Bilayer t-J⊥-V Model in Nickelate Superconductors

• The bilayer t-J⊥-V model is a theoretical framework that characterizes strongly correlated bilayer systems by combining intralayer hopping, interlayer magnetic exchange, and Coulomb repulsion into a unified Hamiltonian.
• It demonstrates that strong interlayer exchange (J⊥) and constrained kinetic processes drive robust s-wave pairing and a parabolic superconducting dome that contrasts with conventional cuprate behavior.
• The model also predicts competing orders, including charge-density waves and magnetic stripe phases, offering critical insights into phase transitions in pressurized nickelate superconductors.

The bilayer $t$–$J_{\perp}$–$V$ model is a theoretical framework designed to describe strongly correlated bilayer systems, such as pressurized nickelate superconductors (e.g., La$_3$Ni$_2$O$_7$), with a particular focus on the interplay between kinetic energy, magnetic exchange, and interlayer interactions. Its relevance encompasses both the elucidation of unconventional superconductivity mechanisms and the detailed paper of density and magnetic ordering phenomena in complex oxides and cold atom lattices. The model unifies themes from the two-orbital $t$–$J$ physics, bilayer Kondo-lattice limits, and constrained Hilbert space approaches that highlight the role of interlayer magnetic exchange and interlayer repulsion in driving, suppressing, or transforming collective electronic states.

1. Model Definition and Structure

The bilayer $t$–$J_{\perp}$–$V$ model describes a system of electrons on a square-lattice bilayer, subject to both in-plane (intralayer) and out-of-plane (interlayer) couplings. The minimal Hamiltonian in the form applicable to both nickelates and cold atom realizations is

$$H = -t\sum_{\ell, \langle ij \rangle, \sigma} P\, c_{\ell i\sigma}^\dagger c_{\ell j\sigma}\,P + J_{\perp} \sum_i S_{1i}\cdot S_{2i} + V \sum_i n_{1i}n_{2i}$$

where

• $t$ is the intralayer hopping amplitude,
• $J_{\perp}$ is the interlayer (rung) antiferromagnetic exchange coupling,
• $V$ is the interlayer Coulomb repulsion,
• $P$ is the projector onto the subspace with no double occupancy (or, in derived models, onto an “Empty–Singlon–Doublon” Hilbert space),
• $c_{\ell i\sigma}^\dagger$ creates an electron of spin $\sigma$ at site $i$ in layer $\ell$,
• $S_{\ell i}$ and $n_{\ell i}$ are the spin and number operators at site $i$ in layer $\ell$.

Extensions relevant for $d$-orbital nickelates may augment this with multiorbital couplings, Hund’s rule terms, and orbital-dependent $t$ or $J$, and more detailed models integrate oxygen $p$-orbital physics or nearest-neighbor in-plane exchange ($J_\parallel$) and three-site terms (Gu et al., 2023, Kaneko et al., 14 Apr 2025).

A reduced form, derived in the strong-coupling (double Kondo-lattice) and strong $J_\perp$ limit, is the so-called ESD (Empty, Singlon, Doublon) model: $$\hat{H}_\text{ESD} = -t \sum_{\ell, \langle ij \rangle, \sigma} P c_{\ell i\sigma}^\dagger c_{\ell j\sigma} P + \epsilon\sum_i (n_{h,i} + n_{d,i})$$ where only the nearest-neighbor kinetic term over the projected Hilbert space remains and $\epsilon$ is an on-site energy dependent on $J_\perp$ and $V$ (Oh et al., 11 Nov 2024).

2. Physical Regimes and Microscopic Justification

The model is motivated by the electronic structure of systems such as La$_3$Ni$_2$O$_7$, where two active Ni $e_g$ orbitals ($d_{x^2-y^2}$ and $d_{z^2}$) exist per site in a bilayer geometry. DFT studies (Gu et al., 2023, Kaneko et al., 14 Apr 2025) and Schrieffer–Wolff analysis (Kaneko et al., 14 Apr 2025) show that:

• $d_{z^2}$ orbitals are strongly coupled across layers via large interlayer hybridization ($t_{\perp}^{aa}$ up to 0.664 eV) and antiferromagnetic superexchange, forming “rung singlets” or acting as localized $S=1$ spins in the large Hund's coupling limit.
• $d_{x^2-y^2}$ orbitals are responsible for mobile carriers, with weak direct interlayer hopping but sizeable induced $J_\perp$ through Hund’s coupling.

In the strong Hund’s coupling regime, a local high-spin state is stabilized, giving rise to a “type-II $t$–$J$” model with a Hilbert space spanned by spin-1 triplets and spin-½ “singlons.” The effective interlayer exchange “shared” via Hund’s coupling transmits the strong magnetic correlations from the $d_{z^2}$ to the $d_{x^2-y^2}$ channel, even when the latter does not participate directly in interlayer hopping (Oh et al., 2023, Oh et al., 2 Sep 2025). Charge-transfer scenarios further generalize the model to include Zhang–Rice spin-½ states formed with ligand holes on oxygen $p$ orbitals rather than on-site Ni $d$-orbitals, providing a physical justification for suppression of direct interlayer hopping and emergence of specific effective Hilbert space constraints (Oh et al., 30 Apr 2024).

3. Superconductivity and Pairing Mechanisms

The $t$–$J_{\perp}$–$V$ model (and descendants) realizes robust superconductivity, particularly under large $J_{\perp}$ and negligible interlayer hopping ($t_\perp \approx 0$):

• Interlayer $s$-wave pairing: Strong $J_\perp$ directly favors formation of rung singlets (spin singlets across two layers), leading to an $s$-wave superconducting order parameter

$$\Delta_z = \frac{1}{\sqrt{2}} \langle c_{i,1,\uparrow}^\dagger c_{i,2,\downarrow}^\dagger + c_{i,2,\uparrow}^\dagger c_{i,1,\downarrow}^\dagger \rangle$$

which is uniform across the plane (Qu et al., 2023, Chen et al., 2023).

• Kinetic-energy-driven pairing: In the projected Hilbert space, kinetic processes (“hopping resonance”) between empty, singly, and doubly occupied states can generate pairing even in the absence of explicit attractive interactions; the superconducting pairing gap and phase stiffness both increase monotonically with doping, a behavior in sharp contrast with the pseudogap dome of cuprates (Oh et al., 11 Nov 2024).
• Induced and orbital-selective mechanisms: In multi-orbital and charge-transfer regimes, density fluctuations or Schrieffer-Wolff–generated effective interactions can induce attractive intralayer pairing even without bare on-site attraction (Vanhala et al., 2014); orbital-selective models predict much stronger SC order for mobile ($d_{x^2-y^2}$ or in-plane ligand) channels than for those with significant interlayer hopping (where Pauli blocking suppresses coherent pairing) (Chen et al., 2023).

Simulations (DMRG, iPEPS, tanTRG) yield a robust zero-temperature order parameter ($\bar\Delta_z \sim 0.1$ for $J_\perp = 2t$, much larger than the single-layer value) and a parabolic “superconducting dome” versus doping, with optimal doping distinctly higher ($x \approx 0.4-0.5$) than that in cuprates (Qu et al., 2023, Oh et al., 30 Apr 2024). Phase stiffness calculations predict the possibility of transition temperatures as high as $T_c \sim 0.5t$, suggesting room-temperature superconductivity is plausible if materials with sufficiently large $J_\perp/t$ can be engineered (Oh et al., 11 Nov 2024).

4. Competing and Coexisting Orders

The model supports rich phase behavior depending on parameters and external control (e.g., doping $x$, chemical potential $\mu$, interlayer polarization $\epsilon$, or pressure):

• Density order: Attractive interlayer interactions ($V$) in extended Hubbard or $t$–$J$–$V$ variants can stabilize checkerboard density (CDW) phases characterized by $$D = \frac{1}{N}|\sum_{\sigma,\ell,i} \text{sgn}(i) \langle n_{\sigma\ell i}\rangle|$$ near half-filling (Vanhala et al., 2014).
• Magnetism and stripes: At weaker $J_\perp$ and in the presence of a third-neighbor in-plane exchange $J_2$, the system exhibits antiferromagnetic (AFM) or double spin-stripe phases that can coexist with superconductivity. Increasing $t_\perp$ or $J_\perp$ leads to a transition from $d$-wave pairing and AFM order to $s$-wave pairing coexisting with double stripe magnetism (Tian et al., 23 Dec 2024).
• Flux phases: Time-reversal symmetry breaking flux phases, featuring staggered loop currents and split zero-bias conductance peaks, are stabilized at surfaces by competition between suppressed superconductivity and interlayer couplings. In bilayer systems with both $J_\perp$ and $V$, novel flux patterns (type B: opposite direction in two layers) arise, which can suppress the observability of spontaneous magnetic fields at the macroscopic level while retaining TRS violation (Kuboki, 2015).
• Normal and pseudogap metals: The kinetic regime at high $J_\perp$ supports an unusual symmetric pseudogap phase (or “second Fermi liquid”) in the underdoped regime, with a Fermi surface that partially violates the conventional Luttinger relation (Oh et al., 2 Sep 2025).

First-order phase transitions separate density-ordered, superconducting (interlayer or intralayer dominated), and normal states (Vanhala et al., 2014). Pressure can tune the system between optimal pairing regimes (e.g., favoring in-plane oxygen orbitals and high $T_c$ at moderate pressure, then suppressing pairing once $p_z$-like ligand orbitals become active at higher pressure) (Oh et al., 30 Apr 2024).

5. Derivation, Limitations, and Extensions

The $t$–$J_{\perp}$–$V$ model has been microscopically derived from two-orbital Hubbard Hamiltonians using Schrieffer–Wolff transformation (Kaneko et al., 14 Apr 2025). Key results include:

• Exchange parameters $J^{(\mu\nu)}_{R,R'}$ expressed in terms of hopping integrals and on-site/interorbital Coulomb energies; quantitative values for La$_3$Ni$_2$O$_7$ (e.g., $t_\perp^{aa} \approx 0.664$ eV, $t_\parallel^{bb} \approx 0.491$ eV) permit direct comparison to experiment.
• In strongly correlated regimes, local states per site are reduced to a minimal set—four spin-½ (“singlon” $d^7$) and three spin-1 (“doublon” $d^8$) states for type-II $t$–$J$ models (Oh et al., 2023), or the ESD basis for effective projected kinetic models (Oh et al., 11 Nov 2024, Oh et al., 2 Sep 2025).

Limitations include the validity of projections to a restricted Hilbert space (requiring large $J_\perp$ and Hund’s coupling), neglect of possible weak interlayer hybridization for $d_{x^2-y^2}$, and accurate description of magnetic order at pressures or dopings far from the ideal limit. Incorporation of explicit oxygen physics is essential for capturing charge-transfer regimes and high-doping/pressure phases (Oh et al., 30 Apr 2024, Kaneko et al., 14 Apr 2025).

Extensions under current investigation involve trilayer analogues for La$_4$Ni$_3$O$_{10}$, rare-earth substitution to further enhance $J_\perp$ and $t_\parallel$, and application of the model to cold atom bilayers, where interlayer spin exchange and kinetic constraints can be readily tuned (Chen et al., 2023, Qu et al., 2023).

6. Experimental Relevance and Future Directions

The bilayer $t$–$J_{\perp}$–$V$ model provides a detailed quantitative and qualitative framework for understanding pressurized La$_3$Ni$_2$O$_7$ and related nickelate superconductors. The predicted robust $s$-wave interlayer pairing, parabolic doping dependence of the pairing gap, and suppression of AFM order with enhanced $J_\perp$ are all consistent with experimental observations (Qu et al., 2023, Chen et al., 2023, Oh et al., 30 Apr 2024). The model further accounts for the lack of surface spontaneous fields in cuprates (due to cancellation from staggered flux patterns), charge-density order at near half-filling, and the sharply different optimal doping compared to cuprates.

Promising directions include:

• Material engineering to increase $J_\perp/t$ (e.g., reducing out-of-plane lattice constants) for higher $T_c$ (Oh et al., 11 Nov 2024, Oh et al., 2 Sep 2025).
• Precision ARPES and STM studies to further test predictions about Fermi surface topology, gap structure, and symmetry.
• Cold atom experiments to simulate kinetic-constraint driven pairing in bilayer optical lattices with tunable exchange and hopping.
• Investigation of charge-transfer regime effects and potential generalization to trilayer and multilayer systems.

Ongoing studies aim to chart the full phase diagram, exploring interplay between superconductivity, magnetism, charge order, and non-Fermi-liquid metallic states in strongly correlated bilayer materials.