Optimal array geometries for kinetic magnetism and Nagaoka polarons

TL;DR

This work addresses how finite-size connectivity governs kinetic magnetism and Nagaoka polarons in Hubbard systems realized as quantum-dot arrays. It combines exact diagonalization with graph-theoretic analysis, showing that the algebraic connectivity $\lambda_2$ and Katz centrality (KC) quantify hole delocalization and site occupations, yielding a scaling $t_c/U \sim \lambda_2^2$ for square geometries and guiding the design of optimal geometries. It further reveals that optimized geometries support stronger Nagaoka polarons and can extend the ferromagnetic regime, while applying a perpendicular magnetic flux introduces Aharonov-Bohm phases that can destroy NFM at a critical flux $\phi_c$ and induce AFM-like correlations, with $\phi_c$ depending on $t/U$ and geometry. The findings provide concrete connectivity-based design rules for engineering kinetic ferromagnetism in finite QD clusters and motivate extensions to directed graphs with complex weights to model flux effects.

Abstract

Quantum dot (QD) platforms have enabled the direct observation of Nagaoka ferromagnetism (NFM) in small arrays and non-infinite interaction strength. However, optimizing the cluster connectivity characteristics that yield a ground state with maximal spin and their robustness against magnetic fields remains unexplored. Employing exact diagonalization of the Hubbard Hamiltonian, we find a connection between the existence of kinetic ferromagnetism and graph theory descriptions. Algebraic connectivity ($λ_2$) and Katz centrality (KC) are shown to be related to the spin-correlation over the system. In square arrays, the onset of NFM is found to be $t_c/U\simeq λ_2^2$. In optimal cluster geometries, large $λ_2$ and low KC fluctuation per site are found to enhance $t_c/U$, extending the NFM phase while diminishing the strength of spin correlation clouds. A perpendicular magnetic field introduces Aharonov-Bohm phases, and a critical flux for which NFM is destroyed. We further find that tuning the flux phase to $π$ results in a ground state that exhibits antiferromagnetic correlations (counter-Nagaoka state). Our results illustrate how NFM and polaron formation can be predicted from the array's connectivity ($λ_2$ and KC), and how the introduction of flux results in the counterintuitive destruction of kinetic ferromagnetism in the system.

Figures (9)

• Figure 1: (a) Lowest energy manifold in a $3 \times 3$ square array with 8 electrons (one less than half-filled) as a function of $t/U$. Transition to the NFM regime occurs for $t/U < t_{c}/U \approx 0.0144$. (b) Hole-spin-spin correlator $C_{hss}$ in ($t/U=0.01$), and beyond ($t/U=0.02$) the kinetic ferromagnetic regime, showing the relative spin alignments in the vicinity of the hole at central site (white circle).
• Figure 1: Algebraic connectivity as a function of $L$ in square arrays of $L \times L$ is found to be $\lambda_2 \simeq \pi^2 /L^2$. See text.
• Figure 1: Hole distribution per site in the proposed clusters with ten sites at $t<t_c$. The color of each circle conveys $\langle h_{i} \rangle$, while its diameter is proportional to the Katz centrality (KC). Notice that in the b-array, the hole is mostly concentrated in the bulk-site (larger diameter), whereas in the a-array, the hole is distributed over more sites. In the e-array, the hole is distributed more evenly among sites, corresponding to a low fluctuation in its KC (site's diameter).
• Figure 2: Hole distribution per site in the clusters of Table \ref{['tab:geometries_8M']} at $t<t_c$. The color of each circle conveys $\langle h_{i} \rangle$, while its diameter is proportional to the Katz centrality (KC) of the site. Notice that in the L-array, the hole is mostly concentrated in the central bulk-site (larger diameter), whereas in the 2-Square array (right-most graph), the hole is distributed uniformly (all sites have the same diameter).
• Figure 3: Nagaoka polarons in clusters with different connectivities, in and out of the kinetic ferromagnetic regime. $\tilde{C}_{hss}$ determines the total spin-spin correlation in the vicinity of the hole (white dot). In the Nagaoka regime, upper row, $\tilde{C}_{hss}$ is shown for $t<t_c$ to explore the signature ferromagnetic correlations; notice its higher value for the L-array (left-most). The signature ferromagnetic correlations vanish throughout for values of $t>t_c$, as shown in the lower row just outside the NFM regime (at $t/U = 0.011, 0.013, 0.018, 0.027$ from left to right).