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arXiv:cond-mat/0611170 · Physical Review B

Two-dimensional spin-1 frustrated Heisenberg model with valence-bond ground states

Zi Cai, Shu Chen, Su-Peng Kou, Yupeng Wang

Published Aug 24, 2007• 6 citations
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Abstract

In this paper, we propose a method to understand the nature of the quantum disorder phase of the two-dimensional (2D) high-spin frustrated model. The ground state and excitation properties of a fully frustrated 2D spin-1 model are studied based on a model whose ground state can be found exactly. By analogy to the pseudopotential approach in the fractional quantum Hall effect, we conclude that the ground states of the fully frustrated spin-1 model are doubly degenerate valence-bond solid (VBS) states along the horizontal or vertical direction of the square lattice. We also find that the ground state could be characterized by a nonzero string order, which rarely happened in the 2D case. The method that we used reveals the connection between the fractional quantum Hall effect and the frustrated 2D antiferromagnetism system. The VBS states capture the main character of the disordered phase in the 2D spin-1 frustrated system and can be verified by a numerical method.

DOI
10.1103/physrevb.76.054443

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In this paper, we propose a method to understand the nature of the quantum disorder phase of the two-dimensional (2D) high-spin frustrated model. The ground state and excitation properties of a fully frustrated 2D spin-1 model are studied based on a model whose ground state can be found exactly. By analogy to the pseudopotential approach in the fractional quantum Hall effect, we conclude that the ground states of the fully frustrated spin-1 model are doubly degenerate valence-bond solid (VBS) states along the horizontal or vertical direction of the square lattice. We also find that the ground state could be characterized by a nonzero string order, which rarely happened in the 2D case. The method that we used reveals the connection between the fractional quantum Hall effect and the frustrated 2D antiferromagnetism system. The VBS states capture the main character of the disordered phase in the 2D spin-1 frustrated system and can be verified by a numerical method.
We outline the recent results on the ground state for a class of one- and two-dimensional frustrated quantum spin models with competing ferro(F)- and antiferromagnetic (AF) interactions. Frustrated spin systems are known to have many interesting properties due to large quantum fluctuations. As a result of these fluctuations usual mean-field approach gives quite crude (if not false) description of these systems. Therefore, exactly solvable models are very instructive at investigations of such systems. The exact ground state wave function of proposed models has a structure of the valence-bond state (VBS) type. One of the 1D model describes the transition line between the F and AF phase. The exact singlet ground state on this line has a double-spiral ordering. Using different approximation methods we study the magnetization curve in the AF phase. The second considered set of the 1D and 2D models has an exact non-degenerate ground state with exponentially decaying spin correlations. We also proposed the 1D and 2D electronic models with exact ground state represented in terms of singlet bond functions which are the generalization of the RVB functions including ionic states.
Using the coupled cluster method for high orders of approximation and complementary exact diagonalization studies we investigate the ground state properties of the spin-1/2 J1−−J2{J}_{1}--{J}_{2}J1​−−J2​ frustrated Heisenberg antiferromagnet on the square lattice. We have calculated the ground state energy, the magnetic order parameter, the spin stiffness, and several generalized susceptibilities to probe magnetically disordered quantum valence-bond phases. We have found that the quantum critical points for both the N\'eel and collinear orders are ${J}_{2}^{c1}\ensuremath{\approx}(0.44\ifmmode\pm\else\textpm\fi{}0.01){J}_{1}$ and ${J}_{2}^{c2}\ensuremath{\approx}(0.59\ifmmode\pm\else\textpm\fi{}0.01){J}_{1}$, respectively, which are in good agreement with the results obtained by other approximations. In contrast to the recent study by [Sirker et al. Phys. Rev. B 73, 184420 (2006)], our data do not provide evidence for the transition from the N\'eel to the valence-bond solid state to be first order. Moreover, our results are in favor of the deconfinement scenario for that phase transition. We also discuss the nature of the magnetically disordered quantum phase.
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.
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The nature of the intermediate ground‐state phase in the spin‐1/2 frustrated square lattice model has long been debated. We investigate the phase diagram of this model using cluster density matrix embedding theory (CDMET). Two antiferromagnetic phases are directly identified from our analysis of the ground‐state wave function: the Néel antiferromagnetic phase for J 2 ≤ 0.45 and the stripe antiferromagnetic phase for J 2 ≥ 0.65. Although our analysis of the ground state finds no direct evidence of an internal phase transition within the intermediate phase, an examination of the first excited state reveals a continuous quantum phase transition therein, with a critical point at J 2 = 0.55 that separates the intermediate phase into a plaquette valence bond solid (PVBS) and a (P + C) valence bond solid (VBS).