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arXiv:cond-mat/9402002

Block spins and Chirality in Heisenberg model on Kagome and triangular lattices

V. Subrahmanyam

Published Feb 1, 1994• 0 citations
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Abstract

The Spin-1/2 Heisenberg Model (HM) is investigated using a block-spin renormalization approach on Kagome and triangular lattices. In both cases, after coarse graining the triangles on original lattice and truncation of the Hilbert space to the triangular ground state subspace, HM reduces to an effective model on a triangular lattice in terms of the triangular-block degrees of freedom i.e. the spin and the chirality quantum numbers. The chirality part of the effective Hamiltonian captures the essential difference between the two lattices. It is seen that simple eigenstates can be constructed for the effective model whose energies serve as upper bounds on the exact ground state energy of HM, and chiral-ordered variational states have high energies compared to the other variational states.

DOI
10.48550/arxiv.cond-mat/9402002

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The Spin-1/2 Heisenberg Model (HM) is investigated using a block-spin renormalization approach on Kagome and triangular lattices. In both cases, after coarse graining the triangles on original lattice and truncation of the Hilbert space to the triangular ground state subspace, HM reduces to an effective model on a triangular lattice in terms of the triangular-block degrees of freedom i.e. the spin and the chirality quantum numbers. The chirality part of the effective Hamiltonian captures the essential difference between the two lattices. It is seen that simple eigenstates can be constructed for the effective model whose energies serve as upper bounds on the exact ground state energy of HM, and chiral-ordered variational states have high energies compared to the other variational states.
The spin-1/2 frustrated Heisenberg model (HM) is investigated using a block-spin perturbation approach on kagome$aai--- and triangular lattices. In both cases, after coarse graining the triangles on the original lattice and truncation of the Hilbert space to the triangular ground-state subspace, HM reduces to an effective model on a triangular lattice in terms of the triangular-block degrees of freedom, viz. the spin and the chirality quantum numbers. The chirality part of the effective Hamiltonian captures the essential difference between the two lattices. It is seen that chiral-ordered variational states have high energies compared to the other variational states.
We study low-energy properties of the S=1/2 kagome-triangular-strip Heisenberg model. Using exact diagonalization, we calculate magnetization curves of the model in the ferromagnetic kagome coupling regime, where the singlet ground state and 1/3 plateau emerge. In order to analyze formation of the singlet ground state, we further derive a low-energy effective ladder model, focusing on the parity degrees of freedom for the two degenerating doublets of the unit triangle. We then discuss the entanglement between the spin and parity degrees of freedom.
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.
We present a comprehensive analytical theory of quantum spin liquids (QSLs) on the kagome lattice with Heisenberg exchange J1, next-nearest-neighbor exchange J2, Dzyaloshinskii-Moriya interaction D, and scalar spin chirality interaction λ. The work contains three principal contributions. First, we derive new mean-field analytic relations connecting the chiral interaction λ to the spinon gap, low-temperature thermodynamics, and thermal Hall response, with explicit spin-S dependence and generalization across kagome, triangular, and honeycomb lattices. The central result is Δspinon ∝ S³λ/J12 with lattice-dependent prefactors derived from self-consistent mean-field equations. Second, we derive a controlled effective model by projecting the kagome Hamiltonian onto the triangle-doublet subspace, obtaining an effective spin-chirality model on the triangular lattice with complex-valued interactions arising from the C3 symmetry of the chirality eigenstates. We compute all matrix elements of the effective spin operator in the chirality basis and characterize the inter-chirality coupling mediated by shared lattice sites. Third, we provide a rigorous map of obstructions to exact analytical solutions, systematically examining eight classes of proof techniques and identifying the precise mathematical reason each fails for the kagome Hamiltonian. The Lieb-Schultz-Mattis theorem provides the single rigorous exact result: any gapped ground state on the kagome lattice with spin-1/2 per site must exhibit topological order. All other results are classified by derivation category with explicit uncertainty quantification. AI-Generated Content Statement: This manuscript was generated primarily with the assistance of artificial intelligence language models, including ChatGPT. The work represents an exploratory demonstration of AI-assisted knowledge generation and prompt engineering methods rather than traditional experimental research.
The spin half Heisenberg antiferromagnet on the Kagome lattice, is mapped by Contractor Renormalization to a Spin-Pseudospin Hamiltonian on the triangular superlattice. Variationally, we find a ground state with columnar dimer order. Dimer orientation fluctuations are described by an effective O(2) model at energies above an exponentially suppressed clock mass scale. Our results explain the large density of low energy singlets observed numerically, and the non magnetic T-squared specific heat observed experimentally.
We consider the low-energy spectrum of spin-12\frac{1}{2}21​ two-dimensional triangular lattice models subject to a ferromagnetic Heisenberg interaction and a three spin chiral interaction of variable strength. Initially, we consider quasi-one-dimensional ladder systems of various geometries. Analytical results are derived that yield the behavior of the ground states, their energies, and the transition points. The entanglement properties of the ground state of these models are examined and we find that the entanglement depends on the lattice geometry due to frustration effects. To this end, the chirality of a given quantum state is used as a witness of tripartite entanglement. Finally, the two-dimensional model is investigated numerically by means of exact diagonalization and indications are presented that the low energy sector is a chiral spin liquid.