Symmetry breaking and competing valence bond states in the star lattice Heisenberg antiferromagnet

We investigate the ground state phase diagram of the spin-$1/2$ antiferromagnetic Heisenberg model on the star lattice using infinite projected entangled pair states (iPEPS) and high-order series expansions. The model includes two distinct couplings: $J_d$ on the dimer bonds and $J_t$ on the trimer bonds. While it is established that the system hosts a valence bond solid (VBS) phase for $J_d \ge J_t$, the ground state phase diagram for $J_d < J_t$ has remained unsettled. Our iPEPS simulations uncover a first-order phase transition at $J_d/J_t \approx 0.18$, significantly lower than previously reported estimates. Beyond this transition, we identify a close competition between two valence bond crystal (VBC) states: a columnar VBC and a $\sqrt{3} \times \sqrt{3}$ VBC, with the latter consistently exhibiting lower energy across all finite bond dimensions. The high-order series expansion supports this by finding that the $\sqrt{3} \times \sqrt{3}$ VBC state indeed becomes energetically favorable, but only at sixth order in perturbation theory, revealing the subtle nature of the competition between candidate states.

💡 Research Summary

The paper investigates the ground‑state phase diagram of the spin‑½ antiferromagnetic Heisenberg model on the star lattice, a two‑dimensional Archimedean lattice with the lowest coordination number (3). The Hamiltonian contains two inequivalent nearest‑neighbour couplings: Jₜ on the “trimer” bonds that form triangles, and Jₙ on the “dimer” bonds that connect the triangles. While it is well established that for Jₙ ≥ Jₜ the system resides in a dimer valence‑bond solid (VBS) with strong singlets on the dimer bonds, the nature of the ground state for Jₙ < Jₜ has been controversial. Earlier studies reported a wide range of critical ratios (Jₙ/Jₜ ≈ 0.42, 0.77, 0.91) and even suggested the absence of a transition, leaving the phase diagram in this regime unsettled.

To resolve this, the authors employ infinite projected entangled‑pair states (iPEPS), a variational tensor‑network ansatz capable of representing wavefunctions directly in the thermodynamic limit, together with high‑order series‑expansion perturbation theory. In the iPEPS implementation each trimer is treated as a single physical tensor of dimension d = 8, leading to a honeycomb‑lattice network of tensors with virtual bond dimension D. The simple‑update algorithm is used for imaginary‑time evolution, and the environment is contracted via a corner‑transfer‑matrix renormalization group (CTMRG) adapted to the honeycomb geometry. Convergence is ensured by using environment bond dimensions χ up to D(D + 2)+16, achieving energy precision of 10⁻⁶ for D ranging from 2 to 11.

The iPEPS calculations reveal a first‑order quantum phase transition out of the dimer VBS at a much lower coupling ratio, Jₙ/Jₜ = 0.185(5), than previously reported. Beyond this transition, two competing valence‑bond crystal (VBC) states emerge: a √3 × √3 VBC (three‑fold degenerate, preserving a 120° rotational pattern) and a columnar VBC with a six‑site unit cell (also three‑fold degenerate). By constructing two‑trimer unit‑cell iPEPS setups that respect only the symmetries of each candidate, the authors find that for every finite bond dimension D the √3 × √3 VBC has a slightly lower energy, the difference being of order 10⁻⁵ per site. Because this energy gap is extremely small, the authors supplement the tensor‑network results with a perturbative linked‑cluster expansion.

In the perturbative approach the Hamiltonian is deformed into isolated “bow‑tie” clusters (each consisting of two triangles linked by a dimer bond) with intra‑cluster couplings Jₜ and Jₙ, while inter‑cluster couplings are treated as a small parameter λ. The bow‑tie ground state is non‑degenerate, allowing the use of Löwdin partitioning to compute the ground‑state energy as a power series in λ. Graph‑based linked‑cluster calculations are performed up to seventh order. Remarkably, the energies of the √3 × √3 and columnar VBCs are identical up to fifth order; the first splitting appears at sixth order, precisely matching the tiny 10⁻⁵ energy difference observed in iPEPS. Higher orders amplify this splitting, confirming that the √3 × √3 VBC is energetically favored in the thermodynamic limit.

The combined evidence therefore establishes a clear phase diagram: (i) a robust dimer VBS for Jₙ ≥ Jₜ, (ii) a first‑order transition at Jₙ/Jₜ ≈ 0.185, and (iii) a √3 × √3 VBC as the true ground state for Jₙ < 0.185 Jₜ. The work resolves previous discrepancies, demonstrates the power of combining iPEPS with high‑order perturbation theory to detect minute energy differences, and highlights the delicate balance of competing orders in highly frustrated quantum magnets. The authors suggest that experimental realizations in iron acetate or hybrid copper sulfate compounds, as well as complementary numerical methods such as DMRG or variational Monte Carlo, could be used to test these predictions and explore possible quantum critical behavior near the identified transition.