Emergent Shastry-Sutherland network from square-kagome Heisenberg antiferromagnet with trimerization

TL;DR

This paper addresses the low-energy physics of the $S=\tfrac{1}{2}$ square-kagome Heisenberg antiferromagnet under trimarization. By treating the inter-trimer coupling $\eta$ as a perturbation around the trimerized limit, the authors derive a Kugel-Khomskii-type Hamiltonian on a Shastry-Sutherland network that couples spin and chirality, with explicit first-order form $H^{(1)} = \frac{\eta}{9} \sum h^{\tau(\gamma)} h^{\sigma}$. They construct a dimer-covering mean-field ansatz inspired by the Shastry-Sutherland dimer phase and compute a lower bound energy $E_{\min}$ for this MF state, but exact diagonalization on small clusters shows the true ground state lies below $E_{\min}$, indicating the MF dimer picture is not the exact ground state. The results suggest the ground state is an entangled spin-chirality state not captured by the simple dimer-covering ansatz, and point to the need for larger-scale numerics and entanglement analyses to fully characterize the low-energy sector. Overall, the work links clusterized trimerization to known dimer physics and lays groundwork for exploring cluster-based quantum states in frustrated magnets.

Abstract

We study the $S=1/2$ square-kagome lattice Heisenberg antiferromagnet with the trimarized modulation. In the trimerized limit, each trimer hosts the four-fold degenearte ground states characterized by the spin and chirality degrees of freedom. We find that, within the first-order perturbation theory with respect to the inter-trimer coupling, the effective Hamiltonian is the Kugel-Khomskii-type model on a Shastry-Sutherland lattice. Based on a mean-field decoupling, we propose a dimer-covering ansatz for the effective Hamiltonian; however, the validity of these states in the low-energy sector remains an open question.

Figures (3)

• Figure 1: (a) Trimerized square-kagome lattice. The exchange couplings are 1 for the red solid bonds and $\eta$ for the blue dashed bonds. The gray shade represent the unit cell, and the orange arrows represent the lattice vectors. (b) Network arising from the first-order perturbation from the trimerized limit. The open circles represent the trimers, and the colors of the bonds correspond to $\gamma$ of Eq. (\ref{['eq:ham_perturv']}).
• Figure 2: Comparison with the ED results for the 12-site square-kagome lattice model (gray dots) with the dimer-covering anstaz for (a) $\eta \in [0,0.2]$. Panel (b) is its zoom-up for $\eta \in [0,0.07]$.
• Figure 3: (a) 24-site cluster (black dots in a yellow shade) used for the exact diagonalization. (b) Comparison with the ED results (gray dots) with $E_{\rm min}$ (a red line).