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arXiv:cond-mat/0009336 · Canadian Journal of Physics

Exact diagonalization studies of two-dimensional frustrated anti-ferromagnet models

C. Lhuillier, Philippe Sindzingre, J.-B. Fouet

Published Jan 1, 2001• 19 citations
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Abstract

We describe the four kinds of behavior found in two-dimensional isotropic quantum anti-ferromagnets. Two of them display long-range order at T = 0: the Néel state and the valence bond crystal. The last two are spin liquids. Properties of these different states are briefly described and open questions are underlined. PACS Nos.: 75.10Jm, 75.50Ee, 75.40-s

DOI
10.1139/cjp-79-11-12-1525

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We describe the four kinds of behavior found in two-dimensional isotropic quantum anti-ferromagnets. Two of them display long-range order at T = 0: the Néel state and the valence bond crystal. The last two are spin liquids. Properties of these different states are briefly described and open questions are underlined. PACS Nos.: 75.10Jm, 75.50Ee, 75.40-s
Exact diagonalizations of the spin-1/2 Heisenberg model on the checkerboard lattice have been performed for sizes up to N=36N=36N=36 in the full Hilbert space and N=40N=40N=40 in the restricted subspace of first neighbor dimers. This antiferromagnet does not break SU(2) symmetry and displays long-range order in four-spin S=0S=0S=0 plaquettes. Both the symmetry properties of the spectrum and various correlations functions are extensively studied. At variance with the kagom\'e antiferromagnet, the Heisenberg quantum model on a checkerboard lattice is a valence bond crystal. Some results concerning the three-dimensional spin-1/2 pyrochlore magnet (for sizes 16 and 32) are also shown: this system could behave differently from its two-dimensional analog.
Exact diagonalizations indicate that the effective 1-dimensional behavior (sliding Luttinger liquid phase) of the frustrated spin-1/2 crossed chain model, predicted by Starykh, Singh and Levine [Phys. Rev. Lett. 88, 167203 (2002)], persists for a wide range of transverse couplings. The extension of the other phases (plaquette valence bond crystal and Néel long range order) is precised. No clear indication of a coexistence of these two phases is found, at variance with a suggestion of Sachdev and Park (cond-mat/0108214).
I use an improved version of the two-step density-matrix renormalization group method to study ground-state properties of the two-dimensional (2D) Heisenberg model on the checkerboard lattice. In this version, the Hamiltonian is projected on a tensor product of two-leg ladders instead of chains. This allows investigations of 2D isotropic models. I show that this method can describe both the magnetically disordered and ordered phases. The ground-state phases of the checkerboard model as J2{J}_{2}J2​ increases are (i) N\'eel with $Q=(\ensuremath{\pi},\ensuremath{\pi})$, (ii) a valence-bond crystal (VBC) of plaquettes, (iii) N\'eel with $Q=(\ensuremath{\pi}∕2,\ensuremath{\pi})$, and (iv) a VBC of crossed dimers. In agreement with previous results, I find that at the isotropic point J2=J1{J}_{2}={J}_{1}J2​=J1​, the ground state is made of weakly interacting plaquettes with a large gap $\ensuremath{\Delta}\ensuremath{\approx}0.67{J}_{1}$ to triplet excitations. The same approach is also applied to the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model. There is no evidence of a columnar dimer phase in the highly frustrated regime.
Exact diagonalizations of the spin-1/2 Heisenberg model on the checkerboard lattice have been performed for sizes up to N=36 in the full Hilbert space and N=40 in the restricted subspace of first neighbor dimers. This antiferromagnet does not break SU(2) symmetry and displays long range order in 4-spin S=0 plaquettes. Both the symmetry properties of the spectrum and various correlations functions are extensively studied. At variance with the kagom{é} antiferromagnet, the Heisenberg quantum model on a checkerboard lattice is a Valence Bond Crystal. Some results concerning the 3-dimensionnal spin-1/2 pyrochlore magnet (for sizes 16 and 32) are also shown: this system could behave differently from its 2-dimensional analog.