuremath{'}}$, the preferred period of the stripe decreases. For the values of ${t}^{\ensuremath{'}}$ predicted for cuprate high-Tc{T}_{c}Tc​ superconductors, we find stripes with a period 4 in the charge order, in agreement with experiments. Superconductivity in the period 4 stripe is suppressed at 1/81/81/8 doping. Only at larger doping, $0.18\ensuremath{\lesssim}\ensuremath{\delta}<0.25$, the period 4 stripe exhibits coexisting ddd-wave superconducting order. The uniform ddd-wave state is only favored for sufficiently large positive ${t}^{\ensuremath{'}}$.
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.
Whether the ground state of the square lattice Hubbard model exhibits superconductivity remains a major open question, central to understanding high temperature cuprate superconductors and ultra-cold fermions in optical lattices. Numerical studies have found evidence for stripe-ordered states and superconductivity at strong coupling but the phase diagram remains controversial. Here, we show that one can resolve the subtle energetics of metallic, superconducting, and stripe phases using a new class of neural quantum state (NQS) wavefunctions that extend hidden fermion determinant states to Pfaffians. We simulate several hundred electrons using fast Pfaffian algorithms allowing us to measure off-diagonal long range order. At strong coupling and low hole-doping, we find that a non-superconducting filled stripe phase prevails, while superconductivity coexisting with partially-filled stripes is stabilized by a negative next neighbor hopping t-prime, with |t-prime| > 0.1. At larger doping levels, we introduce momentum-space correlation functions to mitigate finite size effects that arise from weakly-bound pairs. These provide evidence for uniform d-wave superconductivity at U = 4, even when t-prime = 0. Our results highlight the potential of NQS approaches, and provide a fresh perspective on superconductivity in the square lattice Hubbard model.