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Physical Review

Linear Antiferromagnetic Chain

Sergio Rodríguez

Published Dec 15, 1959• 56 citations
Publisher

Abstract

It is shown that a linear chain consisting of a large number of atoms of spin \textonehalf{} with nearest neighbor ferromagnetic or antiferromagnetic interactions is mathematically equivalent to a one-dimensional Fermi gas with two-body forces. This equivalence is used to construct a wave function for which the expectation value of the energy lies between the two approximations obtained by Hulth\'en. Also, a perturbation treatment is given which permits, in principle, to obtain the exact antiferromagnetic ground state.

DOI
10.1103/physrev.116.1474

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It is shown that a linear chain consisting of a large number of atoms of spin \textonehalf{} with nearest neighbor ferromagnetic or antiferromagnetic interactions is mathematically equivalent to a one-dimensional Fermi gas with two-body forces. This equivalence is used to construct a wave function for which the expectation value of the energy lies between the two approximations obtained by Hulth'en. Also, a perturbation treatment is given which permits, in principle, to obtain the exact antiferromagnetic ground state.
The distribution of 'unbound' states in the Bethe formalism for the linear chain of spin 1/2 atoms with a Heisenbert exchange interaction between nearest neighbors is investigated in the vicinity of the antiferromagnetic ground state, utilizing the spin wave states discovered by des Cloizeaux and Pearson. An upper bound is obtained for the free energy of the anti ferromagnetic chain at very low temperatures. Plausible arguments are presented to show that the 'unbound' states actually make a negligible contribution to the partition function of an infinite chain, at least in the absence of a magnetic field.
The one-dimensional quantum spin-1/2 model with nearest-neighbor ferromagnetic and next-nearest-neighbor antiferromagnetic interaction is considered. The Hamiltonian is firstly rewritten in a form with rotated spin operators, then bosonized by using the linear spin wave approximation and then treated by using the Green function approach. An integral expression of the quantum correction to the classical ground state energy is derived. The critical behavior of the ground state energy in the vicinity of the transition point from the ferromagnetic to the singlet ground state is analyzed by numerical calculation and the result is −8γ 2 .
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.
The one-dimensional quantum spin-1/2 model with nearest-neighbor ferromagnetic and next-nearest-neighbor antiferromagnetic interaction is considered. The Hamiltonian is first bosonized by using the linear spin wave approximation, and then is treated by using the Green's function approach. An integral expression of the quantum correction to the classical ground state energy is derived. The critical behavior of the ground state energy in the vicinity of the transition point from the ferromagnetic to the singlet ground state is analyzed by numerical calculation, and the result is −8γ2-8γ^2−8γ2.
We study the ground-state and finite-temperature properties of the one-dimensional isotropic spin-1/2 Heisenberg magnet with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions which compete with each other. Extrapolating the exact results for finite-size systems of up to 20 spins, we estimate the ground-state energy and the ground-state two-spin correlation function in the infinite-size limit. Using the cluster transfer matrix method, we also calculate the temperature dependence of the internal energy and the specific heat and that of the inverse correlation length and the wave number, by both of which the asymptotic behavior of the two-spin correlation function in the long-distance limit is characterized. We compare the results of these calculations with those obtained previously for the case where the nearest-neighbor interaction is antiferromagnetic.