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Physical Review B

Néel and valence-bond crystal phases of frustrated two-dimensional Heisenberg models

S. Moukouri

Published Feb 28, 2008• 17 citations
Publisher

Abstract

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 ${J}_{2}$ 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 ${J}_{2}={J}_{1}$, 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.

DOI
10.1103/physrevb.77.052408

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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=(\ensuremathπ,\ensuremathπ)Q=(\ensuremath{\pi},\ensuremath{\pi})Q=(\ensuremathπ,\ensuremathπ), (ii) a valence-bond crystal (VBC) of plaquettes, (iii) N'eel with Q=(\ensuremathπ∕2,\ensuremathπ)Q=(\ensuremath{\pi}∕2,\ensuremath{\pi})Q=(\ensuremathπ∕2,\ensuremathπ), 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 \ensuremathΔ\ensuremath≈0.67J1\ensuremath{\Delta}\ensuremath{\approx}0.67{J}_{1}\ensuremathΔ\ensuremath≈0.67J1​ to triplet excitations. The same approach is also applied to the J1\ensuremath-J2{J}_{1}\text{\ensuremath{-}}{J}_{2}J1​\ensuremath-J2​ model. There is no evidence of a columnar dimer phase in the highly frustrated regime.
I use an improved version of the two-step density matrix renormalization group method to study ground-state properties of the 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 J2J_2J2​ increases are: (i) Néel with Q=(π,π)Q=(π,π)Q=(π,π), (ii) a valence bond crystal (VBC) of plaquettes, (iii) Néel with Q=(π/2,π)Q=(π/2,π)Q=(π/2,π), and (iv) a VBC of crossed dimers. In agreement with previous results, I find that at the isotropic point J2=J1J_2=J_1J2​=J1​, the ground state is made of weakly interacting plaquettes with a large gap Δ≈0.67J1Δ\approx 0.67 J_1Δ≈0.67J1​ to triplet excitations.
We use the recently developed tensor network algorithm based on infinite projected entangled pair states (iPEPS) to study the phase diagram of frustrated antiferromagnetic J1{J}_{1}J1​-J2{J}_{2}J2​ Heisenberg model on a checkerboard lattice. The simulation indicates a N\'eel ordered phase when J2<0.88J1{J}_{2}<0.88{J}_{1}J2​<0.88J1​, a plaquette valence bond solid state when 0.88<J2/J1<1.110.88<{J}_{2}/{J}_{1}<1.110.88<J2​/J1​<1.11, and a stripe phase when J2>1.11J1{J}_{2}>1.11{J}_{1}J2​>1.11J1​, with two first-order transitions across the phase boundaries. The calculation shows that the cross dimer state proposed before is unlikely to be the ground state of the model, although such a state indeed arises as a metastable state in some parameter region.
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.
We present a comprehensive computational study of the phase diagram of the frustrated S=1/2S=1/2S=1/2 Heisenberg antiferromagnet on the honeycomb lattice, with second-nearest (J2)({J}_{2})(J2​) and third-neighbor (J3)({J}_{3})(J3​) couplings. Using a combination of exact diagonalizations (EDs) of the original spin model, of the Hamiltonian projected into the nearest-neighbor short-range valence-bond basis, and of an effective quantum dimer model, as well as a self-consistent cluster mean-field theory, we determine the boundaries of several magnetically ordered phases in the region ${J}_{2},{J}_{3}\ensuremath{\in}[0,1]$, and find a sizable magnetically disordered region in between. We characterize part of this magnetically disordered phase as a plaquette valence-bond crystal phase. At larger J2{J}_{2}J2​, we locate a sizable region in which staggered valence-bond crystal correlations are found to be important, either due to genuine valence-bond crystal (VBC) ordering or as a consequence of magnetically ordered phases, which break lattice rotational symmetry. Furthermore, we find that a particular parameter-free Gutzwiller projected tight-binding wave function has remarkably accurate energies compared to finite-size extrapolated ED energies along the transition line from conventional N\'eel to plaquette VBC phases, a fact that points to possibly interesting critical behavior---such as a deconfined critical point---across this transition. We also comment on the relevance of this spin model to model the spin liquid region found in the half filled Hubbard model on the honeycomb lattice.
We study the ground-state properties of the two-dimensional spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on a square lattice, within diagrammatic approximations using an auxiliary-fermion formulation with exact projection. In a first approximation, we assume a phenomenological width of the pseudofermion spectral function to calculate the magnetization, susceptibilities, and the spin-correlation length within random-phase approximation, demonstrating the appearance of a paramagnetic phase between the N\'eel-ordered and Collinear-ordered phases, at sufficiently large pseudofermion damping. Second we use a functional renormalization-group formulation. We find that the conventional truncation scheme omitting three-particle and higher-order vertices is not sufficient. We therefore include self-energy renormalizations in the single-scale propagator as recently proposed by Katanin, to preserve Ward identities in a better way. We find N\'eel order at $g={J}_{2}/{J}_{1}\ensuremath{\lesssim}{g}_{\text{c}1}\ensuremath{\approx}0.4\dots{}0.45$ and Collinear order at $g\ensuremath{\gtrsim}{g}_{\text{c}2}\ensuremath{\approx}0.66\dots{}0.68$, which is in good agreement with results obtained by numerical studies. In the intervening quantum paramagnetic phase, we find enhanced columnar dimer and plaquette fluctuations of equal strength.
We study the ground state properties of the two-dimensional spin-1/2 J_1-J_2-Heisenberg model on a square lattice, within diagrammatic approximations using an auxiliary fermion formulation with exact projection. In a first approximation we assume a phenomenological width of the pseudofermion spectral function to calculate the magnetization, susceptibilities and the spin correlation length within RPA, demonstrating the appearance of a paramagnetic phase between the Neel ordered and Collinear ordered phases, at sufficiently large pseudo fermion damping. Secondly we use a Functional Renormalization Group formulation. We find that the conventional truncation scheme omitting three-particle and higher order vertices is not sufficient. We therefore include self-energy renormalizations in the single-scale propagator as recently proposed by Katanin, to preserve Ward identities in a better way. We find Neel order at g = J_2/J_1 0.66 ... 0.68, which is in good agreement with results obtained by numerical studies. In the intervening quantum paramagnetic phase we find enhanced columnar dimer and plaquette fluctuations of equal strength.