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arXiv:1109.6758 · Journal of the Physical Society of Japan

Entanglement Perturbation Theory for Antiferromagnetic Heisenberg Spin Chains

Lihua Wang, S. G. Chung

Published Oct 23, 2012• 3 citations
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Abstract

A recently developed numerical method, entanglement perturbation theory (EPT), is used to study the antiferromagnetic Heisenberg spin chains with z-axis anisotropy $λ$ and magnetic field B. To demonstrate the accuracy, we first apply EPT to the isotropic spin-1/2 antiferromagnetic Heisenberg model, and find that EPT successfully reproduces the exact Bethe Ansatz results for the ground state energy, the local magnetization, and the spin correlation functions (Bethe ansatz result is available for the first 7 lattice separations). In particular, EPT confirms for the first time the asymptotic behavior of the spin correlation functions predicted by the conformal field theory, which realizes only for lattice separations larger than 1000. Next, turning on the z-axis anisotropy and the magnetic field, the 2-spin and 4-spin correlation functions are calculated, and the results are compared with those obtained by Bosonization and density matrix renormalization group methods. Finally, for the spin-1 antiferromagnetic Heisenberg model, the ground state phase diagram in $λ$ space is determined with help of the Roomany-Wyld RG finite-size-scaling. The results are in good agreement with those obtained by the level-spectroscopy method.

DOI
10.1143/jpsj.81.114712

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A recently developed numerical method, entanglement perturbation theory (EPT), is used to study the antiferromagnetic Heisenberg spin chains with z-axis anisotropy λλλ and magnetic field B. To demonstrate the accuracy, we first apply EPT to the isotropic spin-1/2 antiferromagnetic Heisenberg model, and find that EPT successfully reproduces the exact Bethe Ansatz results for the ground state energy, the local magnetization, and the spin correlation functions (Bethe ansatz result is available for the first 7 lattice separations). In particular, EPT confirms for the first time the asymptotic behavior of the spin correlation functions predicted by the conformal field theory, which realizes only for lattice separations larger than 1000. Next, turning on the z-axis anisotropy and the magnetic field, the 2-spin and 4-spin correlation functions are calculated, and the results are compared with those obtained by Bosonization and density matrix renormalization group methods. Finally, for the spin-1 antiferromagnetic Heisenberg model, the ground state phase diagram in λλλ space is determined with help of the Roomany-Wyld RG finite-size-scaling. The results are in good agreement with those obtained by the level-spectroscopy method.
A recently developed numerical method, entanglement perturbation theory (EPT), is used to study the antiferromagnetic Heisenberg spin chains with z-axis anisotropy λ and magnetic field B. To demonstrate the accuracy, we first apply EPT to the isotropic spin1 2 antiferromagnetic Heisenberg model, and find that EPT successfully reproduces the exact Bethe Ansatz results for the ground state energy, the local magnetization, and the spin correlation functions (Bethe ansatz result is available for the first 7 lattice separations). In particular, EPT confirms for the first time the asymptotic behavior of the spin correlation functions predicted by the conformal field theory, which realizes only for lattice separations larger than 1000. Next, turning on the z-axis anisotropy and the magnetic field, the 2-spin and 4-spin correlation functions are calculated, and the results are compared with those obtained by Bosonization and density matrix renormalization group methods. Finally, for the spin-1 antiferromagnetic Heisenberg model, the ground state phase diagram in λ space is determined with help of the Roomany-Wyld RG finite-size-scaling. The results are in good agreement with those obtained by the level-spectroscopy method.
In this dissertation, we use the recently developed Entanglement Perturbation Theory (EPT) to solve antiferromagnetic spin chain problems, for both spin 1/2 and spin 1. We firs use EPT-g (EPT algorithm for ground state) to get the ground state properties. We calculated precisely energies, magnetization and spin-spin correlations. The precision of the long range spin-spin correlation functions for spin 1/2 chains is unprecedented. Due to its special structure, we also use EPT-g to calculate the firs excited state properties for spin 1 chains. Hence we plotted the phase diagram for spin 1 xxz chains. Also a generalization of EPT (EPT-e) for elementary excitation in one dimension determined the spin-triplet magnon spectrum for spin 1 chain for the firs time for the entire Brillouin zone, including the Haldane gap at k = π. The spin chains, the 2D&3D Ising models, the 1D&2D Hubbard models and the Bose-Hubbard model whose study by EPT is recently started represent a variety of strongly correlated many body systems with translational symmetry. Namely EPT is a general method for them. On the other hand, the successful application of EPT-e indicates EPT's potential to study the inhomogeneous quantum systems.
We develop entanglement perturbation theory (EPT) for infinite Quasi-1D quantum systems. The spin-1/2 Heisenberg chain with ferromagnetic nearest neighbor (NN) and antiferromagnetic next nearest neighbor (NNN) interactions with an easy-plane anisotropy is studied as a prototypical system. The obtained phase diagram is compared with a recent prediction [Phys. Rev. B 81, 094430 (2010)] that dimer and Néel orders appear alternately as the XXZ anisotropy Δ approaches the isotropic limit Δ = 1. The first and second transitions (across dimer, Néel and dimer phases) are detected with improved accuracy at Δ ≈ 0.722 and 0.930. The third transition (from dimer to Néel phases), previously predicted to be at Δ ≈ 0.98, is not detected at this Δ in our method, strongly indicating that the second Néel phase is absent.
Two-dimensional spin 1/2 antiferromagnetic Heisenberg models are numerically studied using Entanglement Perturbation Theory, where the ground state wave function is described by a product of local matrices defined at every site, and each matrix is optimized variationally to minimize the energy. We first apply this method for the spin 1/2 antiferromagnetic Heisenberg model on the square lattices to reproduce correctly the known ground state energy and the spin structure factors. Then, we study the spin 1/2 antiferromagnetic Heisenberg model on the triangular lattice with spatially anisotropic nearest neighbor couplings, J and J ', to establish the ground state phase diagram as a function of J'/ J.
We consider the properties of finite isotropic antiferromagnetic Heisenberg chains with S=1/2,1,3/2S=1/2,1,3/2S=1/2,1,3/2 spins when a weak magnetic field is applied on a few sites, using White's density-matrix renormalization-group method. For the S=1S=1S=1 chain there exists only one length scale in the system which determines the behavior of the one- and two-point correlation functions both around the local perturbation and near the free boundary. For the critical half-odd-integer spin cases the exponent of the spin-spin correlation function was found to be $\ensuremath{\eta}=1$ and the exponent of the decay of the site magnetization around the perturbed site is ${x}_{m}=\ensuremath{\eta}/2.$ Close to a free boundary, however, the behavior is completely different for S=1/2S=1/2S=1/2 and S>1/2.S>1/2.S>1/2.