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arXiv:1108.2532 · International Journal of Modern Physics B

Entanglement perturbation theory for infinite quasi-1D quantum systems

Lihua Wang, S. G. Chung

Published Dec 31, 2014• 2 citations
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Abstract

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.

DOI
10.1142/s0217979215500423

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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.
We study the zero-temperature phase diagram of the J 1 XXZ – J 2 XXZ Heisenberg model for spin-1 particles on an infinite square lattice interacting via nearest-neighbour ( ) and next-nearest-neighbour ( J 2 >0) bonds. The two bonds have the same XXZ -type anisotropy in spin space. The effects on the quasiclassical Néel-ordered and collinear stripe-ordered states of varying the anisotropy parameter Δ are investigated using the coupled cluster method carried out up to high orders. By contrast with the case for spin- particles studied previously, no intermediate disordered phase between the Néel and collinear stripe phases, for any value of the frustration J 2 / J 1 , for either the z -aligned (Δ>1) or xy -planar-aligned (0≤Δ<1) states, is predicted here. The quantum phase transition is determined as first order for all values of J 2 / J 1 and Δ. The position of the phase boundary J 2 c (Δ) is determined accurately. It is observed to deviate most from its classical position (for all values of Δ>0) at the Heisenberg isotropic point (Δ = 1), where J 2 c (1) = 0.55 ± 0.01. By contrast, at the XY isotropic point (Δ = 0), we find J 2 c (0) = 0.50 ± 0.01. In the Ising limit ( ), as expected.
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.
We study here different regions in phase diagrams of the spin-1/2, spin-1 and spin-3/2 one-dimensional antiferromagnetic Heisenberg systems with frustration (next-nearest-neighbor interaction J2) and dimerization (δ). In particular, we analyze the behaviors of the bipartite entanglement entropy and fidelity at the gapless to gapped phase transitions and across the lines separating different phases in the J2-δ plane. All the calculations in this work are based on numerical exact diagonalizations of finite systems.
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We calculate the quantum phase diagram of an XXZ chain with nearest-neighbor (NN) J1{J}_{1}J1​ and next-NN exchange J2{J}_{2}J2​ with anisotropies ${\ensuremath{\Delta}}_{1}$ and ${\ensuremath{\Delta}}_{2},$ respectively. In particular we consider the case ${\ensuremath{\Delta}}_{1}=\ensuremath{-}{\ensuremath{\Delta}}_{2},$ to interpolate between the XX chain $({\ensuremath{\Delta}}_{i}=0)$ and the isotropic model with ferromagnetic phase J2.{J}_{2}.J2​. For ${\ensuremath{\Delta}}_{1}<\ensuremath{-}1,$ a ferromagnetic phase and two antiferromagnetic phases exist. For $|{\ensuremath{\Delta}}_{i}|<1,$ the boundary between the dimer and spin fluid phases is determined by the method of crossing of excitation spectra. For large J2/J1,{J}_{2}{/J}_{1},J2​/J1​, this method seems to indicate the existence of a second spin fluid critical phase. However, an analysis of the spin stiffness and magnetic susceptibility for ${\ensuremath{\Delta}}_{1}={\ensuremath{\Delta}}_{2}=1$ suggests that a small gap is present.