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Journal of the Physical Society of Japan

The Real-Space Renormalization-Group Study of a Heisenberg Antiferromagnetic Chain of Spin S

Yasushi Ishii, Yukito Tanabe

Published Jul 15, 1981• 0 citations
Publisher

Abstract

The ground state energy, the correlation functions and the excitation energy of a Heisenberg antiferromagnetic chain of spin S are investigated by the real-space renormalization-group method. The asymptotic form ( S →∞) of the ground state energy is compared with the result obtained by the spin wave theory. The calculated correlation function < S ( i ) · S ( i + l ) > shows no long range order for a system of finite S . The excitation spectrum is found to have singular dips at q =(odd integer) ×π/3 n a . This is associated with the shortcoming of the three-site renormalization.

DOI
10.1143/jpsj.50.2224

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The ground state energy, the correlation functions and the excitation energy of a Heisenberg antiferromagnetic chain of spin S are investigated by the real-space renormalization-group method. The asymptotic form ( S →∞) of the ground state energy is compared with the result obtained by the spin wave theory. The calculated correlation function < S ( i ) · S ( i + l ) > shows no long range order for a system of finite S . The excitation spectrum is found to have singular dips at q =(odd integer) ×π/3 n a . This is associated with the shortcoming of the three-site renormalization.
The S=2 antiferromagnetic Heisenberg open chain is analyzed with the real-space numerical renormalization-group method improved by White. The low-ly ing spectrum and the one-point function are evaluated to show the degeneracy and the boundary effect on the ground state. Generalized string correlation funct ions were calculated for θ=π and θ=π 2. The θ=π 2 string correlation is found to indicate the long-range hidden order of the ground state. These results are quite consistent with the S=2 VBS argument.
A zero-temperature renormalization group (RG) approach is applied to the one-dimensional, spin-1/2 antiferromagnetic Heisenberg dimerized (alternating) chain. Specifically, the ground state energy and lowest-lying spectral excitations are examined. The calculation indicates the existence of a gap in the spectrum of the dimerized chain which vanishes only in the limit of a uniform spin chain, in contrast to a recent Green’s function approach. The RG results are in reasonable agreement with numerical extrapolations on the exact eigenvalue spectrum of finite chains of up to 12 spins. Both methods are compared with several other approximate treatments of the Heisenberg system, and tested by comparison with exact results for the spin-1/2 XY dimerized chain.
We present results of a numerical renormalization-group study of the isotropic S=1 Heisenberg chain. The density-matrix renormalization-group techniques used allow us to calculate a variety of properties of the chain with unprecedented accuracy. The ground-state energy per site of the infinite chain is found to be e0{\mathit{e}}_{0}e0​\ensuremath{\simeq}-1.401 484 038 971(4). Open-ended S=1 chains have effective S=1/2 spins on each end, with exponential decay of the local spin moment away from the ends, with decay length \ensuremath{\xi}\ensuremath{\simeq}6.03(1). The spin-spin correlation function also decays exponentially, and although the correlation length cannot be measured as accurately as the open-end decay length, it appears that the two lengths are identical. The string correlation function shows long-range order, with g(\ensuremath{\infty})\ensuremath{\simeq}-0.374 325 096(2). The excitation energy of the first excited state, a state with one magnon with momentum q=\ensuremath{\pi}, is the Haldane gap, which we find to be \ensuremath{\Delta}\ensuremath{\simeq}0.410 50(2). We find many low-lying excited states, including one- and two-magnon states, for several different chain lengths. The magnons have spin S=1, so the two-magnon states are singlets (S=0), triplets (S=1), and quintuplets (S=2). For magnons with momenta near \ensuremath{\pi}, the magnon-magnon interaction in the triplet channel is shown to be attractive, while in the singlet and quintuplet channels it is repulsive.
A simple real-space renormalization method yields the ground-state energy of the Heisenberg antiferromagnet. We find the ground-state energy per spin for s=12s=\frac{1}{2}s=21​ (-0.4438 in 1D, -0.6723 in 2D) and s=1s=1s=1 (-1.388 in 1D and -1.907 in 2D) to three-figure accuracy, using properties of relatively small odd-numbered clusters. Our results provide reasonable proof for long-range order in the ground state of the s=12s=\frac{1}{2}s=21​ Heisenberg antiferromagnet in 2D.
We study the ground state of one- and two-dimensional (square-lattice) spin-1/2 quantum antiferromagnets using a numerical real-space renormalization-group (RG) approach. In our RG approach we consider blocks of various sizes but with an odd number of sites; we retain only the doublet ground state and we integrate out the higher-energy states by means of second-order quasidegenerate perturbation theory. That is, we assume that the role of the excited states of a block, in the RG iteration process, is to renormalize the effective coupling parameters between blocks. We compute the ground-state energy of a spin-1/2 linear chain for various block sizes and find close agreement with the Bethe-ansatz exact solution. In the case of the spin-1/2 square-lattice quantum antiferromagnet, the obtained ground-state energy is in reasonable agreement with the available numerical estimates.
A real space renormalization group method is applied to study the spin-1/2 two-dimensional anisotropic Heisenberg antiferromagnets. We carry out the calculation on a 5×5 cluster by using a variational approach. The ground-state energy per site is estimated as e0=−0.6845±0.0005 which is in good agreement with other numerical estimates. We also calculate the different ground state energies when anisotropy changes. By comparing with 3×3 cluster calculation the resulting threshold for long range order drops and the curve of the renormalization group parameter vs. anisotropy shows no hint of discontinuity over a wide range of anisotropic parameters. Hence, our calculation suggests that long-range order exists for the ground-state of the spin-1/2 Heisenberg antiferromagnets in two dimensions.