Geometric and Orbital Control of Correlated States in Small Hubbard Clusters

TL;DR

The paper addresses how to deterministically engineer local electron pairing in quantum-dot arrays described by a multi-orbital Hubbard model. It develops a predictive design by decoupling three control levers—lattice geometry, orbital hybridization, and external electric fields—and uses ED and self-consistent HF across cluster geometries to map double occupancy $D$ as a pairing proxy, with $t_0$ normalized to unity by maximizing the non-interacting bandwidth $W$. Key contributions include three design principles: geometric hierarchy governed by the coordination number $Z$, orbital hybridization via $t_{\mathrm{orb}}$ that enhances pairing at moderate $U$, and field squeezing by $\mathbf{E}$ that robustly increases pairing, with a filling-dependent phase diagram. These results provide a practical blueprint for bottom-up quantum hardware design, enabling targeted control of charge and spin correlations in quantum-dot platforms through geometry, orbital engineering, and local fields.

Abstract

Arrays of semiconductor quantum dots provide a powerful platform to design correlated quantum matter from the bottom up. We establish a predictive framework for engineering local electron pairing in these artificial molecules by systematically deploying three control levers: lattice geometry, orbital hybridization, and external electric fields. Using Hartree-Fock simulations on canonical 3D clusters from the tetrahedron (Z = 3) to the FCC lattice (Z = 12), at and near half-filling, we uncover three fundamental design principles. (i) Geometric Hierarchy: The resilience to Coulomb repulsion U is dictated by the coordination number Z, which controls kinetic delocalization. (ii) Orbital Hybridization: Counter-intuitively, inter-orbital hopping t_orb acts not as a simple suppressor of pairing, but as a sophisticated control knob that enhances double occupancy at moderate U by engineering the on-site energy landscape. (iii) Field Squeezing: An electric field robustly induces pairing by forcing charge localization, an effect most potent in low-connectivity clusters. These principles form a blueprint for deterministically targeting charge and spin correlations in quantum-dot-based quantum hardware.

Figures (5)

• Figure 1: Geometric control of pairing vs. Hubbard U. Total double occupancy ($D$) as a function of Hubbard repulsion ($U/t_0$) for five different lattice geometries. (a) At half-filling, a clear hierarchy emerges where higher coordination ($Z$) sustains greater pairing against $U$. (b) In the doped case (one electron removed), the hierarchy is less rigid, and the suppression of $D$ is more gradual.
• Figure 2: Control of pairing via inter-orbital hopping $t_{\mathrm{orb}}$. Total double occupancy ($D_{\text{total}}$) versus $t_{\mathrm{orb}}/t_0$ for all geometries at weak ($U=4t_0$), intermediate ($U=8t_0$), and strong ($U=12t_0$) coupling. (Top Panel) At half-filling, $t_{\mathrm{orb}}$ strongly enhances pairing at moderate $U$, while its effect is quenched at large $U$. (Bottom Panel) In the doped case, the enhancement is more modest but follows a similar trend.
• Figure 3: Thermal effects on pairing at strong coupling ($U=8t_0$). Double occupancy ($D$) versus temperature. (a) At half-filling, all geometries exhibit a sharp thermal crossover from a low-pairing Mott state to a high-temperature disordered state, universally saturating at $D=0.25$. (b) In the doped case, the system saturates to a lower, geometry-dependent value of $D$, reflecting the reduced particle density.
• Figure 4: Electric field induced pairing at moderate coupling ($U=4t_0$). Double occupancy ($D$) as a function of E-field amplitude. (a) At half-filling, the field robustly enhances pairing. The effect is dramatically more pronounced in low-coordination lattices (Tetrahedron, Octahedron), which act as effective "quantum squeezers." (b) In the doped case, the overall enhancement is more modest as mobile holes provide alternative charge rearrangement pathways.
• Figure 5: Pairing phase diagram for the Octahedron. Total double occupancy ($D_{\text{total}}$) as a function of Hubbard repulsion $U$ and inter-orbital hopping $t_{\mathrm{orb}}$. (a) At half-filling, a clear transition from a high-pairing (bright) to a low-pairing (dark) Mott state is driven by $U$. (b) The transition is smoother in the doped case, consistent with more metallic behavior.