Tensor-network study of the ground state of maple-leaf Heisenberg antiferromagnet

TL;DR

The paper studies the ground-state phase diagram of the spin-$\frac{1}{2}$ nearest-neighbor Heisenberg antiferromagnet on the maple-leaf lattice with $J_h = J_t = J$ as a function of the dimer coupling $J_d$. It applies infinite projected entangled pair states (iPEPS) with a $C_3$-symmetric CTMRG to access the thermodynamic limit and determine phase boundaries. The main finding is that only two phases exist—the magnetically ordered canted-$120^rac{ ext{2}}{ ext{3}}$ phase and an exact dimer singlet product state—with a first-order transition at $J_d/J \approx 1.45$ and small finite moments in the ordered region, while quantum fluctuations renormalize the canting angle away from the classical prediction across most of the ordered regime. The work demonstrates the capability of iPEPS+CTMRG for frustrated lattices and provides openly available data, contributing to a clearer understanding of the maple-leaf Heisenberg model and benchmarking against CCM and other numerical methods.

Abstract

We study the quantum phase diagram of the spin-$1/2$ nearest-neighbor Heisenberg model on the maple-leaf lattice using infinite projected entangled pair states (iPEPS) combined with a corner transfer matrix renormalization group scheme adapted to $C_3$-symmetric lattices. Focusing on the fully antiferromagnetic $J$-$J_d$ model with $J_h = J_t := J$, we map out the ground-state phase diagram as a function of the dimer coupling $J_d$. Our results show that the system hosts only two phases: a magnetically ordered canted-$120^\circ$ phase and an exact dimer singlet product phase. We identify a first-order transition between these two phases at $J_d/J \approx 1.45$. Within the magnetically ordered phase, we observe small but finite magnetic moments. We also resolve the quantum renormalization of the canting angle, which deviates from the classical prediction over almost the entire magnetically ordered phase.

Figures (4)

• Figure 1: (a) Maple-leaf lattice (MLL) with nearest-neighbor interactions. The green dashed bonds denote the dimer couplings $J_d$, while the thick violet and thin red bonds represent the inter-dimer couplings $J_t$ and $J_h$, respectively. (b) Ground-state phase diagram of the Heisenberg model on the maple-leaf lattice with $J_h = J_t = 1$ as a function of $J_d$, obtained using different numerical techniques, including the present work. Both CCM Farnell2011 and our iPEPS simulations find that the system realizes only two phases, namely the canted-$120^\circ$ ordered phase and the product dimer singlet phase, separated by a first-order transition at $J_d \approx 1.45$. The iDMRG Beck2024 and NQS calculations Beck2024 also identify these two phases, but with an intervening region of uncertainty between them, indicated by the dashed lines. In contrast, the pseudofermion functional renormalization group study Gresista2023 reports a quantum spin liquid phase between the canted-$120^\circ$ ordered phase and the product dimer singlet phase.
• Figure 2: (a) Coarse-grained three-site tensors. Each $J_t$ trimer of the lattice is represented by a tensor with physical dimension $d = 8$. The thick black lines indicate virtual bonds of dimension $D$. (b) Two-dimensional iPEPS wave function defined on a honeycomb network with a six-site unit cell. The gray lines denote the physical legs. (c) The wave-function overlap is represented by the contraction of an infinite tensor network. (d) This contraction is approximated by an effective environment composed of row and corner tensors. The thicker black lines indicate environment bonds of dimension $\chi$.
• Figure 3: Scaling of the ground-state energy per site, $e_g$, obtained from iPEPS as a function of $1/D$ at $J_h = J_t = 1$ for (a) $J_d = 0$ and (b) $J_d = 1$. Energies obtained using different numerical approaches, namely CCM Farnell2011, iDMRG Beck2024, and entropy method Hutsak_maple, are also indicated. The dark solid lines show the $D \to \infty$ extrapolations obtained by fitting the data. The shaded regions represent estimates of the error bars, where the upper bounds correspond to the energies at $D = 12$, and the lower bounds are obtained from a linear extrapolation of the $D = 11$ and $D = 12$ data. Scaling of the average local magnetic moment, $m$, obtained from iPEPS as a function of $1/D$ at $J_h = J_t = 1$ for (c) $J_d = 0$ and (d) $J_d = 1$. Conservative error bar estimates for $m$, indicated by the shaded regions, are obtained in the same manner as for the energy.
• Figure 4: (a) Energy of the canted-$120^\circ$ ordered ground state as a function of $J_d$, obtained from iPEPS calculations for different bond dimensions $D$. The dashed line indicates the energy of the exact dimer singlet product state. The transition between the canted-$120^\circ$ phase and the exact dimer phase is clearly first order, as can be seen in the inset. (b) Dependence of the local spin moment, $m$, in the canted-$120^\circ$ phase on $J_d$, obtained from iPEPS calculations for various bond dimensions $D$. (c) Canting angle $\alpha$ as a function of $J_d$. The definition of the canting angle is shown in the inset. The dashed curve corresponds to the classical canting angle $\alpha_\text{cl}$ [see Eq. \ref{['eq:alpha']}].