Microscopic mechanism of high-temperature superconductivity revealed by ab initio studies on hole-doped multilayer cuprates HgBa$_2$Ca$_2$Cu$_3$O$_8$ under pressure

Abstract: Triple-layer cuprate superconductor $\mathrm{HgBa_2Ca_2Cu_3O_8}$ (Hg1223) keeps the record of the highest superconducting (SC) critical temperature $T_{c}\sim 134$K among all the existing materials at ambient pressure. $T_{c}$ further increases under pressure up to $T_{c}\sim 160$K. However, its microscopic mechanism remains to be elucidated. We solve {\it ab initio} Hamiltonians for Hg1223 using a variational solver supplemented by a neural network. The pressure dependence of the $d$-wave SC order parameter and estimated $T_{c}$ show a dome-like structure in essential agreement with the experimental indications. The origin of the strong SC amplitude at ambient pressure is identified as strong local Coulomb repulsion $U$ attributed to poor screening. Further increase in $T_{c}$ under pressure is understood from interplay of three elements, namely increased electron hopping $t$, decreased $U$ and more importantly, strongly reduced offsite Coulomb repulsion $V$ with increasing pressure. Pairing mechanism is identified as the emergent local attraction counterintuitively generated from the originally strong local repulsion $U$. The emergent attraction is interpreted from attraction from reduced repulsion'', originating from the release of the fluctuating doubly-occupied sites characterized from thefalse vacuum'' in the Mott insulator to the double-occupation-free $d$-wave SC states upon carrier doping. This instantaneous attraction is in contrast with the conventional BCS SC mediated by bosonic glues. The local attraction is consistent with the electron fractionalization supported in experimental analyses. The coexistence of the SC and antiferromagnetic order is also demonstrated as a characteristic feature of the multi-layer system. The microscopic understanding of Hg1223 offers a new route explicitly using this emergent attraction to design and optimize SC materials.

• The paper identifies a dome-like Tc evolution under pressure, revealing an emergent local attraction as the key mechanism driving Cooper pairing in Hg1223.
• Ab initio cGW Hamiltonian derivations combined with variational Monte Carlo techniques capture layer-resolved self-doping and the coexistence of superconducting and antiferromagnetic orders.
• Results benchmark experimental Tc trends and propose materials design strategies by optimizing the balance between onsite repulsion and reduced off-site Coulomb interactions.

Microscopic Mechanism of High-Temperature Superconductivity in HgBa$_2$Ca$_2$Cu$_3$O$_8$: Ab Initio Insights Under Pressure

Introduction

The paper performs an ab initio computational analysis of the triple-layer cuprate $\mathrm{HgBa_2Ca_2Cu_3O_8}$ (Hg1223), which holds the record for highest ambient pressure superconducting critical temperature ($T_c$) among all known cuprates. The study utilizes recently developed constrained GW (cGW)-based Hamiltonian derivation and highly accurate variational Monte Carlo (VMC) solvers augmented with neural networks. The investigation encompasses ambient and pressurized Hg1223, treating layer-resolved properties, self-doping phenomena, and competition between superconducting (SC) and antiferromagnetic (AF) orders.

A primary finding is the identification of a dome-like pressure dependence of $T_c$, consistent with experiment, and the uncovering of a local, instantaneous attraction emerging from strong onsite Coulomb repulsion as the root of Cooper pairing in this material. The results are benchmarked against other cuprates, single-, double-, and infinite-layer, and interpreted in terms of interaction parameters $U/|t_1|$ and the role of non-local Coulomb repulsion $V$.

Ab Initio Hamiltonian Formulation and Numerical Techniques

The effective Hamiltonian is derived using cGW with self-interaction correction and level renormalization feedback, yielding parameter-free layer-resolved model descriptions for the antibonding states formed by strongly hybridized Cu $3d_{x^2-y^2}$ and O $_2$0 orbitals. The Hamiltonian structure includes:

• $_2$1: Long-range hopping terms capturing intralayer and interlayer transfers;
• $_2$2: Layer-dependent onsite Coulomb repulsion;
• $_2$3: Long-range off-site Coulomb interactions;
• $_2$4: Layer-dependent chemical potentials reflecting self-doping.

The ground state is solved using variational wavefunctions with Gutzwiller, Jastrow, doublon-holon, and RBM factors. System sizes up to $_2$5 are accessed, facilitating extrapolation to the thermodynamic limit and accurate treatment of competing orders.

Figure 1: Crystal structure of Hg1223, revealing the arrangement of inner and outer CuO$_2$6 planes and primitive cell directions $_2$7.

Ambient Pressure Results: Layer-Resolved Self-Doping, SC Domes, and Order Competition

A distinct aspect of multilayer cuprates like Hg1223 is self-doping, leading to differential hole concentrations between inner (IP) and outer (OP) CuO$_2$8 layers. Ab initio calculations quantitatively reproduce this effect, finding lower hole density in the IP.

Figure 2: Inner and outer-layer hole densities versus total hole density $_2$9, illustrating self-doping in Hg1223 at ambient pressure.

The SC order parameter $_3$0 exhibits a conventional dome structure, with its maximum for IP ($_3$10.14) exceeding that of other cuprates (e.g., Bi2212). The doping position of the dome aligns with experimental measurements. A key observation is the layer proximity effect—despite the IP and OP having differing intrinsic optimal doping, strong interlayer coupling synchronizes their SC domes.

Figure 3: Doping dependence of the SC order parameter $_3$2 for IP and OP after size extrapolation.

A characteristic feature is the coexistence of AF and SC orders in the underdoped regime, especially in the IP close to half-filling. When chemical potential differences are appropriately tuned, proximity-induced coexistence is observed, consistent with experiment.

Figure 4: Spin structure factors and SC correlations in the IP and OP, supporting AF–SC coexistence in multi-layer Hg1223.

Figure 5: Size extrapolation of AF order parameter $_3$3 and SC order parameter $_3$4 in the IP, both nonzero in the ground state.

Pressure Dependence: Origin and Quantitative Analysis of Enhanced $_3$5

Ab initio calculations under pressure reveal dome-like $_3$6 evolution, peaking near 30 GPa (experimental maximum $_3$7 K), and confirm that the observed enhancement is not solely attributable to increased bandwidth or reduced $_3$8. The role of off-site Coulomb repulsion $_3$9 is pronounced: pressure reduces $_8$0 more efficiently than $_8$1. The interplay of increased hopping, reduced $_8$2, and substantially decreased $_8$3 yields the observed pressure-dependent dome.

Figure 6: Pressure dependence of SC order parameter in the IP, showing stability up to $_8$430 GPa then reduction at higher pressures; inset tracks $_8$5, $_8$6, $_8$7 across pressures.

A universal scaling for $_8$8 is found to apply: $_8$9, confirmed for Hg1223 under pressure and ambient conditions. Comparison across cuprates, including theoretical and experimental values, demonstrates the necessity of accounting for $\mathrm{HgBa_2Ca_2Cu_3O_8}$0 reduction under pressure for accurate $\mathrm{HgBa_2Ca_2Cu_3O_8}$1 prediction.

Figure 7: $\mathrm{HgBa_2Ca_2Cu_3O_8}$2 versus $\mathrm{HgBa_2Ca_2Cu_3O_8}$3 across cuprates; Hg1223 under pressure deviates from ambient-pressure universal trends, highlighting reduced $\mathrm{HgBa_2Ca_2Cu_3O_8}$4.

Figure 8: Comparison of theoretical and experimental $\mathrm{HgBa_2Ca_2Cu_3O_8}$5 versus pressure for Hg1223, exhibiting the pressure-induced $\mathrm{HgBa_2Ca_2Cu_3O_8}$6 dome.

Figure 9: Cross-material comparison of theoretical and experimental $\mathrm{HgBa_2Ca_2Cu_3O_8}$7, demonstrating quantitative accuracy of ab initio approach.

Emergent Local Attraction: Origin of Cooper Pairing

The paper provides a microscopic mechanism for Cooper pairing: “attraction by reduced repulsion” in a strongly correlated, doped Mott insulator. Quadratic fits to layer-resolved onsite energy $\mathrm{HgBa_2Ca_2Cu_3O_8}$8 yield an effective attraction parameter $\mathrm{HgBa_2Ca_2Cu_3O_8}$9, peaking near optimal $T_c$0 ($T_c$19 for cuprates, $T_c$26-8 for Hubbard model). The dependence of $T_c$3 on $T_c$4, and further on $T_c$5, is emphasized. Pressure effects are consistent with enhanced SC by reduced off-site repulsion.

Figure 10: $T_c$6 (effective attraction) versus $T_c$7 for Hg1223 and CaCuO$T_c$8, demonstrating universality in emergent attraction among cuprates.

Figure 11: $T_c$9 versus $T_c$0 at various dopings, marking the optimal interaction and pressure-induced enhancement.

Figure 12: Product $T_c$1 as a function of interaction, capturing the dome-like structure of $T_c$2.

The instantaneous, local attraction, derived from negative curvature in $T_c$3, is distinct from BCS retarded pairing via bosonic mediators. Dynamical spin correlations are discussed as broader contributors, but the analysis emphasizes spatial-temporally local origins.

Discussion: Implications and Materials Design

The results imply that $T_c$4 optimization within the cuprate paradigm hinges on maximizing $T_c$5 near the optimal value, while simultaneously minimizing $T_c$6 through lattice engineering or external pressure. Higher $T_c$7 could be realized by materials where bandwidth and reduced off-site interactions coexist with strong onsite repulsion.

The findings are consistent with empirical scaling relations (e.g., Uemura plot), but offer quantitative microscopic justification. Comparisons with previous DMFT and ML-guided studies underline the necessity of treating spatial/temporal quantum fluctuations and layer-dependent effects to capture $T_c$8 variations, particularly in multi-layer compounds.

Conclusion

This paper provides an authoritative ab initio account of high-$T_c$9 superconductivity in Hg1223, reproducing experimental trends in $U/|t_1|$0 and SC order under ambient and pressurized conditions. The mechanism is identified as emergent local, instantaneous attraction from reduced repulsion in a strongly correlated background, universal across cuprates. The results refine the materials design strategy for high-$U/|t_1|$1 by targeting optimal $U/|t_1|$2, enhanced hopping, and minimized off-site repulsion. The methodology sets a benchmark for computational studies seeking to unify microscopic theory and materials optimization in unconventional superconductors.