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arXiv:cond-mat/0404556 · Journal of the Physical Society of Japan

Tensor Product State Formulation for the Spin 1/2 Antiferromagnetic XXZ Model on the Checkerboard Lattice

Nobuya Maeshima

Published Jan 15, 2004• 6 citations
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Abstract

Vertical density matrix algorithm (VDMA), a tensor product state formulation of the ``higher-dimensional'' density matrix renormalization group, is applied to the spin 1/2 antiferromagnetic XXZ model on the checkerboard lattice. The VDMA was, in the preceding study, applied to the transverse field Ising model on the square lattice and the three-dimensional classical Ising model. In the present paper, its implementation procedure is modified in order to apply the VDMA to the XXZ model. Numerical accuracy of the VDMA is investigated for the XXZ model on the square lattice, which shows that the method gives reliable results for the ground state energy. In the frustrated region, VDMA results are compared with a simple calculation based on a magnetically disordered state. It is found that the weakly frustrated region is in the Néel ordered phase, while in the strongly frustrated region the realized phase cannot be identified clearly by the obtained results.

DOI
10.1143/jpsj.73.60

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Vertical density matrix algorithm (VDMA), a tensor product state formulation of the ``higher-dimensional'' density matrix renormalization group, is applied to the spin 1/2 antiferromagnetic XXZ model on the checkerboard lattice. The VDMA was, in the preceding study, applied to the transverse field Ising model on the square lattice and the three-dimensional classical Ising model. In the present paper, its implementation procedure is modified in order to apply the VDMA to the XXZ model. Numerical accuracy of the VDMA is investigated for the XXZ model on the square lattice, which shows that the method gives reliable results for the ground state energy. In the frustrated region, VDMA results are compared with a simple calculation based on a magnetically disordered state. It is found that the weakly frustrated region is in the Néel ordered phase, while in the strongly frustrated region the realized phase cannot be identified clearly by the obtained results.
We present a new algorithm to calculate the thermodynamic quantities of three-dimensional (3D) classical statistical systems, based on the ideas of the tensor product state and the density matrix renormalization group. We represent the maximum-eigenvalue eigenstate of the transfer matrix as the product of local tensors that are iteratively optimized by the use of the "vertical density matrix" formed by cutting the system along the transfer direction. This algorithm, which we call vertical density matrix algorithm (VDMA), is successfully applied to the 3D Ising model. Using the Suzuki-Trotter transformation, we can also apply the VDMA to 2D quantum systems, which we demonstrate for the 2D transverse field Ising model.
By means of the recently proposed algorithm based on the tensor product states, the magnetization process of the spin-1/2 antiferromagnetic XXZ model on a square lattice is investigated. In the large spin-anisotropy limit, clear evidence of a first-order spin–flip transition is observed as an external magnetic field is increased. Our findings of the critical field and the discrete jumps in various local order parameters are in good agreement with the quantum Monte Carlo data in the literature. Our results imply that this algorithm can be an accurate and efficient numerical approach in studying first-order quantum phase transitions in two dimensions.
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.
Ground-state properties of a spin-\textonehalf{} XXZ\mathrm{XXZ}XXZ antiferromagnet on the Bethe lattice with three nearest neighbors are investigated; especially, the nature of the phase transition driven by the anisotropy is examined. In actual calculations, we employ the density-matrix renormalization-group method which is so far used in the studies on the one-dimensional quantum systems; its applicability to the higher-dimensional system is thus exhibited in our calculations. Numerical data on local spin correlations imply that the model undergoes the first-order phase transition at the Heisenberg point Jxy=Jz{J}_{\mathrm{xy}}={J}_{z}Jxy​=Jz​ and the ground state is conjectured to be in the Ising-like ordered phase (the ordered phase in XY\mathrm{XY}XY plain) for Jxy<Jz{J}_{\mathrm{xy}}<{J}_{z}Jxy​<Jz​ (Jxy>Jz{J}_{\mathrm{xy}}>{J}_{z}Jxy​>Jz​). We also compare our results with those for other systems, say, the XYZ\mathrm{XYZ}XYZ model on the square lattice to discuss a possible type of symmetry breaking.
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.