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Progress of Theoretical Physics

Magnetic Properties of Antiferromagnetic Heisenberg Linear Chain

Takao Todani, 和子 川崎

Published Oct 1, 1973• 27 citations
Publisher

Abstract

By the Green’s function method, where spin operators are transformed to fermion operators, an antiferromagnetic Heisenberg chain system has been investigated within the Hartree-Fock approximation. The static correlation ≪SzkSz-k> cannot be reduced to the Ornstein-Zernike form and has no singularity at k=π. The absorption function χzz’’(k,ω) cannot be expressed by the δ-function-like spectrum even at 0°K, but rather by a narrow diffused one, which is due to the two-magnon process. The present dispersion relation has a gap at k=π for T=0°K. The second order phase transition, then, may not occur in the linear chain. The results obtained are discussed comparing with those in the X-Y model and experimental data on TMMC.

DOI
10.1143/ptp.50.1216

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Journal of Physics Condensed Matter•Jun 21, 2007•2 citations
By the Green’s function method, where spin operators are transformed to fermion operators, an antiferromagnetic Heisenberg chain system has been investigated within the Hartree-Fock approximation. The static correlation ≪SzkSz-k> cannot be reduced to the Ornstein-Zernike form and has no singularity at k=π. The absorption function χzz’’(k,ω) cannot be expressed by the δ-function-like spectrum even at 0°K, but rather by a narrow diffused one, which is due to the two-magnon process. The present dispersion relation has a gap at k=π for T=0°K. The second order phase transition, then, may not occur in the linear chain. The results obtained are discussed comparing with those in the X-Y model and experimental data on TMMC.
The one-dimensional Heisenberg antiferromagnet with 8=1/2 is treated with the use of the Fermi operators and an additional condition that the long-range order is absent.Our spin wave spectrum of a one-dimensional Heisenberg antiferromagnet without longrange order is expressed by andThis is in good agreement with the exact spin wave spectrum role =1!'/2lsin kl which is obtained by des Cloizeaux and Pearson at T=O.Correlation functions <S1S2> and <SlSs> are in good agreement with the exact value at T=O, also.Thermodynamic properties of this system agree with those of Bulaevskii and those of Katsura and Inawashiro. § I. IntroductionThe one-dimensional Heisenberg antiferromagnet has been investigated by many authors.Most of previous theories have been for either zero or infinite temperatures.Recently various experiment on (CD3) 4NMnCl 3 (TMMC)ll requires further theoretical work of a Heisenberg linear-chain magnet for finite temperatures.The experimental results of linear antiferromagnet show that at low temperatures there are long-lived spin waves in spite of the absence of long-range order.At T= 0, the spin wave spectrum, wqoc I sin ql, is gained approximately by Anderson 2 l and exactly by des Cloizeaux and Pearson.8 l Recently Richards, 4 l Scales and Gersh, 5 l Kondo and Yamaji 6 l obtained the spin wave spectrum of linear antiferromagnet at finite temperatures with the use of the second-order Green's function theory.The spin wave spectrum, however, does not show exactly wqoc I sin ql at finite temperatures.
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.
The spectral properties, and associated critical singularities, of spin-1 Heisenberg antiferromagnetic chains are of high current interest, both theoretical and experimental. Theoretical studies using the single mode approximation and also a Monte Carlo calculation indicate that the spectrum has a gap at k=π, as predicted by Haldane. They also predict the unusual feature that the excited state continuum is asymmetric about the mid-point of the Brillouin zone, k=π/2. However, neither approach is sufficiently accurate to predict reliably the magnitude of the gap at k=0 which was not discussed by Haldane. Our approach is to use static (spectral) calculations of excited states of chains of N=4 through 14 spins for all values of k, a novel feature not associated with previous finite-size calculations, to discuss the dispersion spectra of the S=1 Heisenberg chain. In addition, we have calculated dynamical properties, in particular S(q,ω), exactly for finite chains up to N=14 spins. This allows us to estimate the gap at k=0 to good accuracy, and we find that to within this accuracy the gap at k=0 is twice the Haldane gap at k=π. This result is important in connection with very recent neutron scattering and magnetization measurements. In addition, we have calculated the correlation length and critical singularities associated with dynamical quantities.
Wave functions are calculated for the ground state and low-lying excited states of a linear antiferromagnetic chain of N spins S = 1/2 with alternate strong and weak exchange integrals, using the Fermi operator method (N is even). There are two degenerate ground states with alternating spin distributions, formed by taking a Hartree-Fock state vector composed of ‘alternant spin waves’ with different amplitudes on the two sub-lattices. The normal Hartree-Fock state is unstable and collapses into one or other alternating state. One of these is equivalent to the pseudo-spin ground state of Soos as well as the wave functions used by Ruijgrok and Rodriguez, and Oguchi. The other is derived by the time-reversal transformation or by a translation of the regular chain, and is asymptotically orthogonal to the first when N is large. New states are constructed which have the full translational symmetry of the Hamiltonian, and there is a doubling of the low-lying spectrum of states in which the excitation density is of order 1/N, as compared with Soos' theory. This affects the magnetic susceptibility since the higher states are not doubled and do not remain orthogonal.
We present low-temperature static and dynamic properties of the quantum one-dimensional isotropic integer spin Heisenberg magnet with antiferromagnetic nearest-neighbour (nn) and next-nearest-neighbour (nnn) interactions. The modified spin-wave theory is used to provide quantities such as the spin-wave dispersion relation, the ground-state energy, the gap and its dependence with temperature, and the asymptotic behaviour of the spin–spin correlation function. The ground state energy and the singlet–triplet energy gap are obtained for several values of j, defined as the ratio of the nnn interaction constant to the nn one. Our results show that the ground-state and the gap energies increase with j, in accordance with numerical results available in the literature. The calculation of the dynamic correlation function is performed using the projection operator formalism: the procedure includes up to two-magnon processes. We show that, depending on the values of the frustration parameter j, the wavevector and the temperature, a double-peak structure for the dynamical correlation function can develop.