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Journal of Physics C Solid State Physics

On the ground state energy of the linear antiferromagnetic Heisenberg chain with nearest- and next-nearest-neighbour interactions

P.M. van den Broek

Published Oct 20, 1980• 7 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/13/29/021

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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.
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.
The ground state of chains with antiferromagnetic XY interactions is examined by direct diagonalisation of the Hamiltonian. Attention is focussed on chains with nearest and next-nearest neighbour interactions. It is found that results for finite chains can be readily extrapolated to infinite chains. The dimerised ground state corresponds to a maximum in the energy per spin with respect to the next-nearest neighbour interaction parameter. An alternate normalisation scheme provides additional insight into how the ground state energy depends on the interaction parameters.
Ground-state properties of the Hamiltonian H=12J ∑ i=1N σi·σi+1 + 12Jα ∑ i=1N σi·σi+2(σN+1 ≡ σ1, σN+2 ≡ σ2) are studied for both signs of J and −1 ≤ α ≤ 1 to gain insight into the stability of the ground state with nearest-neighbor interactions only (α = 0) in the presence of the next-nearest-neighbor interaction. Short chains of up to 8 particles have been exactly studied. For J > 0, the ground state for even N belongs always to spin zero, but its symmetry changes for certain values of α. For J < 0, the ground state belongs either to the highest spin (ferromagnetic state) or to the lowest spin and so to zero for even N. The trend of the results suggests that these facts are true for arbitrary N and that the critical value of α is probably zero. Upper and lower bounds to the ground-state energy per spin of the above Hamiltonian are obtained. Such bounds can also be obtained for the square lattice with the nearest- as well as the next-nearest-neighbor interaction.
Starting from the exact result of Majumdar and Ghosh (1969) for the dimerized ground state of a Heisenberg S=1/2 antiferromagnet with competing next-nearest-neighbour interaction J2+1/2J1 on a linear chain, it is shown that this state is the ground state of the same kind of Hamiltonian, if couplings are extended up to 2n-nearest-neighbours, and the relation J1=2n, J2=2n-1, ..., J2n=1 holds. Furthermore, it is proved rigorously that this state is the ground state when n=2 and J1-2J2+J3=0, J3=2J4, in a finite interval of values of the parameter J4. A rigorous lower limit for the extension of this interval is found to be 0<or=J4<or=1/4J1. A comparison with ground-state configurations of the equivalent classical Heisenberg and Ising model is discussed.
We study the properties of a linear chain of spins governed by the Hamiltonian H=J1 ∑ j=1NSj·Sj+1+J2 ∑ j=1NSj·Sj+2and derive equations for an upper bound of the free energy by means of a temperature dependent Hartree-Fock approximation. These equations can be solved at zero temperature, yielding an approximate wavefunction for the ground state, of which the energy is an upper bound for the exact ground state energy. The upper bounds obtained are improvements of the results of Majumdar and Ghosh. There are several different cases to be considered, depending on the relative values of J1 and J2. In none of these is there a gap between the ground state and the first excited state in the thermodynamic limit. Finally, correlation functions are shortly discussed in connection with spiral structures in classical spin systems.