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Journal of Applied Physics

Renormalization group and other calculations for the one-dimensional spin-1/2 dimerized Heisenberg antiferromagnet

J. N. Fields, Henk W. J. Blöte, J. C. Bonner

Published Mar 1, 1979• 50 citations
Publisher

Abstract

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.

DOI
10.1063/1.327176

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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.
A zero-temperature lattice renormalization-group calculation is presented and applied to the antiferromagnetic one-dimensional dimerized spin-\textonehalf{} Heisenberg chain. The calculation indicates the existence of a gap in the spectrum of the dimer system, which vanishes only in the limit of a uniform spin chain. To improve the accuracy of the numerical estimates of the ground-state energy and gap, an iterative extension of finite-chain techniques is also applied. These approximate techniques are checked against the dimerized XY\mathrm{XY}XY chain, for which an exact solution is found.
The one-dimensional spin-1 Heisenberg antiferromagnetic chains are studied by a generalized two-time Green's-function method at finite temperature. 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 is set to zero at each site to guarantee the lack of the long-range order in the one-dimensional Heisenberg system. The Green's functions are decoupled in terms of the correlation functions of spin operators. A set of self-consistent equations of the correlation functions are derived and solved numerically. Correlation functions of spin operators, thermodynamic properties such as the internal energy, and specific heat in the entire temperature region are obtained. The Haldane gap in the excitation spectrum appears naturally in the analytical result. When k is zero, there is a gap 2\ensuremath{\Delta}, which is 1.0J and close to the theoretical result of White and Huse. And there is a broad maximum in the temperature dependence of the specific heat. The properties of the thermodynamic quantities are consistent with the numerical results on the finite chains. \textcopyright{} 1996 The American Physical Society.
A real-space renormalization-group theory is proposed for the anisotropic Heisenberg chain which yields physically reasonable results for all values of the temperature and of the anisotropy. The method is based on a four-spin cluster and a transformation which preserves the sublattice structure of the antiferromagnet. At low temperatures the transformation maps the four lowest energy levels of the four-spin cluster onto the four levels of two spins, giving ground-state energies in excellent agreement with exact results. The specific-heat exponent for the isotropic antiferromagnet is given correctly, but the exponent for larger transverse coupling is somewhat too small. A scaling form is proposed for the free energy near the isotropic antiferromagnetic fixed point, and the crossover to Ising-like behavior is discussed.
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.
We examine the ground state and the excitations of an one-dimensional Heisenberg spin 1/2 antiferromagnet with alternating dimers and four-spin plaquettes (dimer-plaquette chain). The properties of the system depend on the competing dimer and plaquette bonds. Several exact, exact-numerical and perturbational results are presented. We find that the system is gapped for all parameter values. For the latter one we find an effective S=1 Haldane like behaviour in the limit of dominating dimer bonds. The model considered is related to the recently found 1/5-depleted square-lattice Heisenberg system CaV4O9CaV_4O_9CaV4​O9​.