Finite-temperature superconducting transition of the two-dimensional mixed-dimensional bilayer t–J model: BKT critical temperature and pairing signatures with interlayer density coupling

The mixed-dimensional (mixD) bilayer t–J model — two fermionic layers with intralayer hopping and exchange but no interlayer single-particle tunneling, coupled by an interlayer spin exchange $J_\perp$ and a tunable interlayer density coupling $V$ — is the strongly correlated model for which current optical-lattice quantum simulators have the most realistic path to a superconducting state, and it has been proposed as an effective description of pressurized bilayer nickelate superconductors. Its finite-temperature physics in two dimensions is, however, known only through one-dimensional (ladder) density-matrix-renormalization-group evidence and a perturbative strong-coupling estimate $T_c\sim J_\perp/2$ for the Berezinskii–Kosterlitz–Thouless (BKT) transition. The open problem is the quantitative determination of the two-dimensional finite-temperature superconducting phase: the BKT critical temperature as a function of $J_\perp/t_\parallel$ , doping, and $V$ ; the finite-temperature pairing correlation signatures observable in experiments; the critical interlayer repulsion beyond which quasi-long-range order is destroyed; and the observability of this phase in cold-atom experiments at realistic temperatures. Ground-state neural-quantum-state simulations of the 2D mixD bilayer (2026) and mean-field-plus-DMRG critical-temperature estimates for anisotropic mixD arrays (2026) have advanced the zero-temperature and quasi-one-dimensional sides, but the isotropic 2D finite-temperature question with nonzero $V$ remains open, and no experiment has yet observed finite-temperature pairing in a 2D mixD bilayer.

Background

One of the central goals of analog quantum simulation is to realize high-temperature superconductivity in a controlled model system. Fermionic ultracold atoms can be trapped in an optical lattice — a periodic light-field potential whose wells play the role of crystal sites — and made to implement the Fermi–Hubbard model, in which particles hop between neighboring sites with amplitude $t$ and two atoms on the same site repel with energy $U$ . In the strongly correlated limit $U\gg t$ , states with double occupancy are projected out and the system is described by the t–J model: doped carriers (holes) move with hopping $t$ while the remaining singly occupied sites interact via an antiferromagnetic superexchange coupling $J\sim t^2/U$ . Whether the two-dimensional (2D) t–J or Hubbard model supports $d$ -wave superconductivity, and at what temperature, remains one of the most debated open questions of strongly correlated physics, and finite-temperature numerics for doped 2D fermion systems are extremely demanding.

A mixed-dimensional (mixD) bilayer offers a route with much higher energy scales. It consists of two coupled layers with no single-particle tunneling between them ( $t_\perp=0$ ): intralayer hopping $t_\parallel$ and superexchange $J_\parallel$ act within the layers, while the layers are coupled by an interlayer spin-exchange (Kondo) coupling $J_\perp$ and, in the cold-atom realization, a tunable interlayer density-density coupling $V$ . The interlayer exchange binds dopants sitting on top of each other into interlayer singlet pairs, and because single particles cannot hop between layers while the pairs remain mobile within the doped layer, pairing tendencies are strongly enhanced. This geometry was proposed as an effective low-energy description of the pressurized bilayer nickelate superconductor La $_3$ Ni $_2$ O $_7$ , where a large interlayer exchange is transferred through Hund's-rule coupling. Bohrdt, Homeier, Bloch, Demler and Grusdt (Nature Physics 18, 651 (2022)) showed that the mixD bilayer Mott insulator realizes an exceptionally strong pairing mechanism (string-based binding), and Lange, Homeier, Demler, Schollwöck, Grusdt and Bohrdt (Physical Review B 109, 045127 (2024)) found a pairing dome controlled by an emergent Feshbach resonance (a resonance between scattering particles and a bound pair state, which lets an attractive interaction appear effectively tunable) in ladder studies, including a pair charge gap at commensurate doping for sufficiently repulsive $V$ .

Schlömer, Lange, Franz, Chalopin, Bojović, Wang, Bloch, Hilker, Grusdt and Bohrdt (PRX Quantum 5, 040341 (2024)) proposed an optical-lattice implementation of the 2D mixD bilayer t–J model with local control: one layer is hole-doped and the other doublon-doped (a doublon is a site occupied by two fermions of opposite spin), which after a partial particle-hole transformation realizes the hole-doped mixD t–J model with a tunable interlayer density coupling $V$ ; in this transformed basis, superconducting pair-pair correlations become measurable as density-density correlations. Its numerical evidence comes from DMRG (density matrix renormalization group — a tensor-network method yielding essentially exact results for one-dimensional and quasi-one-dimensional systems) on ladders, where any nonzero temperature eventually destroys quasi-long-range order because one-dimensional correlations decay exponentially at finite $T$ : pairing correlations decay algebraically, with Luttinger parameter $K\approx1$ (the dimensionless exponent parameter setting the power-law decay of correlations in one dimension) over a broad range of $V/t_\parallel$ , with exponential decay setting in only above $V_c\approx2.5t_\parallel\approx1.7J_\perp$ (for the experimentally representative $J_\parallel/t_\parallel=0.5$ , $J_\perp/t_\parallel=1.5$ ). For the 2D bilayer the paper expects a Berezinskii–Kosterlitz–Thouless (BKT) transition — the mechanism by which a two-dimensional superconductor develops quasi-long-range order, with pairing correlations decaying algebraically below $T_c$ — around $T_c\sim J_\perp/2$ , an estimate obtained in the earlier ladder study of Schlömer, Schollwöck, Grusdt and Bohrdt (Communications Physics 7 (2024)) only in the perturbative strong-coupling limit where the model maps to hard-core bosons in the XXZ universality class. What does not exist is a quantitative determination of $T_c$ and of the finite-temperature pairing signatures for the full 2D model at intermediate couplings and nonzero $V$ , and a demonstration of this phase in an experiment.

Problem Statement

Determine, quantitatively, the finite-temperature superconducting phase of the two-dimensional mixed-dimensional bilayer t–J model realized in optical-lattice proposals — intralayer hopping $t_\parallel$ and superexchange $J_\parallel$ , interlayer spin exchange $J_\perp$ and interlayer density coupling $V$ , and no interlayer single-particle tunneling. The question has three connected parts: (i) compute the Berezinskii–Kosterlitz–Thouless critical temperature $T_c$ of the 2D mixD bilayer as a function of $J_\perp/t_\parallel$ , doping, and $V$ , establishing how the perturbative strong-coupling estimate $T_c\sim J_\perp/2$ is modified at intermediate couplings and for nonzero (repulsive) $V$ , and determining the critical interlayer repulsion $V_c(T)$ beyond which quasi-long-range superconducting order is destroyed; (ii) compute the finite-temperature pairing correlation function $\langle\hat\Delta^\dagger_i\hat\Delta_j\rangle$ (with $\hat\Delta_i$ the interlayer singlet-pair operator) across the transition, providing the pairing signatures — correlation exponents, correlation lengths, and stiffness-type observables — that experiments at realistically achievable temperatures and system sizes should observe; and (iii) assess experimental observability, including the quantitative effect of the terms neglected in the mapping to the effective single-layer t–J model (three-site and off-resonant virtual processes) on the relevant observables at intermediate $J_\perp/t_\parallel$ . The scientific objective of these parts is a quantitative, experimentally testable account of finite-temperature quasi-long-range superconducting pairing in the 2D mixD bilayer, replacing the perturbative $T_c$ estimate and the one-dimensional/zero-temperature evidence currently available. Determining $d$ -wave long-range order in the single-layer repulsive Fermi–Hubbard model is a separate open problem and is not part of this question.

References

Henning Schlömer, Hannah Lange, Titus Franz, Thomas Chalopin, Petar Bojović, Si Wang, Immanuel Bloch, Timon A. Hilker, Fabian Grusdt, Annabelle Bohrdt, 'Local control and mixed dimensions: Exploring high-temperature superconductivity in optical lattices', PRX Quantum 5, 040341 (2024), DOI 10.1103/PRXQuantum.5.040341, arXiv:2406.02551, https://arxiv.org/abs/2406.02551
Henning Schlömer, Ulrich Schollwöck, Fabian Grusdt, Annabelle Bohrdt, 'Superconductivity in the pressurized nickelate La3Ni2O7 in the vicinity of a BEC–BCS crossover', Communications Physics 7, 366 (2024), DOI 10.1038/s42005-024-01854-9, arXiv:2311.03349, https://arxiv.org/abs/2311.03349
Annabelle Bohrdt, Lukas Homeier, Immanuel Bloch, Eugene Demler, Fabian Grusdt, 'Strong pairing in mixed dimensional bilayer antiferromagnetic Mott insulators', Nature Physics 18, 651–656 (2022), DOI 10.1038/s41567-022-01561-8, arXiv:2108.04118, https://arxiv.org/abs/2108.04118
Hannah Lange, Lukas Homeier, Eugene Demler, Ulrich Schollwöck, Fabian Grusdt, Annabelle Bohrdt, 'Feshbach resonance in a strongly repulsive ladder of mixed dimensionality: A possible scenario for bilayer nickelate superconductors', Physical Review B 109, 045127 (2024), DOI 10.1103/PhysRevB.109.045127, arXiv:2309.15843, https://arxiv.org/abs/2309.15843
Hannah Lange, Ao Chen, Antoine Georges, Fabian Grusdt, Annabelle Bohrdt, Christopher Roth, 'Simulating superconductivity in mixed-dimensional $t_\parallel$ - $J_\parallel$ - $J_\perp$ bilayers with neural quantum states', arXiv:2602.10091 (2026), https://arxiv.org/abs/2602.10091
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Dominik Bourgund, Thomas Chalopin, Petar Bojović, Henning Schlömer, Si Wang, Titus Franz, Sarah Hirthe, Annabelle Bohrdt, Fabian Grusdt, Immanuel Bloch, Timon A. Hilker, 'Formation of individual stripes in a mixed-dimensional cold-atom Fermi–Hubbard system', Nature 637, 57–62 (2025), DOI 10.1038/s41586-024-08270-7, arXiv:2312.14156, https://arxiv.org/abs/2312.14156
Hannah Lange, Eugene Demler, Jan von Delft, Annabelle Bohrdt, Fabian Grusdt, 'Connecting single-layer t-J to Kondo lattice models: Exploration with cold atoms', arXiv:2512.09926 (2025), https://arxiv.org/abs/2512.09926