AKLT Hamiltonian from Hubbard tripods

Abstract

We investigate how the spin-1 Affleck-Kennedy-Lieb-Tasaki (AKLT) Hamiltonian can emerge from a microscopic fermionic model based on half-filled Hubbard tripods. We first show that a single tripod hosts a robust threefold-degenerate low-energy manifold corresponding to an effective $S = 1$ degree of freedom. This manifold prevails over a broad range of interactions and remains stable against moderate disorder. We then combine exact diagonalization with fourth-order quasi-degenerate perturbation theory to derive an effective bilinear-biquadratic spin model for a pair of coupled tripods and identify coupling regimes where the target ratio is approached. In particular, tuning leg-center hopping together with two symmetry-inequivalent leg-leg hoppings yields the characteristic singlet-triplet degeneracy associated with a biquadratic-to-bilinear ratio close to 1/3. Extending the analysis to three tripods, we compare nonequivalent coupling geometries and find a strategy that suppresses unwanted longer-range and multispin terms while preserving the target nearest-neighbor couplings in the weak-coupling regime. These results establish a concrete bottom-up route from Hubbard clusters to valence-bond-solid spin physics in tunable quantum-dot arrays.

Figures (7)

• Figure 1: The left panel depicts the schematics of a single Hubbard tripod with central site $c$ and legs $l_1$, $l_2$, $l_3$. The right panel shows the excitation spectrum of a single Hubbard tripod at half filling, as a function of $U$.
• Figure 2: Excitation spectrum of a single tripod subject to random onsite and bond disorder of characteristic strength $t^*$. Solid lines denote the mean values, and shaded areas indicate the distribution's spread at one standard deviation. The data shown represent statistics from 200 individual disorder configurations.
• Figure 3: Energy spectrum of the bilinear-biquadratic Hamiltonian of the spin-1 dimer as function of the coupling ratio $\beta/J$. For $J>0$ below (above) $\beta/J=1/3$, the non-degenerate singlet (threefold degenerate triplet) is the ground state. At $\beta/J=1/3$, the singlet and triplet states become degenerate, resulting in a fourfold degenerate ground state separated by a gap of $2J$ from the $S=2$ multiplet.
• Figure 4: Inter-tripod couplings. Red solid lines depict hopping from a center-site to a leg-site on a different cluster $H_{\text{leg-center}}$, while the dotted magenta line and the dashed dark blue lines denote inequivalent hoppings between legs of different clusters, $H_{\text{leg-leg-1}}$ and $H_{\text{leg-leg-2}}$, respectively.
• Figure 5: Degeneracy parameter $\sigma_2$ as a function of various coupling strengths. In (a), results are obtained for $t_{l2}=0$, while in (b), $t_{l1}=0$. Diagonal white lines in each plot indicate the sampled section of parameter space in Fig. \ref{['fig:two_tripod_spectra']}. The dashed curve in (b) corresponds to results from fourth-order perturbation theory.