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arXiv:cond-mat/0210122

Study of Low-energy States of Clusters of Spin-1/2 and Spin-1 Triangles with Kagome-like Geometries

Indranil Rudra, S. Ramasesha, Diptiman Sen

Published Oct 6, 2002• 0 citations
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Abstract

We study the low-energy properties of Heisenberg antiferromagnetic spin-1/2 and spin-1 systems on various clusters made up of triangles. Some of the clusters have a geometry similar to representative pieces of the Kagome lattice, while others have the geometry of a sawtooth chain. While the ground state always has the lowest possible spin (0 or 1/2), the nature of the low-energy excitations depends on the geometry and the site spin. For the Kagome clusters with spin-1 sites, the lowest excitations are gapped, with singlet and triplet excitations having similar gaps. This is in contrast to Kagome clusters with spin-1/2 sites where there are several low-energy singlet excitations (possibly gapless in the thermodynamic limit), while triplet excitations have a gap. For the sawtooth chain with spin-1 sites, the lowest excitations are triplets with a gap; the gap to singlet excitations is about twice the triplet gap.

DOI
10.48550/arxiv.cond-mat/0210122

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We study the low-energy properties of Heisenberg antiferromagnetic spin-1/2 and spin-1 systems on various clusters made up of triangles. Some of the clusters have a geometry similar to representative pieces of the Kagome lattice, while others have the geometry of a sawtooth chain. While the ground state always has the lowest possible spin (0 or 1/2), the nature of the low-energy excitations depends on the geometry and the site spin. For the Kagome clusters with spin-1 sites, the lowest excitations are gapped, with singlet and triplet excitations having similar gaps. This is in contrast to Kagome clusters with spin-1/2 sites where there are several low-energy singlet excitations (possibly gapless in the thermodynamic limit), while triplet excitations have a gap. For the sawtooth chain with spin-1 sites, the lowest excitations are triplets with a gap; the gap to singlet excitations is about twice the triplet gap.
Motivated by recent work on Heisenberg antiferromagnetic spin systems on various lattices made up of triangles, we examine the low-energy properties of a chain of antiferromagnetically coupled triangles of half-odd-integer spins. We derive the low-energy effective Hamiltonian to second order in the ratio of the coupling J2{J}_{2}J2​ between triangles to the coupling J1{J}_{1}J1​ within each triangle. The effective Hamiltonian contains four states for each triangle which are given by the products of spin-1/2 states with the states of a pseudospin 1/2. We compare the results obtained by exact diagonalization of the effective Hamiltonian with those obtained for the full Hamiltonian using exact diagonalization and the density-matrix renormalization group method. It is found that the effective Hamiltonian gives an accurate value for the ground-state energy only if the ratio J2/J1{J}_{2}{/J}_{1}J2​/J1​ is less than about 0.2 and that too for the spin-1/2 case with linear topology. The chain of spin-1/2 triangles shows interesting properties like spontaneous dimerization and several singlet and triplet low-energy (possibly gapless) states which lie close to the ground state. We have also studied the spin-3/2 case and find the low-energy effective Hamiltonians (LEH's) to be less accurate there than in the spin-1/2 case. Finally, we have studied nonlinear topologies where the LEH results deviate further from the exact results.
We study the low-energy properties of a sawtooth chain with spin-1's at the bases of the triangles and spin-12′s\frac{1}{2}\mathrm{'}\mathrm{s}21​′sat the vertices of the triangles. The spins have Heisenberg antiferromagnetic interactions between nearest neighbors, with a coupling J2{J}_{2}J2​ between a spin-1 and a spin-12,\frac{1}{2},21​, and a coupling J1=1{J}_{1}=1J1​=1 between two spin-1's. Analysis of the exact diagonalization data for periodic chains containing up to N=12N=12N=12 unit cells shows that the ground state is a singlet for exchange couplings up to approximately J2=3.8,{J}_{2}=3.8,J2​=3.8, whereas for larger J2{J}_{2}J2​ the system exhibits a ferrimagnetic ground state characterized by a net ferromagnetic moment per unit cell of 1/2. In the region of small interactions J2,{J}_{2},J2​, the mixed spin sawtooth chain maps on to an effective isotropic spin model representing two weakly interacting and frustrated spin-12\frac{1}{2}21​ Heisenberg chains composed of spin-12\frac{1}{2}21​ sites at odd and even vertices, respectively. Finally, we study the phenomenon of a macroscopic magnetization jump which occurs if a magnetic field is applied with a value close to the saturation field for J2=2.{J}_{2}=2.J2​=2.
We study the system size dependence of the singlet-triplet excitation gap in the S=1/2S=1/2S=1/2 kagome-lattice Heisenberg antiferromagnet by numerical diagonalization. We successfully obtain a new result of a cluster of 42 sites. The two sequences of gaps of systems with even-number sites and that with odd-number sites are separately analyzed. Careful examination clarifies that there is no contradiction when we consider the system to be gapless.
We analyse the antiferromagnetic spin-1/2 Heisenberg model on a depleted kagome lattice, where some bonds have been reduced to exchange integral J_2 << J_1. The fully depleted system consists of 1D chains, each with a doubly degenerate singlet-pair ground state and a spectral gap (like the Majumdar-Ghosh model). There are localised and itinerant low-energy excitations. The modes from the lowest branch of excitations are incapable of lifting the 2D system's ground state degeneracy at finite J_2/J_1 << 1. Low-energy excitations of the 2D system are dominated by coherently propagating bound states of the 1D excitations.
We study the spin-1∕21∕21∕2 Heisenberg model in a lattice that interpolates between the triangular and the kagome lattices. The exchange interaction along the bonds of the kagome lattice is JJJ, and the one along the bonds connecting kagome and nonkagome sites is ${J}^{\ensuremath{'}}$, so that ${J}^{\ensuremath{'}}=J$ corresponds to the triangular limit and ${J}^{\ensuremath{'}}=0$ to the kagome one. We use variational and exact diagonalization techniques. We analyze the behavior of the order parameter for the antiferromagnetic phase of the triangular lattice, the spin gap, and the structure of the spin excitations as functions of ${J}^{\ensuremath{'}}∕J$. Our results indicate that the antiferromagnetic order is not affected by the reduction of ${J}^{\ensuremath{'}}$ down to ${J}^{\ensuremath{'}}∕J\ensuremath{\simeq}0.2$. Below this value, antiferromagnetic correlations grow weaker, a description of the ground state in terms of a N\'eel phase renormalized by quantum fluctuations becomes inadequate, and the finite-size spectra develop features that are not compatible with antiferromagnetic ordering. However, this phase does not appear to be connected to the kagome phase as well, as the low-energy spectra do not evolve with continuity for ${J}^{\ensuremath{'}}\ensuremath{\rightarrow}0$ to the kagome limit. In particular, for any nonzero value of ${J}^{\ensuremath{'}}$, the latter interaction sets the energy scale for the low-lying spin excitations, and a gapless triplet spectrum, destabilizing the kagome phase, is expected.
A proposed ground state for the kagome lattice Heisenberg model consists of a valence bond crystal (VBC) with a 36-site unit cell. We calculate the low-lying triplet and singlet excitations in the VBC phase for the infinite-lattice model and for the 36-site cluster. For the infinite lattice, the lowest triplet excitation is found to have a spin gap of approximately $0.08\ifmmode\pm\else\textpm\fi{}0.02J$ and a bandwidth of only about 0.01J0.01J0.01J. For the 36-site cluster, which consists of a single unit cell with periodic boundary conditions, there are substantial finite-size effects: the spin gap there is estimated to be approximately 0.2J0.2J0.2J, which is close to the exact diagonalization result of 0.164J0.164J0.164J. The triplet excitations attract one another and form many bound states in the spin-singlet channel. We find a large number of such bound states for the 36-site cluster, many of which appear to lie below the spin gap, which is again in agreement with the results from exact diagonalization.