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

TL;DR

This paper addresses the ground state of the spin-$\frac{1}{2}$ Heisenberg antiferromagnet on the star lattice, where two inequivalent bond types $J_d$ and $J_t$ control dimer and trimer connections. It combines infinite projected entangled pair states (iPEPS) with a high-order perturbative linked-cluster expansion to resolve the $J_d < J_t$ regime, revealing a first-order transition at $J_d/J_t \approx 0.185$ from a dimer VBS to a competing valence bond crystal. The results show that a $\sqrt{3} \times \sqrt{3}$ VBC is energetically favored over a columnar VBC across finite bond dimensions, with the perturbative analysis indicating the energy split emerges at sixth order and aligns with the iPEPS outcomes. This work clarifies the subtle competition among VBC states in a geometrically frustrated 2D lattice and informs experimental exploration of star-lattice magnets, where Dzyaloshinskii–Moriya interactions and material specifics may further influence the phase behavior.

Abstract

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.

Figures (9)

• Figure 1: (a) The star lattice with its six-site unit cell, and the two symmetry-inequivalent nearest-neighbor bonds with Heisenberg exchange couplings $J_d$ ('dimer bonds') and $J_t$ ('trimer bonds'). (b)-(d) The candidate valence bond ground states of \ref{['eq-hamil']} for $J_d,J_t>0$. The blue ellipses indicates strong singlet amplitudes (also throughout the article). (b) Fully-symmetric dimer VBS state. (c) $C_3$ symmetry-breaking VBC with $\sqrt{3}\times\sqrt{3}$ order. (d) $C_3$ breaking columnar VBC with six-site unit-cell. The $\sqrt{3}\times\sqrt{3}$ VBC and the columnar VBC are both three-fold degenerate. (e) Summary of previously reported phase diagrams for the system, alongside our result, as a function of $J_d/J_t$. ED predics a transition at $J_d/J_t \approx 0.77$Star-ED; Gutzwiller-projected wavefunction and bond-operator approaches suggest a first-order phase transition at $J_d/J_t \approx 0.42$Star_PSG; iPEPS studies report either a continuous transition at $J_d/J_t \approx 0.91$Star-iPEPS1 or no transition at all down to $J_d/J_t \approx 0.33$Star-iPEPS2. The first two studies propose a $\sqrt{3} \times \sqrt{3}$ VBC as the favored state for $J_t \gg J_d$, while the iPEPS study Ref. Star-iPEPS1 finds a resonating VBS with a six-site unit cell. Our result indicates a first-order transition out of the dimer VBS phase at $J_d/J_t = 0.185(5)$ into the $\sqrt{3} \times \sqrt{3}$ VBC.
• Figure 2: (a) iPEPS representation of a 2D wave function on the honeycomb lattice with a six-site unit cell. The black lines represent the virtual legs of dimension $D$ while the gray lines represent the physical legs of dimension $d=8$. (b) The overlap of wavefunctions is represented as the contraction of an infinite two-dimensional tensor network. The overlap of the tensors $a^{[\mathbf{x}]\dagger}a^{[\mathbf{x}]}$ is mapped to a local tensor $A^{[\mathbf{x}]}$ with dimensions $D^2 \times D^2 \times D^2$. This contraction is approximated using an environment composed of row and corner tensors. The bold black lines indicate the environment bond dimension $\chi$.
• Figure 3: The two-trimer unit based setups that hosts only (a) the columnar VBC and (b) the $\sqrt{3}\times\sqrt{3}$ VBC state. Note that in the first setup, the two tensors, $a$ and $b$, are arranged in an $a$-$b$-$a$-$b$-$a$-$b$ sequence. In contrast, the second setup follows the ordering $a$-$b$-$R(a)$-$R(b)$-$R^2(a)$-$R^2(b)$ in counterclockwise direction, where $R$ represents a rotation by $2\pi/3$ about the center of the tensors.
• Figure 4: The deformation of the Hamiltonian \ref{['eq-hamil']} for the $\sqrt{3}\times\sqrt{3}$ and columnar VBC states into bowties with the usual $J_t$ and $J_d$ couplings and the couplings between the bowties, $\lambda$, as a small perturbations
• Figure 5: Scaling of the GS energies per site, $e_g$, obtained in iPEPS as a function of the bond dimension $D$ at the isotropic point, $J_d = J_t = 1$. The spin-spin correlations on the two types of bonds are also shown. The GS is a uniform dimer VBS state. The excellent energy convergence for the larger values of $D$ is due to the low entanglement of the GS.