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arXiv:2510.24405 · Frontiers of Physics

Charge stripe and superconductivity tuned by interlayer interaction in a sign-problem-free bilayer extended Hubbard model

Runyu Ma, Zenghui Fan, Hongxing Liu, Yichen Yao, Tianxing Ma, Hai-Qing Lin

Published Jan 1, 2026• 0 citations
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Abstract

Competing orders represent a central challenge in understanding strongly correlated systems. In this work, we employ projector quantum Monte Carlo simulations to study a sign-problem-free bilayer extended Hubbard model. In this model, a charge stripe phase, characterized by a peak at (0±2πδ,0) is induced by highly anisotropic interlayer antiferromagnetic spin-exchange coupling Jz, and strongly suppressed by the decreasing spin-exchange anisotropy (i.e., introducing the spin-flip term J⊥); in contrast, the introduction of J⊥ favors the emergence of interlayer pairing superconductivity. We further demonstrate that the anisotropy of the interlayer spin-exchange directly governs the competition between these two phases, while the on-site interaction U plays a complex role in tuning both the charge stripe and superconductivity. Furthermore, against the background of charge stripes we discover the spin stripes, characterized by twice the period of charge stripes, indicating the profound internal connection between them. Our work identifies the key factors driving charge stripe formation, highlights the sensitivity of both the charge stripe and superconducting phases to interaction parameters, and thereby provides valuable insights into competing orders in strongly correlated systems.

DOI
10.15302/frontphys.2026.115202

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Competing orders represent a central challenge in understanding strongly correlated systems. In this work, we employ projector quantum Monte Carlo simulations to study a sign-problem-free bilayer extended Hubbard model. In this model, a charge stripe phase, characterized by a peak at (0±2πδ,0) is induced by highly anisotropic interlayer antiferromagnetic spin-exchange coupling Jz, and strongly suppressed by the decreasing spin-exchange anisotropy (i.e., introducing the spin-flip term J⊥); in contrast, the introduction of J⊥ favors the emergence of interlayer pairing superconductivity. We further demonstrate that the anisotropy of the interlayer spin-exchange directly governs the competition between these two phases, while the on-site interaction U plays a complex role in tuning both the charge stripe and superconductivity. Furthermore, against the background of charge stripes we discover the spin stripes, characterized by twice the period of charge stripes, indicating the profound internal connection between them. Our work identifies the key factors driving charge stripe formation, highlights the sensitivity of both the charge stripe and superconducting phases to interaction parameters, and thereby provides valuable insights into competing orders in strongly correlated systems.
The competition between antiferromagnetism and superconductivity is one of the central questions in research involving strongly correlated systems. In this paper, we utilize a double-layer model containing Hubbard interaction and interlayer Heisenberg interaction to reveal their competition. This model is free of the sign problem under certain conditions, and we perform projector quantum Monte Carlo simulations to extract the ground-state correlations of magnetism and superconductivity. Our results show that the superconductivity emerges when the antiferromagnetism is suppressed by tuning the filling or the anisotropy of the interlayer Heisenberg interaction. This model can be seen as an analog of unconventional superconductors and may help us to understand the transition from an antiferromagnetic insulator to a superconductor.
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.
We carry out a sign-problem-free quantum Monte Carlo calculation of a bilayer model with a repulsive intralayer Hubbard interaction and a ferromagnetic interlayer interaction. The latter breaks the global SU(2)SU(2)SU(2) spin rotational symmetry but preserves a $U(2)\ifmmode\times\else\texttimes\fi{}U(2)$ invariance under mixing of same-spin electrons between layers. We show that despite the difference in symmetry, the bilayer model exhibits the same qualitative features found in the single-layer Hubbard model. These include stripe phases, whose nature is sensitive to the presence of next-nearest-neighbor hopping, a maximum in the Knight shift that moves to lower temperatures with increasing hole doping, and lack of evidence for intralayer ddd-wave superconductivity. Instead, we find a superconducting phase, coexisting with stripes, whose critical temperature traces a dome as a function of doping and is due to interlayer spin-polarized pairing that is induced by the ferromagnetic interaction.
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.
Abstract The relation between d -wave superconductivity and stripes is fundamental to the understanding of ordered phases in high-temperature cuprate superconductors 1–6 . These phases can be strongly influenced by anisotropic couplings, leading to higher critical temperatures, as emphasized by the recent discovery of superconductivity in nickelates 7–10 . Quantum simulators with ultracold atoms provide a versatile platform to engineer such couplings and to observe emergent structures in real space with single-particle resolution. Here we show, to our knowledge, the first signatures of individual stripes in a cold-atom Fermi–Hubbard quantum simulator using mixed-dimensional (mixD) settings. Increasing the energy scale of hole–hole attraction to the spin exchange energy, we access the interesting crossover temperature regime in which stripes begin to form 11 . We observe extended, attractive correlations between hole dopants and find an increased probability of forming larger structures akin to individual stripes. In the spin sector, we study correlation functions up to the third order and find results consistent with stripe formation. These observations are interpreted as a precursor to the stripe phase, which is characterized by interleaved charge and spin density wave ordering with fluctuating lines of dopants separating domains of opposite antiferromagnetic order 12–14 .
Charge stripes have been widely observed in many different types of unconventional superconductors, holding varying periods ( P{{\mathcal{P}}}P ) and intensities. However, a general understanding on the interplay between charge stripes and superconducting properties is still incomplete. Here, using large-scale unbiased numerical simulations on a general inhomogeneous Hubbard model, we discover that the charge-stripe period P{{\mathcal{P}}}P , which is variable in different real material systems, could dictate the pairing symmetries—d wave for P≥4,s{{\mathcal{P}}}\ge 4,sP≥4,s and d waves for P≤3{{\mathcal{P}}}\le 3P≤3 . In the latter, tuning hole doping and charge-stripe amplitude can trigger a d-s wave transition and magnetic-correlation shift, where the d-wave state converts to a pairing-density wave state, competing with the s wave. These interesting phenomena arise from an unusual stripe-induced selection rule of pairing symmetries around on-stripe region and within inter-stripe region, giving rise to a critical point of P=3{{\mathcal{P}}}=3P=3 for the phase transition. In general, our findings offer important insights into the differences in the superconducting pairing mechanisms across many P{{\mathcal{P}}}P -dependent superconducting systems, highlighting the decisive role of charge stripe. Charge stripes have been observed in various unconventional superconductors, but their interplay with the superconducting properties is not fully understood. Here, the authors show, via large-scale numerical simulations based on an inhomogeneous Hubbard model, that the charge stripe period can dictate the superconducting pairing symmetry.