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Zeitschrift für Naturforschung A

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

Samuel Nyckees, Pratyay Ghosh, Frédéric Mila

Published Mar 11, 2026• 0 citations
Publisher

Abstract

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° phase and an exact dimer singlet product phase. We identify a first-order transition between these two phases at J d / J ≈ 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.

DOI
10.1515/zna-2025-0409

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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° phase and an exact dimer singlet product phase. We identify a first-order transition between these two phases at J d / J ≈ 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.
We study the quantum phase diagram of the spin-1/21/21/2 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 C3C_3C3​-symmetric lattices. Focusing on the fully antiferromagnetic JJJ-JdJ_dJd​ model with Jh=Jt:=JJ_h = J_t := JJh​=Jt​:=J, we map out the ground-state phase diagram as a function of the dimer coupling JdJ_dJd​. Our results show that the system hosts only two phases: a magnetically ordered canted-120∘120^\circ120∘ phase and an exact dimer singlet product phase. We identify a first-order transition between these two phases at Jd/J≈1.45J_d/J \approx 1.45Jd​/J≈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.
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 microscopically analyze the nearest-neighbor Heisenberg model on the maple leaf lattice through neural quantum state (NQS) and infinite density matrix renormalization group (iDMRG) methods. Embarking to parameter regimes beyond the exact dimer singlet ground state with a dimer bond spin exchange coupling Jd{J}_{d}Jd​ varied against the exchange strength JJJ of all other bonds, the iDMRG (NQS) method finds a dimer state paramagnetic phase for Jd/J>1.464{J}_{d}/J>1.464Jd​/J>1.464 (Jd/J>1.39{J}_{d}/J>1.39Jd​/J>1.39) and a canted ${120}^{\ensuremath{\circ}}$ magnetic order for Jd/J<1.419{J}_{d}/J<1.419Jd​/J<1.419 (Jd/J<1.23{J}_{d}/J<1.23Jd​/J<1.23). Assessing training convergence inaccuracies of the NQS method and the influence of finite cylindric circumference in the iDMRG method, we discuss the possible existence of an intermediate phase between the magnet and the dimer paramagnet.
We study the spin-1 Heisenberg model on the square lattice with the antiferromagnetic nearest-neighbor J1{J}_{1}J1​ and the next-nearest-neighbor J2{J}_{2}J2​ couplings by using the infinite projected entangled-pair state (iPEPS) ansatz and the density matrix renormalization group (DMRG) calculation. The iPEPS simulation, which studies the model directly in the thermodynamic limit, finds a crossing of the ground state from the N\'eel magnetic state to the stripe magnetic state at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.549$, showing a direct phase transition. In the finite-size DMRG calculation on the cylinder geometry up to the cylinder width Ly=10{L}_{y}=10Ly​=10, we find that around the same critical point the N\'eel and the stripe orders are strongly suppressed, which implies the absence of an intermediate phase. Both calculations identify that the stripe order comes with a first-order transition at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.549$. Our results indicate that unlike the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ square model, quantum fluctuations in the spin-1 model may not be strong enough to stabilize an intermediate nonmagnetic phase.
Quantum antiferromagnets on geometrically frustrated lattices have long attracted interest for the formation of quantum disordered states and the possible emergence of quantum spin liquid (QSL) ground states. Here we turn to the nearest-neighbor spin-1/21/21/2 Heisenberg antiferromagnet on the maple-leaf lattice, which is known to relieve frustration by the formation of canted ${120}^{\ensuremath{\circ}}$ magnetic order or valence bond crystal order when varying the bond anisotropy. Employing a pseudofermion functional renormalization group approach to assess its ground state phase diagram in detail, we present evidence for a QSL regime sandwiched between these two limiting phases. The formation of such a QSL might signal proximity to a possible deconfined quantum critical point from which it emerges, and that is potentially accessible by tuning the exchange couplings. Our conclusions are based on large-scale simulations involving a careful finite-size scaling analysis of the behavior of magnetic susceptibility and spin-spin correlation functions under renormalization group flow.
We study the ground-state phase diagram of the spin-1/21/21/2 antiferromagnetic Heisenberg model on the square-kagome lattice using infinite projected entangled-pair states (iPEPS). By systematically varying the ratio of exchange couplings on triangular and square plaquettes, we establish a complete quantum phase diagram in the thermodynamic limit. In the intermediate-coupling regime, we identify four distinct nonmagnetic phases that are unambiguously characterized as valence-bond crystals (VBCs) by their symmetry-inequivalent patterns of strong and weak spin-spin correlations. These include a plaquette crossed-dimer VBC, a loop-six VBC stabilized around the isotropic point, a generalized pinwheel VBC with reduced rotational symmetry, and a decorated loop-six VBC proximate to ferrimagnetic order. We determine the phase boundaries using a combination of bond-resolved correlation functions, entanglement entropy, and magnetization. For transitions not accompanied by sharp entanglement signatures, we extract the spin gap from finite-field simulations, allowing us to distinguish gapped and gapless VBC phases. At larger coupling ratios, the system undergoes transitions into imperfect and perfect ferrimagnetic states. Our results resolve long-standing ambiguities in the square-kagome Heisenberg model and provide a quantitatively reliable reference for future theoretical and experimental studies of frustrated quantum magnets.