Predicting critical temperature in quantum simulators for high-$T_c$ superconductivity: the matrix product state plus mean field approach

Abstract: Quantum simulation based on ultra cold atomic lattice gases is one of the most promising platforms to investigate high-$T_c$ superconductivity beyond the limited capabilities of quantum many body numerics on classical computers. Yet, despite enormous progress since the field's inception, realizing a high-$T_c$ superconducting state still remains out of reach. The present work lays the groundwork to purpose the recently proposed, and already partly realized, mixed-dimensional (mixD) models, towards this end. These systems offer the proven capability to realize very high pairing energies while retaining appreciable mobility of pairs. We specifically investigate the potential of 2D mixD-models with anisotropic tunneling, using the matrix product state plus mean field theory (MPS+MF) for fermions, and show that these models may enter a high-$T_c$ superconducting phase. These simulations in turn are based on a comprehensive characterization of the 1D mixD-systems, which are the sub-units of which the 2D system is comprized. In this, we cover the range of currently experimentally relevant system sizes, and establish practical heuristics to determine when finite\hyp size effects preclude the use of a 1D mixD-system to build the 2D ones.

• The paper develops a matrix product state plus mean field (MPS+MF) framework that combines DMRG calculations of isolated t–J ladders with self-consistent inter-ladder coupling to estimate the bilayer critical temperature.
• The paper finds that increasing the intra-ladder tunneling raises pairing energy only above density n = 0.5 while reducing the spin gap, and identifies a parameter sweet spot near t_x ≈ 3.5J_y and n = 0.5–0.75.
• The paper predicts significant fractions of T_c/J_y across simulated densities, with T_c decreasing as density rises and scaling approximately with t_z²/E_p, while cautioning that MPS+MF may overestimate T_c by roughly a constant factor of four.

Overview and strategy

The paper by Köhler and Kantian lays the groundwork for using mixed-dimensional (mixD) bilayer systems—realizable in ultracold atomic lattice gases—as quantum simulators of high-$T_c$ superconductivity. Rather than attacking the 2D Hubbard model directly, the authors adopt the matrix product state plus mean field (MPS+MF) framework for fermions, in which a 2D bilayer is treated as an array of weakly coupled t–J ladders. The constituent 1D mixD ladders, one version of which has already been realized experimentally [Hirthe2023], possess an analytically understood pairing mechanism based on string-mediated chargon pairing. The work makes three contributions: a comprehensive characterization of isolated 1D mixD ladders at experimentally relevant system sizes, a pilot MPS+MF calculation of the superconducting critical temperature $T_c$ of the resulting 2D bilayer, and a practical heuristic for detecting finite-size breakdown of the TLL-based inputs.

The model

The starting point is an effective t–J Hamiltonian for a 2D bilayer viewed as a stack of ladders along $z$, with intra-ladder tunneling $t_x$, vanishing intra-bilayer single-fermion tunneling ($t_y=0$), spin exchanges $J_x \ll J_y$ (with $J_y$ set to 1), and inter-ladder tunneling $t_z$ along the stack. The metastable-state preparation scheme of Rabl, Kantian and collaborators, demonstrated in [Hirthe2023], suppresses inter-layer single-particle tunneling via an energy offset while permitting exceptionally large inter-layer spin exchange $J_y$.

Within MPS+MF, the full Hamiltonian is replaced by an effective single-ladder Hamiltonian augmented by self-consistent particle–particle (pp) amplitudes $\alpha^R_d$ and particle–hole (ph) amplitudes $T_c$0, with the pp amplitudes carrying the prefactor $T_c$1. The validity of this reduction requires $T_c$2, where $T_c$3 is the pairing energy and $T_c$4 the spin gap of an isolated ladder; the factor of two on $T_c$5 reflects that, in the ultracold-atom setting, pair-breaking by pseudospin flips requires exchanging atoms between ladders, a process gapped by $T_c$6. The authors note this factor does not apply in solid-state realizations, where $T_c$7 alone constrains $T_c$8.

Method

The workflow is semi-analytical. DMRG at fixed $T_c$9 and $z$0 yields the ground-state energies of the isolated ladder in adjacent particle-number and spin sectors, from which $z$1, $z$2, the compressibility $z$3, the charge velocity $z$4, the Luttinger parameter $z$5, and the stiffness $z$6 are extracted. The critical temperature then follows from the gap $z$7 between the ground and first excited state of the effective (mean-field-coupled) Hamiltonian via the ratio function $z$8, so that $z$9. This avoids the far costlier brute-force route of computing thermal density matrices at many temperatures; prior work established both routes agree closely. DMRG calculations were extrapolated in bond dimension (up to $t_x$0, with at least three converged values).

A notable practical result is that $t_x$1, $t_x$2, $t_x$3, $t_x$4, $t_x$5 and $t_x$6 become remarkably stable beyond low ladder lengths; for $t_x$7 the differences are smaller than plot symbols. This justifies using $t_x$8 data—matching quantum-gas-microscope linear sizes—as representative.

Performance metrics of the isolated ladder

The survey over $t_x$9 and densities $t_y=0$0 yields a substantially more nuanced picture than earlier work. The pairing energy does grow with $t_y=0$1—following the predicted $t_y=0$2 scaling—but only for densities $t_y=0$3, and this growth comes at the direct expense of the spin gap, which decreases with the same scaling. The "sweet spot" where $t_y=0$4 and $t_y=0$5 coincide lies near $t_y=0$6 and $t_y=0$7. Consequently, simply increasing $t_y=0$8 beyond the value of the initial experiment is not a viable route to high $t_y=0$9.

The transport-related quantities add a further tension: the stiffness $J_x \ll J_y$0 and sound velocity $J_x \ll J_y$1 increase with $J_x \ll J_y$2 but their maxima do not coincide with those of $J_x \ll J_y$3, while the Luttinger parameter $J_x \ll J_y$4—which governs the superconducting susceptibility, diverging as $J_x \ll J_y$5—is maximal at low $J_x \ll J_y$6 and is essentially independent of $J_x \ll J_y$7. Thus a blanket increase of $J_x \ll J_y$8 improves some superconductivity-related metrics ($J_x \ll J_y$9, $J_y$0, $J_y$1) while significantly degrading others ($J_y$2, $J_y$3).

Critical temperature of the 2D bilayer

For the pilot calculation the authors use parameters close to those of the realized experiment ($J_y$4, $J_y$5 neglected) with $J_y$6 and $J_y$7–$J_y$8. Three findings emerge. First, $J_y$9 reaches significant fractions of $t_z$0 across all simulated densities, meaning the implemented system would realize the analogue of a repulsively mediated high-$t_z$1 superconducting state—subject to three caveats the authors state explicitly: competing instabilities have not been ruled out; entropies per particle, the actual figure of merit, have not been computed; and MPS+MF systematically overestimates $t_z$2 by an approximately constant factor, possibly on the order of 4 based on prior work. The constant-factor nature of the overestimate means parameter trends and optimization targets remain meaningful, and MPS+MF is far better behaved than pure mean-field treatments, which increasingly overestimate $t_z$3 as pairs become heavy.

Second, and contrary to the naive expectation that maximizing $t_z$4 maximizes $t_z$5, $t_z$6 decreases monotonically with density even though $t_z$7 increases: the system behaves like other high-$t_z$8 superconductors limited by phase stiffness, where pairs are strongly bound and heavy, and shedding pairing energy by lowering density raises $t_z$9. The data collapse shows $J_y$0 scaling linearly in $J_y$1 over most of the explored range, not yet entering the superlinear regime found in earlier work.

Third, finite-size effects in $J_y$2 are modest and shrink as $J_y$3 grows, with the data for different $J_y$4 collapsing at higher densities.

A heuristic for finite-size validity

Because $J_y$5—and hence $J_y$6 and $J_y$7—is extracted from the first excited state in the $J_y$8 sector, contamination by the gapped antisymmetric density sector is a serious risk at large $J_y$9 and small $\alpha^R_d$0. Comparing DMRG with exact diagonalization, the authors identify a kink in the tilted energy of the first excited state, signaling a level crossing, accompanied by a sharp change in the local density profile from an excited-density shape to one resembling the absolute ground state. The crossing point moves to larger $\alpha^R_d$1 as $\alpha^R_d$2 grows, confirming it as a finite-size artifact. This heuristic delineates the region of the $\alpha^R_d$3–$\alpha^R_d$4 plane where the TLL-based inputs—and therefore the $\alpha^R_d$5 predictions—remain trustworthy.

Limitations and open questions

The authors are explicit about what remains unresolved. Competing ordered phases (e.g., insulating instabilities), which routinely decide the fate of doped repulsive-fermion systems, have not been computed here, though MPS+MF tools demonstrated for simpler systems could be adapted. Entropies per particle at the predicted $\alpha^R_d$6—the decisive quantity for experimental reachability—remain uncalculated but are in principle accessible via state purification. The exact overestimation factor of MPS+MF in 2D is unknown; it is larger than in 3D, and incorporating quadratic fluctuations around the mean-field amplitudes could reduce it. The calculations are restricted to $\alpha^R_d$7; behavior near the isotropic limit $\alpha^R_d$8, and whether $\alpha^R_d$9 levels off or deviates from the known scaling as $T_c$00, is left open. Independent numerical validation is likewise indeterminate, with phase-free auxiliary-field QMC a candidate only if it can accommodate the no-double-occupancy constraint.

Conclusion

This work establishes the quantitative foundation for engineering high-$T_c$01 superconductivity in 2D mixD bilayers in ultracold-atom quantum simulators. Its central, somewhat counterintuitive findings—that high pairing energy comes at the cost of the spin gap and superconducting susceptibility, and that $T_c$02 of the bilayer is highest when the pairing energy is relatively low—reshape the optimization strategy for experiments away from simply increasing $T_c$03. Combined with the finite-size heuristic and the demonstrated stability of ladder observables at experimentally feasible lengths, the paper identifies concrete, currently realizable parameter regimes and delineates precisely which calculations (competing orders, entropies, fluctuation corrections) must follow to convert these predictions into an experimental protocol.