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arXiv:2604.24055 · Zeitschrift für Naturforschung A

Spin excitation of the Heisenberg antiferromagnet with frustration: from the bounce-lattice antiferromagnet through the maple-leaf-lattice antiferromagnet to the exact-dimer system

Hiroki Nakano, Tôru Sakai

Published May 20, 2026• 1 citation

Abstract

Abstract The spin- S Heisenberg antiferromagnet on the two-dimensional lattice is investigated for S = 1/2 and S = 1. We consider interaction at isolated dimers ( J d ) and interaction bonds that form the bounce lattice ( J b ). For J d = J b , the system is reduced to the maple-leaf-lattice antiferromagnet. We primarily conduct highly parallelized numerical diagonalization to examine the spin excitation gap above the ground state for various J b / J d cases. For S = 1/2, we report calculations for a 42-site cluster that has not been previously treated. The S = 1 case is examined for the first time for clusters up to 24 sites. Regardless of whether S = 1/2 or 1, we find that the system has a gapped nature for small J d / J b and becomes gapless at J d / J b ∼ 1.4. For S = 1, we also find that another gapped region appears between the gapless case at J d / J b ∼ 1.4 and the boundary of the exact-dimer phase.

DOI: 10.1515/zna-2025-0389

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Abstract The spin- S Heisenberg antiferromagnet on the two-dimensional lattice is investigated for S = 1/2 and S = 1. We consider interaction at isolated dimers ( J d ) and interaction bonds that form the bounce lattice ( J b ). For J d = J b , the system is reduced to the maple-leaf-lattice antiferromagnet. We primarily conduct highly parallelized numerical diagonalization to examine the spin excitation gap above the ground state for various J b / J d cases. For S = 1/2, we report calculations for a 42-site cluster that has not been previously treated. The S = 1 case is examined for the first time for clusters up to 24 sites. Regardless of whether S = 1/2 or 1, we find that the system has a gapped nature for small J d / J b and becomes gapless at J d / J b ∼ 1.4. For S = 1, we also find that another gapped region appears between the gapless case at J d / J b ∼ 1.4 and the boundary of the exact-dimer phase.

The S = 1/2 Heisenberg antiferromagnet on the two-dimensional pyramid lattice is studied by the numerical-diagonalization method. This lattice is obtained by the combination of the Lieb lattice and the square lattice. It is known that when interaction on the square lattice is increased from the ferrimagnetic limit of strong interaction on the Lieb lattice, this system shows gradual decrease and disappearance of spontaneous magnetization in the ground state. The present study treats the region near the case of the square-lattice antiferromagnet accompanied by isolated spins by numerical-diagonalization calculations of finite-size clusters with the maximum size of 39 sites. Our numerical results suggest the existence of a new phase with small but nonzero spontaneous magnetization between two zero-spontaneous-magnetization phases.

The S = 1/2 Heisenberg antiferromagnet on the two-dimensional pyramid lattice is studied by the numerical-diagonalization method. This lattice is obtained by the combination of the Lieb lattice and the square lattice. It is known that when interaction on the square lattice is increased from the ferrimagnetic limit of strong interaction on the Lieb lattice, this system shows gradual decrease and disappearance of spontaneous magnetization in the ground state. The present study treats the region near the case of the square-lattice antiferromagnet accompanied by isolated spins by numerical-diagonalization calculations of finite-size clusters with the maximum size of 39 sites. Our numerical results suggest the existence of a new phase with small but nonzero spontaneous magnetization between two zero-spontaneous-magnetization phases.

We examine the ground state and the excitations of an one-dimensional Heisenberg spin 1/2 antiferromagnet with alternating dimers and four-spin plaquettes (dimer-plaquette chain). The properties of the system depend on the competing dimer and plaquette bonds. Several exact, exact-numerical and perturbational results are presented. We find that the system is gapped for all parameter values. For the latter one we find an effective S=1 Haldane like behaviour in the limit of dominating dimer bonds. The model considered is related to the recently found 1/5-depleted square-lattice Heisenberg system

CaV_4O_9

.

Abstract The spin-S Heisenberg antiferromagnet on the orthogonal-dimer lattice is studied. In most of the investigations of the antiferromagnet on this lattice beginning with the study by Shastry and Sutherland, the S = 1 / 2 case has intensively been treated. In the present study, the cases of S = 1 and 3/2 are particularly treated by the numerical-diagonalization method based on the Lanczos algorithm applied to finite-size clusters. We successfully capture the edge of the exact-dimer phase and that of the Néel-ordered phase. A significant finding is the existence of an intermediate phase between both the edges irrespective of spin-S.

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^\circ

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.