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A CLASS OF ENERGY LEVELS FOR THE HEISENBERG LINEAR CHAIN. 2. LEVELS NEAR THE ANTIFERRO MAGNETIC GROUND STATE

Robert B. Griffiths

Published Jan 1, 1963• 0 citations
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Abstract

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.

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arXiv:1104.2946•Apr 15, 2011•0 citations
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 highest and lowest energies as a function of the total spin are computed for the class of "unbound" states in the Bethe formalism for the linear chain of spin-\textonehalf{} atoms with a Heisenberg exchange interaction between nearest neighbors. The lowest energies are used to compute the magnetization curve for the infinite antiferromagnetic chain in the limit of zero temperature. At zero temperature and in zero field, the magnetic susceptibility has the value 0.050661 $\frac{{g}^{2}{\ensuremath{\mu}}^{2}}{J}$, where $\ensuremath{\mu}$ is the Bohr magneton, ggg the electron ggg factor, and the interaction between neighboring spins is of the form $2J{\mathbf{S}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{2}$.
Antiferromagnetic ground-state energies for infinite Heisenberg linear chains with up to third-nearest-neighbor interactions have been determined by means of numerical extrapolations. On the other hand, new conditions to produce a ferromagnetic ground state have been established for finite chains. These conditions are expressed as lower bounds for the coupling constants. The particular case in which the coupling constant for third-nearest-neighbor interactions is zero is compared in two different ways with previous calculations by other authors, showing good correspondence.
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 author calculates lower bounds for the ground state energy of the linear antiferromagnetic Heisenberg chain with nearest- and next-nearest-neighbour interactions. From these lower bounds new upper bounds are derived. Several approximations of the ground state energy given in the literature are compared with these upper and lower bounds. It is found that the best approximation is given by extrapolation from finite cyclic chains. This extrapolation, however, is restricted to certain values of the relative strength of the interactions. Here he extrapolates the ground state energy from finite open chains where no such restriction occurs.
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 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.