Thermodynamics of the Heisenberg antiferromagnet on the maple-leaf lattice

TL;DR

The study tackles the thermodynamics of the isotropic Heisenberg antiferromagnet on the maple-leaf lattice, a geometrically frustrated 2D system with three inequivalent bonds, by integrating NLCE as the core method with PMFRG, FTLM, and classical Monte Carlo for cross-validation. The authors report a two-peak specific-heat structure at $T_1 \approx 0.479\,J$ and $T_2 \approx 0.131\,J$, and obtain ground-state-energy estimates via zero-temperature NLCE expansions that point to a short-range correlated paramagnetic ground state organized around resonating hexagonal motifs. The equal-time structure factor reveals a pronounced twist near intermediate temperatures due to the lattice’s lack of reflection symmetry, a feature that persists down to $T\to 0$ and is corroborated across methods. Overall, the work provides a coherent finite-temperature picture for a highly frustrated lattice, benchmarks NLCE against PMFRG, FTLM, and classical MC, and offers guidance for experimental neutron-scattering studies.

Abstract

We study the Heisenberg antiferromagnet on the maple-leaf lattice using several numerical approaches, focusing on the numerical linked-cluster expansion (NLCE), which exhibits an unconventional convergence extending to low and even zero temperatures. We evaluate thermodynamic properties as well as spin-spin correlations through the equal-time structure factor. Within NLCE the specific heat capacity reveals a two-peak structure at $T_1 \approx 0.479\,J$ and $T_2 \approx 0.131\,J$, reminiscent of the corresponding result for the triangular lattice. At intermediate temperatures, the spin-spin structure factor develops features that reflect the absence of reflection symmetry in the lattice. The zero-temperature convergence of NLCE enables reliable estimates of the ground-state energy and points to a short-range correlated paramagnetic ground state composed of resonating hexagonal motifs. The NLCE results are benchmarked against Pseudo-Majorana Functional Renormalization Group, finite-temperature Lanczos, and classical Monte Carlo simulations.

Figures (17)

• Figure 1: Depiction of the maple-leaf lattice with its three symmetry-inequivalent nearest-neighbor bonds: triangular (red), hexagonal (blue), and diagonal (green). The figure also shows the Bravais lattice vectors $\mathbf{ a}_1$ and $\mathbf{ a}_2$. The lattice derives its name from its coordination number of five, which allows the use of maple leaves (shown in black) to highlight the nearest-neighbor structure.
• Figure 2: Specific heat capacity as a function of temperature. We show NLCE data from the triangle-based expansion (blue/green; up to order 16) and from the hexagon-based expansion (red; order 3). FTLM data are shown for a cluster with $N = 36$ sites and periodic boundary conditions in black. The positions of the specific-heat peaks of the triangle expansion are indicated at $T_1 \approx 0.479\,J$ and $T_2 \approx 0.131\,J$, as well as the minimum at $T_{\mathrm{min}} \approx 0.198\,J$. For comparison, the high-temperature expansion, along with the Padé approximation and the interpolation to zero temperature using the entropy method (EM) from Hutak hutak_thermodynamics_2025, is shown in yellow.
• Figure 3: Entropy as a function of temperature. We include NLCE results from the triangle-based expansion (blue/green; up to order 16) and from the hexagon-based expansion (red; order 3). FTLM data for a cluster with $N = 36$ sites and periodic boundary conditions are also shown in black. Markers indicate the positions of the two peaks and the minimum in the specific heat capacity, as presented in \ref{['fig:cv']}.
• Figure 4: Energy as a function of temperature. It shows the NLCE data from the triangle-based (blue/green; up to order 16) and hexagon-based (red; order 3) expansions are shown, together with FTLM results for a periodic $N = 36$ cluster (black). The positions of the specific-heat peaks from \ref{['fig:cv']} are marked with black dots. The inset in the energy panel displays the ground-state energy estimates. It contains the literature results shown in \ref{['tab:energies']} from Refs. schmoll_bathing_2025hutak_thermodynamics_2025farnell:2011beck:2024 and NLCE estimates from the triangle expansion (order 16) and the hexagon expansion (order 6), shown in bold fonts.
• Figure 5: Equal-time spin structure factor computed using different numerical methods and at various temperatures. The first, second, and third columns correspond to $T = 2.5\,J$, $T=0.8\,J$, and the Schottky peak at $T = T_1 \approx 0.47\,J$, respectively. Panels (a–c) show the bare NLCE first-order hexagon expansion (decoupled-hexagons limit), panels (d–f) display the third-order hexagon expansion, and panels (g–i) present the 15th-order triangular expansion using Euler resummation, panels (j-l) show PMFRG results, and panels (m-o) refer to the classical Monte Carlo Data results. For details on how we compare with classical Monte Carlo data, see \ref{['app:MC']}.