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arXiv:2511.21806 · Zeitschrift für Naturforschung A

Thermodynamics of the Heisenberg antiferromagnet on the maple-leaf lattice

Robin Schäfer, Paul L. Ebert, Noah Hassan, Johannes Reuther, David J. Luitz, Alexander Wietek

Published Feb 4, 2026• 2 citations
arXivHTMLPDF

Abstract

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 ≈ 0.479 J and T 2 ≈ 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.

DOI
10.1515/zna-2025-0382

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Journal of Physics Conference Series•Dec 20, 2012•0 citations
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 ≈ 0.479 J and T 2 ≈ 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.
The Heisenberg antiferromagnet on the maple-leaf lattice has recently gathered a great deal of attention. Competition between three nonequivalent bond interactions results in various ground-state quantum phases, with the exact dimer-product singlet ground state being among them. The thermodynamic properties of this model are much less understood. We used high-temperature expansion up to the 18th order to study the thermodynamics of the S = 1 / 2 Heisenberg model on the uniform maple-leaf lattice with the ground state exhibiting a six-sublattice 120 ∘ long-range magnetic order. Padé approximants allow us to get reliable results up to the temperatures of about T ≈ 0.4 . To study thermodynamics for arbitrary temperatures, we made the interpolation using the entropy method. Based on the analysis of close Padé approximants, we find ground-state energy e 0 = − 0.53064 ... − 0.53023 in good agreement with numerical results. The specific heat c ( T ) has a typical maximum at rather low temperatures T ≈ 0.379 and the uniform susceptibility χ ( T ) at T ≈ 0.49 . We also estimate the value of χ ( T ) at zero temperature χ 0 ≈ 0.05 ... 0.06 . The ground-state order manifests itself in the divergence of the so-called generalized Wilson ratio.
Motivated by the search for unconventional orders in frustrated quantum magnets, we present a multi-method investigation into the nature of the quantum phase diagram of the spin-1/21/21/2 Heisenberg model on the maple-leaf lattice with three symmetry-inequivalent nearest-neighbor interactions. It has been argued that the parameter regime with antiferromagnetic couplings on hexagons JhJ_hJh​ and ferromagnetic couplings on triangles JtJ_tJt​ and dimer JdJ_dJd​ bonds, is potentially host to a cornucopia of emergent phases with unconventional orders. Our analysis indeed identifies an extended region where any conventional dipolar magnetic order is absent. A hexagonal singlet state is found in the region around Jd=Jt=0J_{d}=J_{t}=0Jd​=Jt​=0, while a dimerized hexagonal singlet order of a lattice nematic character appears proximate to the phase boundary with the c120∘120^\circ120∘ antiferromagnetic order. Interestingly, upon traversing the bulk of the paramagnetic (PM) region, we find a variety of distinct correlation profiles, which are qualitatively different from those of the hexagonal singlet and dimerized hexagonal singlet orders but feature no appreciable spin-nematic response, while the boundary with the ferromagnetic phase shows evidence of spin-nematic order. This PM region is thus likely host to an ensemble of nonmagnetic phases which could putatively include quantum spin liquids. Our phase diagram is built from a complementary application of state-of-the-art implementations of the cluster mean-field and pseudo-fermion functional renormalization group approaches, together with an unconstrained Luttinger-Tisza treatment of the model providing insights from the semi-classical limit.
We establish a remarkably rich ground state phase diagram in the maple-leaf lattice spin-1/21/21/2 Heisenberg antiferromagnet as a function of the three symmetry-inequivalent nearest-neighbor bonds using exact diagonalization and tower-of-states analysis on clusters up to N=36N=36N=36 sites. Besides a hexagonal plaquette state, a star-shaped valence bond solid state is discovered in close vicinity to the (canted) 120∘120^\circ120∘ magnetic phase, strongly reminiscent of a de-confined critical point or Dirac spin liquid scenario on the triangular lattice antiferromagnets. Moreover, an exact dimer product-state is observed next to a collinear Néel-state, similar to the Shastry-Sutherland model. All identified phases compete in a parameter regime close to the isotropic point, providing a promising region for spin liquids to emerge. By analyzing Gutzwiller-projected wave-functions we identify a sliver of parameter regime where a gapped Z2\mathbb{Z}_{2}Z2​ spin liquid Ansatz is in astonishing agreement with the exact N=36N=36N=36 ground state. This rich competition of paramagnetic phases demonstrates that the maple-leaf antiferromagnet is a promising platform for exotic states of matter and quantum critical phenomena.
We study Yang-Lee zeros as they approach the thermodynamic limit of the two-dimensional (2D) nearest-neighbor antiferromagnetic Ising model on square and triangular lattices. We employ the numerical linked cluster expansion (NLCE) equipped with exact enumeration (EE) of the partition function to estimate the Laplacian of the free energy, which is proportional to the zeros density. Using a modified NLCE, where the expansion can be carried directly on the Yang-Lee zeros of the involved clusters, we estimate the density of Yang-Lee zeros in the thermodynamic limit. NLCE gives significantly more zeros than EE in the complex field plane providing more insights on how the root curves look in the thermodynamic limit. For the square lattice at $T\ensuremath{\ll}{T}_{c}$, the results suggest that two vertical lines at $\ifmmode\pm\else\textpm\fi{}{h}_{c}(T)$ in the complex field plane (i.e., two concentric circles in the complex fugacity plane) are the thermodynamic root curves. A similar picture is found for the triangular lattice for phase transitions at large values of magnetic field while further study is needed for phase transitions at smaller values of magnetic field. The convergence of the NLCE and (EE) calculations of the partition function to the thermodynamic limit is studied in both lattices and the temperature-field phase diagram is obtained from Yang-Lee zeros using both methods. This NLCE-based approach will facilitate the study of different types of phase transitions using Yang-Lee zeros in future research.
The spin-1/2 Heisenberg antiferromagnet on the two-dimensional ruby and maple-leaf lattices provides a stringent test case for frustrated quantum magnetism, where semiclassical magnetic order competes closely with quantum-paramagnetic states. We study the generalized maple-leaf model along the axis interpolating between the isotropic ruby lattice and the isotropic maple-leaf lattice, using two independent variational infinite projected entangled-pair state ansätze: one defined on the mapped square lattice and one directly on the native triangular lattice. Our lowest-energy variational states show no magnetic order throughout the region connecting the two isotropic limits, and the local bond correlations recover the 120∘120^\circ120∘ lattice rotation symmetry without imposing it explicitly. Magnetic-field simulations at the two endpoints further indicate a finite zero-magnetization plateau in both cases, with a gradual onset of magnetization on the ruby lattice and a sharper onset on the maple-leaf lattice. These results establish a broad gapped non-magnetic regime in a minimal nearest-neighbor Heisenberg model on lattices of direct relevance to both frustrated quantum magnets and programmable ruby-lattice simulators.
Motivated by experimental results for the thermodynamic properties of the Rb 2 Cu 3 SnF 12 material and the discovery of its valence-bond solid ground state, we utilize the numerical linked-cluster expansions (NLCEs) and devise an expansion tailored to solve the Heisenberg model on a pinwheel-distorted kagome lattice. Using the exchange interactions that are relevant to Rb 2 Cu 3 SnF 12 , we calculate its uniform spin susceptibility and find a very good agreement with experiment. Next, we focus on the ground state of a simplified model of the distorted kagome lattice and take advantage of a zero-temperature Lanczos-based NLCE to study the approach to the ground state of the uniform kagome lattice Heisenberg model using a tuning parameter. Together with results from exact diagonalization of finite clusters, we find evidence that a phase transition occurs before the uniform limit is reached.