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arXiv:cond-mat/0501460 · Physical Review B

Effective quantum dimer model for trimerized kagomé antiferromagnet

Mike E. Zhitomirsky

Published Jun 16, 2005• 19 citations
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Abstract

An effective spin-orbit Hamiltonian is derived for a spin-$1∕2$ trimerized kagom\'e antiferromagnet in the second-order of perturbation theory in the ratio of two coupling constants. Low-energy singlet states of the obtained model are mapped to a quantum dimer model on a triangular lattice. The quantum dimer model is dominated by dimer resonances on a few shortest loops of the triangular lattice. The characteristic energy scale for the dimer model constitutes only a small fraction of the weaker exchange coupling constant.

DOI
10.1103/physrevb.71.214413

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An effective spin-orbit Hamiltonian is derived for a spin-1∕21∕21∕2 trimerized kagom'e antiferromagnet in the second-order of perturbation theory in the ratio of two coupling constants. Low-energy singlet states of the obtained model are mapped to a quantum dimer model on a triangular lattice. The quantum dimer model is dominated by dimer resonances on a few shortest loops of the triangular lattice. The characteristic energy scale for the dimer model constitutes only a small fraction of the weaker exchange coupling constant.
The spin-1/2 Heisenberg antiferromagnet on the kagomé 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 nonmagnetic T2 specific heat observed experimentally.
In analogy to the quantum dimer model on the square lattice introduced by Rokhsar and Kivelson, we identify the low-energy Hilbert space of the spin-1/2 kagom\'e Heisenberg antiferromagnet with the set of dimerized states (nearest-neighbor singlets) and derive an effective Hamiltonian which describes resonance and includes fluctuations into next-nearest-neighbor singlets. Complete diagonalization of this effective Hamiltonian for systems as large as 36 spins makes possible the study of the low-energy thermodynamics of this quantum antiferromagnet and strongly supports the double peak feature in the heat capacity.
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 study spin-1/2 Heisenberg antiferromagnets on a diamond-like-decorated square lattice perturbed by two kinds of further neighbor couplings. In our previous study [J. Phys. Soc. Jpn. 85, 094002 (2016)], the second-order effective Hamiltonian for the Heisenberg model perturbed by a further neighbor coupling was found to be a square-lattice quantum-dimer model with a finite hopping amplitude, t>0, and no dimer-dimer interaction, v=0. In this study, we introduce another kind of further neighbor coupling and show that it leads to an attractive interaction between dimers, which suggests the stabilization of the columnar phase of the square-lattice quantum-dimer model. The calculated v/t is presented as a function of the ratio of the two exchange parameters in the Heisenberg model.
Through exact diagonalization study of the spin - 1/2 Heisenberg model on Kagome lattice with ring-exchange coupling JrJ_{r}Jr​, we find the pure Heisenberg model with Jr=0J_{r}=0Jr​=0 stands as a quantum critical point, as evidenced by avoided level crossing and divergence of the second derivative of the ground state energy with respect to JrJ_{r}Jr​ at Jr=0J_{r}=0Jr​=0. The pure Heisenberg model should thus be gapless in the thermodynamic limit, contrary to common beliefs. At the same time, the ring exchange coupling is found to drive the system into a state with more short ranged spin correlation and with a local spin correlation pattern equivalent to that of the antiferromagnetic Heisenberg model on triangular lattice(the peak of spin structure factor moves to the momentum q⃗=(4π3,0)\vec{q}=(\frac{4π}{3},0)q​=(34π​,0)). The resemblance of state with the Marston-Zeng spin Pereils solid state(in terms of dimer-dimer correlation) is also much enhanced by the ring exchange coupling, although it is unclear if such correlation would solidify into static order in the thermodynamic limit.