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arXiv:1006.4299 · Journal of Physics A Mathematical and Theoretical

Exact ground states for coupled spin trimers

Heinz–Jürgen Schmidt, Johannes Richter

Published Sep 16, 2010• 12 citations

Abstract

We consider a class of geometrically frustrated Heisenberg spin systems which admit exact ground states. The systems consist of suitably coupled antiferromagnetic spin trimers with integer spin quantum numbers s and their ground state Φ will be the product state of the local singlet ground states of the trimers. We provide linear equations for the inter-trimer coupling constants which are equivalent to Φ being an eigenstate of the corresponding Heisenberg Hamiltonian and sufficient conditions for Φ being a ground state. The classical case s → ∞ can be completely analyzed. For the quantum case we consider a couple of examples, where the critical values of the inter-trimer couplings are numerically determined. These examples include chains of corner sharing tetrahedra as well as certain spin tubes. Φ is proven to be gapped in the case of trimer chains. This follows from a more general theorem on quantum chains with product ground states.

DOI: 10.1088/1751-8113/43/40/405205

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We consider a class of geometrically frustrated Heisenberg spin systems which admit exact ground states. The systems consist of suitably coupled antiferromagnetic spin trimers with integer spin quantum numbers s and their ground state Φ will be the product state of the local singlet ground states of the trimers. We provide linear equations for the inter-trimer coupling constants which are equivalent to Φ being an eigenstate of the corresponding Heisenberg Hamiltonian and sufficient conditions for Φ being a ground state. The classical case s → ∞ can be completely analyzed. For the quantum case we consider a couple of examples, where the critical values of the inter-trimer couplings are numerically determined. These examples include chains of corner sharing tetrahedra as well as certain spin tubes. Φ is proven to be gapped in the case of trimer chains. This follows from a more general theorem on quantum chains with product ground states.

We have found the exact ground state for two frustrated quantum spin-

\frac{1}{2}

models on a linear chain. The first model describes the antiferromagnet-ferromagnet transition point. The singlet state at this point has double-spiral ordering. The second model is equivalent to special case of the spin-

\frac{1}{2}

ladder. It has a nondegenerate singlet ground state with exponentially decaying spin correlations, and there is an energy gap. The exact ground-state wave function of these models is presented in a special recurrent form, and a recurrence technique for expectation value calculations is developed.

In this paper, we propose an exotic quantum paramagnetic state in two-dimensional antiferromagnets-the spontaneous trimer state-which is the direct product state of the trimers of spins. Each trimer is a singlet state formed by three neighboring spins with SU(3) symmetry. A frustrated spin-1 Heisenberg model in the kagome lattice is investigated. By analogy to the pseudo-potential approach in the fractional quantum Hall effect (FQHE), we find that the trimer state provides a good description for the exact ground state of this model. Other interesting properties, such as the local excitations as well as magnetization plateaus have also been investigated.

The antiferromagnetic Heisenberg spin systems on the three-leg ladder are investigated. Periodic boundary condition is imposed in the rung direction. The system has an excitation gap for all antiferromagnetic inter-chain coupling (

J_{\perp}>0

). The estimated gap for the strong coupling limit (

J_{\perp}/J_1 \to \infty

) is 0.28

J_1

. Although the interaction is homogeneous and only nearest-neighbor, the ground states of the system are dimerized and break the translational symmetry in the thermodynamic limit. Introducing the next-nearest neighbor coupling (

J_2

), we can see that the system is solved exactly. The ground state wave function is completely dimer-ordered. Using density matrix renomalization group algorithm, we show numerically that the original model (

J_2=0

) has the same nature with the exactly solvable model. The ground state properties of the ladder with a higher odd number of legs are also discussed.

Starting from the exact result of Majumdar and Ghosh for the dimerized ground state of a Heisenberg linear-chain antiferromagnet with competing next-nearest-neighbor interaction

{\mathit{J}}_{2}

=1/2

{\mathit{J}}_{1}

, it is shown that this state is an exact eigenstate of the same Hamiltonian with couplings extended up to fourth-nearest neighbors, if the relations

{\mathit{J}}_{1}

-2

{\mathit{J}}_{2}

+

{\mathit{J}}_{3}

=0 and

{\mathit{J}}_{3}

=2

{\mathit{J}}_{4}

hold. Furthermore, it is proved to all orders in perturbation theory that this state is the ground state of this Hamiltonian in a finite interval of values of the parameter \ensuremath{\lambda}==1/2-

{\mathit{j}}_{2}

around \ensuremath{\lambda}=0. The actual extension of this interval is computed numerically for finite chains and is shown to be significant.