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arXiv:2507.06614 · Physical review. B./Physical review. B

Enhanced superconductivity via layer differentiation in the trilayer Hubbard model

Xun Liu, Mi Jiang

Published Oct 9, 2025• 2 citations
arXivHTMLPDF

Abstract

The multilayer cuprates host the highest superconducting transition temperature while their underlying physical mechanism remains unclear. By employing large-scale dynamical cluster quantum Monte Carlo simulations, the authors show here that the trilayer Hubbard model, with a realistic doping distribution within three layers, exhibits a higher ${T}_{c}$ than the single-layer model. This work also provides strong numerical evidence on the possibility of $d$-wave superconductivity solely hosted in the inner layer.

DOI
10.1103/n28s-ggzc

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The multilayer cuprates host the highest superconducting transition temperature while their underlying physical mechanism remains unclear. By employing large-scale dynamical cluster quantum Monte Carlo simulations, the authors show here that the trilayer Hubbard model, with a realistic doping distribution within three layers, exhibits a higher Tc{T}_{c}Tc​ than the single-layer model. This work also provides strong numerical evidence on the possibility of ddd-wave superconductivity solely hosted in the inner layer.
The relationship between crystal structures and superconducting critical temperatures has attracted considerable attention as a clue to designing higher-TcT_{\rm c}Tc​ superconductors. In particular, the relationship between the number nnn of CuO2_22​ layers in a unit cell of cuprate superconductors and the optimum superconducting transition temperature TcoptT_{\rm c}^{\rm opt}Tcopt​ is intriguing. As experimentally observed in layered cuprates, TcoptT_{\rm c}^{\rm opt}Tcopt​ increases when nnn is increased, up to n=3n=3n=3, and, then, decreases for larger nnn. However, the mechanism behind the nnn dependence of TcoptT_{\rm c}^{\rm opt}Tcopt​ remains elusive although there have been many studies on the nnn dependence. In this paper, we studied a bilayer ttt-t′t't′ Hubbard model to clarify the effects of the adjacent CuO2_22​ layers on the stability of the superconductivity by using a many-variable variational Monte Carlo method. We calculate the superconducting correlation at long distance and zero temperature, and the amplitude of the superconducting gap functions estimated from the momentum distribution as the observables correlated with TcoptT_{\rm c}^{\rm opt}Tcopt​. It is found that the in-plane superconducting correlation is not enhanced in comparison with that in the single-layer ttt-t′t't′ Hubbard model. The superconducting correlations of the bilayer Hamiltonian are significantly small in the overdoped region in comparison with those of single-layer Hamiltonian, which is attributed to the van Hove singularity. In addition, we found that the amplitude of the superconducting gap functions is also similar in both the single-layer and bilayer ttt-t′t't′ Hubbard model at the optimal doping. Therefore, we conclude that the adjacent Hubbard layers are not relevant to the enhancement of TcoptT_{\rm c}^{\rm opt}Tcopt​ in the bilayer cuprates. Possible origins of the enhanced TcoptT_{\rm c}^{\rm opt}Tcopt​ other than the adjacent layers are also discussed.
The single band, two-dimensional Hubbard Hamiltonian has been extensively studied as a model for high temperature superconductivity. While quantum Monte Carlo simulations within the dynamic cluster approximation are now providing considerable evidence for a ddd-wave superconducting state at low temperature, such a transition remains well out of reach of finite lattice simulations because of the ``sign problem.'' We show here that a bilayer Hubbard model, in which one layer is electron doped and one layer is hole doped, can be studied to lower temperatures and exhibits an interesting signal of ddd-wave pairing. The results of our simulations bear resemblance to a recent report on the magnetic and superconducting properties of Ba2Ca3Cu4O8F2{\mathrm{Ba}}_{2}{\mathrm{Ca}}_{3}{\mathrm{Cu}}_{4}{\mathrm{O}}_{8}{\mathrm{F}}_{2}Ba2​Ca3​Cu4​O8​F2​ which contains both electron and hole doped CuO2\mathrm{Cu}{\mathrm{O}}_{2}CuO2​ planes. We also explore the phase diagram of bilayer models in which each sheet is at half-filling.
Numerical evidence for superconductivity in the single-band Hubbard model is elusive or ambiguous despite extensive study, raising the question of whether the single-band Hubbard model is a faithful low-energy effective model for cuprates and whether explicitly including the oxygen ions will recover the properties necessary for a superconducting transition. Here we show, by using numerically exact determinant quantum Monte Carlo simulations of the doped Emery model, that while the single-band model exhibits strikingly TTT-linear resistivity, the three-band model crosses the resistivity of the single-band model from above, indicating a crossover to a more metallic transport regime. The enhanced conductivity is mainly contributed to by a steep increase in the diffusivity of the three-band model at the crossover, suggesting that three-band transport is more coherent than single-band transport at lower temperatures. Below the same crossover temperature in the three-band model, the pair-field susceptibility increases more steeply than at higher temperatures or when compared to the single-band model. This suggests a possible connection between superconductivity and coherent transport, and further implies that coherent transport might be necessary for a model to capture the high-temperature superconductivity observed in hole-doped cuprates.
The competition between ddd-wave superconductivity (SC) and antiferromagnetism (AF) in the high-Tc{T}_{c}Tc​ cuprates is investigated by studying the hole- and electron-doped two-dimensional Hubbard model with a recently proposed variational quantum-cluster theory. The approach is shown to provide a thermodynamically consistent determination of the particle number, provided that an overall shift of the on-site energies is treated as a variational parameter. The consequences for the single-particle excitation spectra and for the phase diagram are explored. By comparing the single-particle spectra with quantum Monte Carlo and experimental data, we verify that the low-energy excitations in a strongly correlated electronic system are described appropriately. The cluster calculations also reproduce the overall ground-state phase diagram of the high-temperature superconductors. In particular, they include salient features such as the enhanced robustness of the antiferromagnetic state as a function of electron doping and the tendency towards phase separation into a mixed antiferromagnetic-superconducting phase at low doping and a pure superconducting phase at high (both hole and electron) doping.
We study the three-band Hubbard model for the copper oxide plane of the high-temperature superconducting cuprates using determinant quantum Monte Carlo and the dynamical cluster approximation (DCA) and provide a comprehensive view of the pairing correlations in this model using these methods. Specifically, we compute the pair-field susceptibility and study its dependence on temperature, doping, interaction strength, and charge-transfer energy. Using the DCA, we also solve the Bethe-Salpeter equation for the two-particle Green's function in the particle-particle channel to determine the transition temperature to the superconducting phase on smaller clusters. Our calculations reproduce many aspects of the cuprate phase diagram and indicate that there is an ``optimal'' value of the charge-transfer energy for the model where Tc{T}_{c}Tc​ is largest. These results have implications for our understanding of superconductivity in both the cuprates and other doped charge-transfer insulators.
Using dynamic cluster quantum Monte Carlo simulations, we study the superconducting behavior of a 1/8 doped two-dimensional Hubbard model with imposed unidirectional stripelike charge-density-wave modulation. We find a significant increase of the pairing correlations and critical temperature relative to the homogeneous system when the modulation length scale is sufficiently large. With a separable form of the irreducible particle-particle vertex, we show that optimized superconductivity is obtained for a moderate modulation strength due to a delicate balance between the modulation enhanced pairing interaction, and a concomitant suppression of the bare particle-particle excitations by a modulation reduction of the quasiparticle weight.