Interplay between Spin Stripes and Pairing Crossover in the Bilayer Hubbard Model

Summary

Motivated by the unresolved question of whether d-wave or s±-wave pairing dominates in the bilayer nickelate La₃Ni₂O₇ and by the unclear connection between its spin order, magnetic excitations, and pairing mechanism, the authors use finite-temperature determinant quantum Monte Carlo simulations of a single-orbital bilayer Hubbard model to study pairing competition under conditions of relatively strong interlayer coupling and comparable intra- and interlayer antiferromagnetic correlation strengths. The results show that with increasing hole doping, spin stripes form spontaneously, accompanied by a crossover of the dominant pairing symmetry from d-wave to s±-wave; this crossover tracks changes in the relative strengths of the two types of spin correlation, with interlayer correlations exceeding intralayer correlations in magnitude. The odd channel of the dynamical spin structure factor further exhibits momentum-dependent reorganization of magnetic excitations, consistent with a fluctuating spin-stripe picture. On this basis, the authors argue that the spin-fluctuation structure is a key link connecting spin stripes and pairing competition and offers a minimal-model perspective for understanding La₃Ni₂O₇; this is a prediction from model calculations rather than an established experimental mechanism.

Materials

• single-orbital bilayer Hubbard model
• La₃Ni₂O₇

Methods

• determinant quantum Monte Carlo (DQMC) simulations
• maximum entropy method for analytic continuation
• pairing susceptibility calculations
• spin susceptibility calculations
• dynamical spin structure factor calculations

Keywords

• spin stripes
• pairing crossover
• d wave pairing
• s± wave pairing
• antiferromagnetic correlations
• spin fluctuations
• magnetic excitations
• hole doping
• interlayer coupling
• fluctuating spin stripes

Highlights

• Finite-temperature DQMC simulations of a minimal single-orbital bilayer Hubbard model are used to study pairing competition at strong interlayer coupling where intralayer and interlayer antiferromagnetic correlations have comparable strengths.
• Real-space staggered spin susceptibility provides clear evidence of antiphase domains and spin stripes emerging within the doping interval where the pairing crossover occurs.
• The odd channel of the dynamical spin structure factor shows momentum-dependent hardening/softening and intensity changes of magnetic excitations with hole doping.
• The results are presented as a minimal-model prediction rather than an established experimental mechanism, and a complete understanding of La₃Ni₂O₇ is noted to require multiorbital physics and structural effects beyond the minimal model.

Conclusions

• Spin stripes develop spontaneously with increasing hole doping, accompanied by a crossover of the dominant pairing symmetry from d-wave to s±-wave.
• The pairing crossover tracks the changing relative strengths of the spin correlations, with interlayer correlations overtaking intralayer counterparts in magnitude.
• The dynamical spin structure factor exhibits a momentum-dependent reorganization of the magnetic excitations, consistent with fluctuating spin stripes.
• The structure of spin fluctuations is identified as a key link between spin stripes and pairing competition, offering a minimal-model perspective for understanding La₃Ni₂O₇.

Main claims

• Spin stripes develop spontaneously with increasing hole doping in the strongly coupled bilayer Hubbard model.
  • Evidence: Real-space staggered spin susceptibility shows antiphase domains at intermediate doping and multiple coexisting domain walls at higher doping,Spin susceptibility peak around (pi,pi) broadens and develops a shallow double-peak structure at higher doping
• The dominant pairing symmetry crosses over from d-wave to s±-wave with increasing hole doping at t_perp = 1.0.
  • Evidence: Effective pairing interaction is most attractive in d-wave at low doping and in s±-wave at higher doping,Extended s-wave pairing remains subleading throughout the parameter range
• The pairing crossover tracks the changing relative strengths of intralayer and interlayer antiferromagnetic correlations, with interlayer correlations overtaking intralayer ones.
  • Evidence: Reversal of relative strengths of intralayer and interlayer antiferromagnetic correlations accompanies the d-wave to s±-wave switch with increasing interlayer hopping,Interlayer correlations exceed intralayer correlations in magnitude within the crossover doping interval
• The dynamical spin structure factor shows a momentum-dependent reorganization of magnetic excitations consistent with fluctuating spin stripes.
  • Evidence: Maximum entropy analytic continuation of DQMC imaginary-time spin susceptibility in the odd channel,Doping-induced hardening/softening and intensity changes of spectral peaks at different momenta
• The structure of spin fluctuations is a key link between spin stripes and pairing competition, providing a minimal-model perspective for La₃Ni₂O₇.
  • Evidence: Coincidence of the doping range of stripe emergence with the pairing and magnetic crossover regime,Qualitative consistency with finite-temperature single-layer Hubbard model results and with neutron-scattering and RIXS observations of stripe magnetic order in La₃Ni₂O₇

Workflow

• model_definition — A minimal single-orbital bilayer Hubbard model retaining local Coulomb interaction and interlayer hybridization captures spin fluctuations and pairing competition in bilayer systems.
  • Materials: single-orbital bilayer Hubbard model; intralayer nearest-neighbor hopping; intralayer next-nearest-neighbor hopping; interlayer hopping; on-site Coulomb interaction; chemical potential
  • Methods: Hamiltonian construction; definition of pairing order parameters for d-wave, extended s-wave, and interlayer s±-wave; definition of pairing susceptibility and pairing vertex
• simulation — DQMC at finite temperature on 2x4 bilayer clusters yields controlled results for spin correlations and pairing susceptibilities in the target coupling regime.
  • Materials: 2x4 bilayer clusters with periodic boundary conditions within each layer
  • Methods: finite-temperature determinant quantum Monte Carlo (DQMC) simulations; 20-50 independently seeded Markov chains with equilibration and measurement sweeps grouped into 5-10 bins; Trotter step chosen so Trotter errors are below statistical uncertainties
  • Observations: fermion-sign problem constrains accessible cluster sizes
• measurement — Spin stripes develop spontaneously with increasing hole doping, accompanied by a crossover of the dominant pairing symmetry from d-wave to s±-wave that tracks the changing relative strengths of intralayer and interlayer spin correlations.
  • Methods: pairing susceptibility per symmetry; uncorrelated pair-field susceptibility and pairing vertex; real-space spin susceptibility and its Fourier transform; nearest-neighbor intralayer and interlayer spin correlations; staggered spin susceptibility
  • Observations: effective pairing interaction is attractive for all symmetries considered and its magnitude decreases monotonically with hole doping; dominant pairing switches from d-wave to s±-wave with increasing interlayer hopping; at t_perp = 1.0 the dominant pairing changes with hole doping; interlayer correlations overtake intralayer correlations in magnitude; antiphase domains appear in real-space staggered spin susceptibility at intermediate doping and domain-wall spacing shrinks with doping
• spectral_analysis — The dynamical spin structure factor exhibits a momentum-dependent reorganization of magnetic excitations consistent with fluctuating spin stripes.
  • Methods: odd channel of the dynamical spin structure factor; analytic continuation of DQMC imaginary-time spin susceptibility via the maximum entropy method
  • Observations: main spectral peak shifts to higher energy and loses intensity with doping; peaks at neighboring momenta harden and lose intensity; peaks at other momenta soften, narrow, and gain intensity, with softening more pronounced at higher doping
• interpretation — The structure of spin fluctuations is a key link between spin stripes and pairing competition, offering a minimal-model baseline for interpreting magnetic excitations of bilayer nickelates.
  • Methods: comparison with finite-temperature single-layer Hubbard model studies; comparison with neutron-scattering and RIXS experiments on La₃Ni₂O₇
  • Observations: doping range of stripe emergence coincides with the pairing and magnetic crossover regime