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Journal of the Physical Society of Japan

Real Space Renormalization Group Study of the Long Range Order in the 2 Dimensional Spin 1/2XXZHeisenberg Model

Shu-ichi Nagasawa

Published Jan 15, 1990• 3 citations
Publisher

Abstract

The presence of the long range order in the ground state of the quantum XXZ Heisenberg model is investigated using real space renormalization group method. Our results suggest the existence of the long range order in the Z component of the spin for | J Z / J X |>1.0, and that in the XY component of the spin for | J Z / J X |<1.0.
DOI
10.1143/jpsj.59.300

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We perform a real-space renormalization-group analysis for the two-dimensional antiferromagnetic Heisenberg model. We first derive an effective model, described by the Hamiltonian: H=J Jj=1L{\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}}Jj=1L​ S(j)\ensuremath{\cdot}S(j+1)+ Jj=1L{\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}}Jj=1L​(-1)j{)}^{\mathit{j}})j\ensuremath{\Vert}J(j)\ensuremath{\Vert}S(0)\ensuremath{\cdot}S(j), where S(0) represents a single spin located in the center of the ring. We show that the renormalized coupling constants \ensuremath{\Vert}J(j)\ensuremath{\Vert} tend to nonzero values as one increases the block size by successive renormalization. Using analytic and numerical arguments, we establish the existence of antiferromagnetic long-range order in the effective model and hence, in the original Heisenberg model. We compare our results with those obtained from perturbation theory, spin-wave theory, and exact-diagonalization calculations.
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A simple modification of the standard Renormalization Group (RG) technique for the study of quantum spin systems is introduced. Our method which takes into account the effect of boundary conditions by employing the concept of superblock, may be regarded as a simple way for obtaining first estimates of many properties of spin systems. By applying this method to the xxz spin 1/2 Heisenberg chain, we obtain the ground state energy with much higher accuracy than the standard RG. We have also obtained the staggered magnetization and the z-component of spin-spin correlation function which confirms the absence of long-range order in the massless region of the 1D xxz model.
A simple modification of the standard Renormalization Group (RG) technique for the study of quantum spin systems is introduced. Our method which takes into account the effect of boundary conditions by employing the concept of superblock, may be regarded as a simple way for obtaining first estimates of many properties of spin systems. By applying this method to the XXZ spin-[Formula: see text] Heisenberg chain, we obtain the ground state energy with much higher accuracy than the standard RG. We have also obtained the staggered magnetization and the z-component of spin–spin correlation function which confirms the absence of long-range order in the massless region of the 1D XXZ model.
We prove the existence of long-range order (LRO) in the ground state of the 2D quantum spin-1/2 XXZ-model for all anisotropies 0≤Δ leq0.220 \leq \Delta\ leq 0.220≤Δ leq0.22 and Δ≥1.47. We achieve this extension of previous proofs by developing an improved variant of the Dyson-Lieb-Simon proof of LRO in quantum spin systems and by deriving improved upper and lower bounds for the ground state energy of the XXZ-model.