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Physical review. B./Physical review. B

Hybridization-induced quantum phase transition in the bilayer Hubbard model

Xun Liu, Mi Jiang

Published Jun 2, 2025• 4 citations
Publisher

Abstract

Inspired by the recent experimental report on the pressure-induced superconductor-insulator transition in cuprate superconductors as well as the superconductivity of the Ruddlesden-Popper-phase ${\mathrm{La}}_{3}{\mathrm{Ni}}_{2}{\mathrm{O}}_{7}$ under high pressure, we systematically investigated the single-orbital bilayer Hubbard model in the regime of large interlayer hybridization to mimic the pressure effects. We map out the phase diagram of interlayer hybridization versus density in the regime of intermediate to strong hybridization. In particular, we found that sufficiently strong hybridization can destroy the ${s}^{\ifmmode\pm\else\textpm\fi{}}$-wave pairing and induces its transition to correlated metallic, pseudogap, and Fermi liquid phases depending on the doping regime. The phase diagram hosted by the bilayer model implies its role as a versatile platform to explore the pressure effects on the two-dimensional to three-dimensional crossover physics of Hubbard-type models.

DOI
10.1103/physrevb.111.245103

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Inspired by the recent experimental report on the pressure-induced superconductor-insulator transition in cuprate superconductors as well as the superconductivity of the Ruddlesden-Popper-phase La3Ni2O7{\mathrm{La}}_{3}{\mathrm{Ni}}_{2}{\mathrm{O}}_{7}La3​Ni2​O7​ under high pressure, we systematically investigated the single-orbital bilayer Hubbard model in the regime of large interlayer hybridization to mimic the pressure effects. We map out the phase diagram of interlayer hybridization versus density in the regime of intermediate to strong hybridization. In particular, we found that sufficiently strong hybridization can destroy the s\ifmmode±\else\textpm\fi{s}^{\ifmmode\pm\else\textpm\fi{}}s\ifmmode±\else\textpm\fi-wave pairing and induces its transition to correlated metallic, pseudogap, and Fermi liquid phases depending on the doping regime. The phase diagram hosted by the bilayer model implies its role as a versatile platform to explore the pressure effects on the two-dimensional to three-dimensional crossover physics of Hubbard-type models.
Inspired by the recent experimental report on the pressure induced superconductor-insulator transition in cuprate superconductors as well as the superconductivity of the Ruddlesden-Popper-phase La3_33​Ni2_22​O7_77​ under high pressure, we systematically investigated the single-orbital bilayer Hubbard model in the regime of large interlayer hybridization to mimic the pressure effects. We map out the phase diagram of interlayer hybridization versus density in the regime of intermediate to strong hybridization. In particular, we found that the sufficiently strong hybridization can destroy the s±s^{\pm}s±-wave pairing and induces its transition to correlated metallic, pseudogap, and Fermi liquid phases depending on the doping regime. The phase diagram hosted by the bilayer model implies its role as the versatile platform to explore the pressure effects on the two-dimensional to three-dimensional crossover physics of Hubbard-type models.
The interaction-driven insulator-to-metal transition has been reported in the ionic Hubbard model (IHM) for moderate interaction UUU, while its metallic phase only occupies a narrow region in the phase diagram. To explore the enlargement of the metallic regime, we extend the ionic Hubbard model to two coupled layers and study the interplay of interlayer hybridization VVV and two types of intralayer staggered potentials ΔΔΔ: one with the same (in-phase) and the other with a πππ-phase shift (anti-phase) potential between layers. Our determinant Quantum Monte Carlo (DQMC) simulations at lowest accessible temperatures demonstrate that the interaction-driven metallic phase between Mott and band insulators expands in the Δ−VΔ-VΔ−V phase diagram of bilayer IHM only for in-phase ionic potentials; while anti-phase potential always induces an insulator with charge density order. This implies possible further extension of the ionic Hubbard model from the bilayer case here to realistic three-dimensional model.
The interaction-driven insulator-to-metal transition has been reported in the ionic Hubbard model (IHM) for moderate interaction UUU, while its metallic phase only occupies a narrow region in the phase diagram. To explore the enlargement of the metallic regime, we extend the ionic Hubbard model to two coupled layers and study the interplay of interlayer hybridization VVV and two types of intralayer staggered potentials $\mathrm{\ensuremath{\Delta}}$: one with the same (in-phase) and the other with a $\ensuremath{\pi}$-phase shift (antiphase) potential between layers. Our determinant quantum Monte Carlo (DQMC) simulations at lowest accessible temperatures demonstrate that the interaction-driven metallic phase between Mott and band insulators expands in the $\mathrm{\ensuremath{\Delta}}\text{\ensuremath{-}}V$ phase diagram of bilayer IHM only for in-phase ionic potentials; while antiphase potential always induces an insulator with charge density order. This implies possible further extension of the ionic Hubbard model from the bilayer case here to a realistic three-dimensional model.
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It has been proposed that the superconducting transition temperature Tc{T}_{\text{c}}Tc​ of an unconventional superconductor with a large pairing scale but strong phase fluctuations can be enhanced by coupling it to a metal. However, the general efficacy of this approach across different parameter regimes remains an open question. Using the dynamical cluster approximation, we study this question in a system composed of an attractive Hubbard layer in the intermediate coupling regime, where the magnitude of the attractive Coulomb interaction ∣U∣|U|∣U∣ is slightly larger than the bandwidth WWW, hybridized with a noninteracting metallic layer. We find that while the superconducting transition becomes more mean-field-like with increasing interlayer hopping, the superconducting transition temperature Tc{T}_{\text{c}}Tc​ exhibits a nonmonotonic dependence on the strength of the hybridization ${t}_{\ensuremath{\perp}}$. This behavior arises from a reduction of the effective pairing interaction in the correlated layer that outcompetes the growth in the intrinsic pair-field susceptibility induced by the coupling to the metallic layer. We find that the largest Tc{T}_{\text{c}}Tc​ inferred here for the composite system is comparable to the maximum value currently estimated for the isolated negative-UUU Hubbard model.
Evolution from intralayer to interlayer superconductivity in a bilayer ttt-JJJ model
Motivated by the bilayer cuprate superconductors and nickelate superconductor La3_33​Ni2_22​O7_77​, we investigate the evolution from intralayer to interlayer superconductivity based on a bilayer two-leg ttt-JJJ-J⊥J_{\bot}J⊥​ model, where ttt is the in-plane electron hopping, JJJ is the in-plane spin interaction, and J⊥J_{\bot}J⊥​ is the inter-plane spin interaction. By means of the density matrix renormalization group calculations, we obtain the quantum phase diagram of the system by tuning J⊥J_{\bot}J⊥​ in a large doping range δ=1/8−1/2δ= 1/8 - 1/2δ=1/8−1/2. We find that a large J⊥J_{\bot}J⊥​ can always drive an interlayer superconductivity by coupling the two layers in both the Luther-Emery liquid and Luttinger liquid states. By coupling two Luther-Emery liquid states, the in-plane superconductivity evolves to inter-plane superconductivity either through an intermediate charge density wave (CDW) phase or directly, depending on doping ratio. This emergent CDW phase, which exists over a finite doping range, appears to develop from the CDW state of the two-leg ladder at δ=1/4δ= 1/4δ=1/4. By coupling two Luttinger liquids, the in-plane Luttinger liquids show a transition to the inter-plane superconducting phase at large J⊥J_{\bot}J⊥​, as reported in previous literature. Interestingly, in the intermediate J⊥J_{\bot}J⊥​ regime we find that while the in-plane Luttinger-liquid features remain stable, the inter-plane superconductivity can develop an enhanced quasi-long-range order with the power exponent KSCzz∼1K^{zz}_{\rm SC} \sim 1KSCzz​∼1. At last, we show that the interlayer superconductivity is also stable by coupling the bilayer three-leg ttt-JJJ ladders by a strong J⊥J_{\bot}J⊥​ interaction, from both the Luther-Emery liquid and Luttinger-liquid states.