Fig. 3
Physical regimes of exchange interaction. a In the Hubbard model, different spin-exchange processes dominate depending on the relative alignment of various single-particle levels (cf. Supplementary Note 7). Specifically, \(\varepsilon _{\mathrm{M}}^ \ast\) is the single-particle energy of the lowest unoccupied orbital in the multielectron dot relative to the left and right orbital, \(\varepsilon ^ \ast = (\varepsilon _{\mathrm{L}}^ \ast - \varepsilon _{\mathrm{R}}^ \ast )/2\) determines the relative detuning between the left and right orbital, and U and εS indicate the charging energy and the level spacing of the multielectron dot. Depending on which processes are energetically allowed or suppressed, we classify different regimes as illustrated. b Measured PS(ε, εM) for the right double dot for fixed interaction time τ = 6 ns. Colored triangles indicate the detuning points used for Figs. 2a and 4b. Dashed lines indicate the location of independently measured charge transitions (see Supplementary Note 4). c, d Same as (b), but for reduced tunnel coupling between the multielectron dot and the inner dots. The interaction time is fixed at τ = 5 and 4 ns in (c) and (d), respectively. e Simulated \(P_{\mathrm{S}}(\varepsilon ^ \ast ,\varepsilon _{\mathrm{M}}^ \ast )\) in the Hubbard model. f Location of charge transitions (dashed lines) in the Hubbard model for the parameters used in (e). The corresponding charge configurations of the four regimes of exchange interaction are schematically indicated by dots in (a)
