Competing Interlayer Loop Currents and Superconductivity in the Bilayer t-J_⊥-V Model

Summary

This paper employs a controlled large-N expansion based on the Hubbard-operator path integral to study the strongly correlated single-orbital d_x²-y² bilayer t-J⊥-V model at the mean-field and beyond-mean-field levels, aiming to characterize pressurized and thin-film bilayer nickelates. Focusing on the interlayer spin exchange J⊥, we find that it simultaneously drives out-of-plane s-wave superconductivity and an out-of-plane bond-order phase z-BOP instability; the self-consistent solution for the complex z-BOP gap is dominated by its imaginary part, which endows the interlayer hopping with a complex phase, thereby generating an interlayer current and inducing an in-plane current, and spontaneously forms a time-reversal-symmetry-breaking loop-current state on the vertical plaquette of the bilayer. Competition between the loop-current phase and superconductivity leads to a doping-dependent dome-shaped superconducting region, with optimal superconductivity appearing near the z-BOP quantum critical point. The phase diagram includes a pure loop-current region, a low-doping coexistence phase of loop current and superconductivity, a higher-doping pure superconducting state, and a correlated metallic state; the Coulomb interaction plays an unimportant role in the formation and stability of the loop current. We conclude that these strongly coupled single-orbital results may provide insights into nickelate physics and into many-body models with spontaneous loop currents that can be quantum simulated on platforms such as optical lattices and superconducting quantum circuits.

Materials

• pressurized bilayer nickelates
• thin-film bilayer nickelates
• one-orbital d_x²-y² bilayer t-J⊥-V model

Methods

• controlled large-N expansion
• Hubbard-operator path integral
• mean-field calculations
• beyond-mean-field calculations
• self-consistent gap equations
• susceptibility calculations
• Chern number calculation

Keywords

• out of plane s wave superconductivity
• out of plane bond order phase (z bop)
• loop current state
• time reversal symmetry breaking
• dome shaped superconducting region
• quantum critical point
• coexistence of loop currents and superconductivity
• correlated metallic state
• interlayer spin exchange j⊥
• coulomb interaction
• topologically protected surface states
• quantum simulation

Highlights

• The imaginary z-BOP channel is the leading instability; the real and imaginary instabilities are nearly degenerate, but the self-consistent solution is purely imaginary with negligible real component.
• The magnetic moment of a single loop is a few times the Bohr magneton, decreases with doping, and vanishes at the quantum critical point.
• The absence of topological surface states is structural rather than numerical because the interlayer pairing carries no momentum dependence and the state is invariant under the product of time reversal and a one-site translation.
• Unlike previous unconstrained t-J⊥-V fermionic Hamiltonians where Coulomb interaction is required to induce loop currents near half-filling, in this framework strong local correlations make additional Coulomb repulsion irrelevant for loop-current formation.
• The model provides a concrete platform for quantum simulation of spontaneous loop currents and their competition with interlayer pairing on ultracold atoms in optical lattices and superconducting quantum circuits.
• The strong-coupling single-orbital results may provide insights into nickelate physics and into many-body models with spontaneous loop currents.

Conclusions

• The out-of-plane spin exchange interaction J⊥ triggers both out-of-plane s-wave superconductivity and an out-of-plane bond-order phase (z-BOP) instability.
• The competition between the real and imaginary components of the z-BOP gap selects the imaginary channel, which imprints a complex interlayer hopping that drives out-of-plane currents and induces in-plane currents, spontaneously forming a time-reversal-symmetry-breaking loop-current state on the vertical plaquettes.
• The competition between the loop-current phase and superconductivity yields a dome-shaped superconducting region as a function of doping, with optimal superconductivity occurring near the z-BOP quantum critical point.
• The resulting phase diagram features a pure loop-current region, a low-doping coexistence phase of loop currents and superconductivity, a pure superconducting state at higher doping, and a correlated metallic state.
• The Coulomb interaction is irrelevant for the formation and stability of the loop current due to screening driven by the local no-double-occupancy constraint.
• The loop-current superconducting state does not support topologically protected surface states or a zero-bias peak; no states appear inside the gap and the Chern number vanishes over the coexistence region.

Main claims

• The interlayer spin exchange J_perp simultaneously triggers out-of-plane s-wave superconductivity and an out-of-plane bond-order phase z-BOP instability.
  • Evidence: Abstract states that focusing on the out-of-plane spin exchange interaction J_perp, we find that it triggers both out-of-plane s-wave superconductivity and an out-of-plane bond-order phase z-BOP instability.,Full text states that out-of-plane s-wave superconductivity is triggered by J_perp and that an out-of-plane bond-order phase z-BOP instability appears below a critical temperature.
• The z-BOP gap is dominated by its imaginary component, which makes the z-axis hopping complex and drives interlayer and in-plane currents, spontaneously forming a time-reversal-symmetry-breaking loop-current state on vertical plaquettes.
  • Evidence: Abstract states that the z-BOP gives rise to a complex z-axis hopping dominated by its imaginary component, which drives out-of-plane currents and induces in-plane ones, spontaneously forming on the vertical plaquettes a loop-current state that breaks time-reversal symmetry.,Full text states that the self-consistent solution is purely imaginary and that the imaginary gap imprints a sublattice-staggered Peierls phase on the interlayer hopping, producing staggered interlayer currents and loop currents; this spontaneous interlayer LC state breaks time-reversal symmetry.
• Competition between the loop-current phase and superconductivity produces a dome-shaped superconducting region, with optimal superconductivity near the z-BOP quantum critical point.
  • Evidence: Abstract states that competition between this loop-current phase and superconductivity yields a dome-shaped superconducting region, with optimal superconductivity occurring near the z-BOP quantum critical point.,Full text states that solving the coupled LC and superconducting gap equations gives a dome behavior for SC with optimal doping close to the QCP.
• The resulting phase diagram contains a pure loop-current region, a low-doping coexistence phase, a pure superconducting state at higher doping, and a correlated metallic state.
  • Evidence: Abstract states that the resulting phase diagram features a pure loop-current region, a low-doping coexistence phase, a pure superconducting state at higher doping, and a correlated metallic state.,Full text describes a broad region where only LCs exist, a coexistence of LCs and SC at low temperature below the QCP, a region of pure SC above the QCP, and a correlated metallic state at large doping and high temperature.
• The Coulomb interaction is irrelevant for the formation and stability of the loop current.
  • Evidence: Summary states that the Coulomb interaction plays an unimportant role in the formation and stability of the loop current.,Full text states that because the Coulomb terms V and V_perp belong to the charge sector, the results show that the Coulomb interaction is irrelevant for the formation and stability of the z-BOP and the corresponding LCs.
• The single-orbital strong-coupling results may provide insights into nickelate physics and into many-body models with spontaneous loop currents that can be quantum simulated on platforms such as optical lattices and superconducting quantum circuits.
  • Evidence: Summary states that these strongly coupled single-orbital results may provide insights into nickelate physics and into many-body models with spontaneous loop currents that can be quantum simulated on platforms such as optical lattices and superconducting quantum circuits.,Full text conclusion states that the model is well-suited for quantum simulation on platforms such as ultracold atoms in optical lattices and superconducting circuits.
• The loop-current superconducting state does not support topologically protected surface states in the coexistence region.
  • Evidence: Full text states that no states appear inside the gap and the Chern number vanishes over the whole coexistence region; the absence is structural because the interlayer pairing carries no momentum dependence and the state remains invariant under time reversal times a one-site translation.

Workflow

• model_definition — The bilayer t-J_perp-V model provides a strongly correlated one-orbital framework for bilayer nickelates.
  • Materials: one-orbital d_x²-y² bilayer t-J_perp-V model; Hubbard operators; no-double-occupancy constraint; six-component bosonic field
  • Methods: strong-coupling Hamiltonian construction; path-integral representation for Hubbard operators
  • Observations: model is mixed-dimensional; local constraint handled by large-N expansion
• large_N_expansion — Large-N expansion reveals instabilities from interlayer spin exchange and shows Coulomb terms are irrelevant for z-BOP and loop currents.
  • Materials: six-component bosonic field; charge sector; spin sector; susceptibility
  • Methods: controlled large-N expansion; mean-field and beyond-mean-field calculations
  • Observations: instabilities from full propagator match those from projection onto spin sector; dual structure decouples local charge excitations from the spin sector
• gap_equation_solution — The imaginary z-BOP component dominates and generates a time-reversal-symmetry-breaking loop-current state.
  • Materials: complex z-BOP gap; superconducting gap; spinor; interlayer and in-plane currents
  • Methods: solve gap equations; self-consistent calculations
  • Observations: real and imaginary z-BOP channels nearly degenerate; imaginary channel selected; in-plane current induced; loop currents thread vertical plaquettes
• phase_diagram_analysis — Competition between loop currents and superconductivity yields a dome-shaped superconducting region with optimal superconductivity near the z-BOP quantum critical point.
  • Materials: coupled LC and superconducting gaps; critical temperatures; phase diagram
  • Methods: solve coupled gap equations; compare critical temperatures and gaps
  • Observations: SC dome with optimal doping near z-BOP QCP; regions: pure LC, low-doping coexistence, pure SC, correlated metal; BCS-like SC gap and BCS ratio
• interpretation — The results provide insights into nickelate physics and propose the model for quantum simulation, while topological surface states are absent in the coexistence region.
  • Materials: phase diagram; topological surface state check; Chern number
  • Methods: interpretation of model results; comparison with nickelate experiments and other theoretical studies; quantum simulation proposal
  • Observations: no topologically protected surface states in LCSC state; no zero-bias peak from loop currents; LC state may relate to TRS-breaking phase in nickelates