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arXiv:cond-mat/9904376 · Physical review. B, Condensed matter

Exact ground state and kink-like excitations of a two-dimensional Heisenberg antiferromagnet

Rahul Siddharthan

Published Oct 1, 1999• 2 citations
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Abstract

A rare example of a two-dimensional Heisenberg model with an exact dimerized ground state is presented. This model, which can be regarded as a variation on the kagom\'e lattice, has several features of interest: it has a highly (but not macroscopically) degenerate ground state; it is closely related to spin chains studied by earlier authors; in particular, it exhibits domain-wall-like ``kink'' excitations normally associated only with one-dimensional systems. In some limits it decouples into noninteracting chains; unusually, this happens in the limit of strong, rather than weak, interchain coupling.

DOI
10.1103/physrevb.60.r9904

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A rare example of a two-dimensional Heisenberg model with an exact dimerized ground state is presented. This model, which can be regarded as a variation on the kagom'e lattice, has several features of interest: it has a highly (but not macroscopically) degenerate ground state; it is closely related to spin chains studied by earlier authors; in particular, it exhibits domain-wall-like ``kink'' excitations normally associated only with one-dimensional systems. In some limits it decouples into noninteracting chains; unusually, this happens in the limit of strong, rather than weak, interchain coupling.
We investigate the quantum phases of a frustrated antiferromagnetic Heisenberg spin-1/2 model Hamiltonian on a kagom\'e-strip chain (KSC), a one-dimensional analog of the kagom\'e lattice, and construct its phase diagram in an extended exchange-parameter space. The isolated unit cell of this lattice comprises of five spin-1/2 particles, giving rise to several types of magnetic ground states in a unit cell: a spin-3/23/23/2 state as well as spin-1/21/21/2 states with and without additional degeneracies. We explore the ground-state properties of the fully connected thermodynamic system using exact diagonalization and density matrix renormalization group methods, identifying several distinct quantum phases. All but one of the phases exhibit gapless spin excitations. The exception is a dimerized spin-gapped phase that covers a large part of the phase diagram and includes the uniformly exchange coupled system. We argue that this phase can be understood by perturbing around the limit of decoupled unit cells where each unit cell has six degenerate ground states. We use degenerate perturbation theory to obtain an effective Hamiltonian, a Kugel-Khomskii model with an anisotropic spin-one orbital degree of freedom, which helps explain the origin of dimerization.
We investigate the quantum phases of a frustrated antiferromagnetic Heisenberg spin-1/2 model Hamiltonian on a Kagome-strip chain (KSC), a one-dimensional analogue of the Kagome lattice, and construct its phase diagram in an extended exchange parameter space. The isolated unit cell of this lattice comprises of five spin-1/2 particles, giving rise to several types of magnetic ground states in a unit cell: a spin-3/23/23/2 state as well as spin-1/21/21/2 states with and without additional degeneracies. We explore the ground state properties of the fully connected thermodynamic system using exact diagonalization and density matrix renormalization group methods, identifying several distinct quantum phases. All but one of the phases exhibit gapless spin excitations. The exception is a dimerized spin-gapped phase that covers a large part of the phase diagram and includes the uniformly exchange coupled system. We argue that this phase can be understood by perturbing around the limit of decoupled unit cells where each unit cell has six degenerate ground states. We use degenerate perturbation theory to obtain an effective Hamiltonian, a Kugel-Khomskii model with an anisotropic spin-one orbital degree of freedom, which helps explain the origin of dimerization.
We study a Heisenberg antiferromagnet (AF) on a one-dimensional strip composed of two rows of corner sharing triangular plaquettes. The geometry of this ladder-like lattice naturally allows for two different exchange couplings, one between pairs of spins on the outer legs (J) and a different one between the spins on the axis and their nearest neighbors on the legs ${(J}^{\ensuremath{'}}).$ For ${J/J}^{\ensuremath{'}}\ensuremath{\lesssim}0.5$ the model is a ferrimagnet. Our main interest is in the region ${J/J}^{\ensuremath{'}}\ensuremath{\gtrsim}0.5,$ where the classical ground state of this system shows the same macroscopic degeneracy as the classical ground state of the antiferromagnet on the kagom\'e lattice. To explore to which extent this similarity between the classical ground states of these two models carries over to their quantum states we have applied exact diagonalization techniques and density-matrix renormalization group (DMRG) methods to our model. Exact diagonalization is restricted to system sizes of up to N=30N=30N=30 sites. For ${J/J}^{\ensuremath{'}}=1,$ the results obtained by this technique, low-energy spectra, correlation functions and the specific heat, agree qualitatively with the results obtained for finite samples of the kagom\'e AF. As in the case of the kagom\'e AF, these finite size results suggest that our model is a spin liquid with a gap between the ground state and the lowest spin excitation. However, extrapolations from DMRG data for strips of up to N=120N=120N=120 sites point towards a vanishing spin gap in a wide range of couplings, $0.5\ensuremath{\lesssim}{J/J}^{\ensuremath{'}}\ensuremath{\lesssim}1.25,$ so that contrary to the above conjecture, our one-dimensional model may in fact be critical in this parameter range. Moreover, our DMRG data indicate that the model undergoes a transition to a gapped state as the ratio ${J/J}^{\ensuremath{'}}$ increases through the value ${J/J}^{\ensuremath{'}}\ensuremath{\simeq}1.25.$ In the vicinity of the transition point, a high density of low-lying singlets develops in the spin gap.
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 CaV4O9CaV_4O_9CaV4​O9​.
We study the quasi-one-dimensional limit of the spin-1/2 quantum Heisenberg antiferromagnet on the kagome lattice. The lattice is divided into antiferromagnetic spin chains (exchange JJJ) that are weakly coupled via intermediate ``dangling'' spins (exchange ${J}^{\ensuremath{'}}$). Using one-dimensional bosonization, renormalization-group methods, and current algebra techniques, the ground state is determined in the limit ${J}^{\ensuremath{'}}⪡J$. We find that the dangling spins and chain spins form a spiral with O(1)O(1)O(1) and $O({J}^{\ensuremath{'}}/J)$ static moments, respectively, atop of which the chain spins exhibit a smaller $O[{({J}^{\ensuremath{'}}/J)}^{2}]$ antiferromagnetically ordered component along the axis perpendicular to the spiral plane.