Results of exact numerical work on the spectra and thermodynamic properties are presented for finite clusters in one and two dimensions, for the isotropic Heisenberg Hamiltonian and the Heisenberg Hamiltonian with longitudinal anisotropy. Eigenvalues of the isotropic Heisenberg loops (that is, those with periodic boundary conditions) of $N\ensuremath{\le}12, 8, 6, 5$, 5 particles, each of spin \textonehalf{}, 1,
23, 2 and
25, respectively, are found. It is shown that the spin-
S linear chain ($N\ensuremath{\rightarrow}\ensuremath{\infty}$) will have ground-state energy per spin $\frac{{\ensuremath{\epsilon}}_{\ensuremath{\infty}}^{s}}{2J}=\ensuremath{-}1.404\ifmmode\pm\else\textpm\fi{}0.002, \ensuremath{-}2.85\ifmmode\pm\else\textpm\fi{}0.03, \ensuremath{-}4.7\ifmmode\pm\else\textpm\fi{}0.4 \mathrm{and} \ensuremath{-}7.1\ifmmode\pm\else\textpm\fi{}0.5$, for
S=1,23,2,and25, respectively. For antiferromagnetic interaction, the lowest excited states of the loops with even number of particles have spin 1 and the familiar double sine spectrum; while for the spin-\textonehalf{} case the eigenvalues are very different from the result of the spin-wave theory, they approach this limit as the spin increases, and for spin
25, the infinite-spin results of the spin-wave theory should be quite accurate. For ferromagnetic loops, there exist spin-wave bound states for higher spins just like those familiar in the spin-\textonehalf{} linear chain. Effective magnetic moments for these loops are tabulated. We also present the moments for the
N=8 spin-\textonehalf{} chain (with open ends), and find the difference to be small. Results on the 3\ifmmode\times\else\texttimes\fi{}4 two-dimensional Heisenberg antiferromagnet are presented and compared with experimental data. Finally, we show some examples of the violation of the nondegeneracy rule, similar to those found by Heilmann and Lieb; levels belonging to different representations of a symmetry group remain permanently degenerate when a parameter is varied.
Full magnetization curves of S=1 antiferromagnetic Heisenberg chains with periodic and open boundaries are investigated by a quantum Monte Carlo method. The boundary condition has a significant effect on the magnetization process especially in the field range where the Zeeman energy is less than the Haldane gap. The magnetization of the periodic chain in the range is strongly suppressed at low temperatures because the ground state is nonmagnetic. On the other hand, the magnetic ground of the open chain bring about finite magnetization. The difference between both the magnetization curves is also investigated in detail by subtracting the magnetization of the periodic chain with L spins from one of the open chains with L+1 spins. It agrees well with the double of the Brillouin function for S=1/2, rather than the Brillouun function for S=1.
We have calculated the two-point correlation functions ${\ensuremath{\omega}}_{\mathrm{il}}$(N)=(4/3)〈S${\ensuremath{\rightarrow}}_{i}$\ensuremath{\cdot}S${\ensuremath{\rightarrow}}_{i+l}$ 〉 and their averages over i,${\ensuremath{\omega}}_{l}$ (N), in the ground state of the one-dimensional antiferromagnetic Heisenberg model for N=4,6,8,...,16 spins. Both periodic (rings) and free-end (chains) boundary conditions are considered. Surprisingly tight lower and upper bounds have been obtained for ${\ensuremath{\omega}}_{l}$(\ensuremath{\infty}) under reasonable assumptions. In addition to showing the rather strong even-l--odd-l alternation in \ensuremath{\Vert}${\ensuremath{\omega}}_{l}$(N)\ensuremath{\Vert}, known from earlier results of Bonner and Fisher for rings with N up to 10, our bounds indicate a smooth behavior in l\ensuremath{\Vert}${\ensuremath{\omega}}_{l}$(\ensuremath{\infty})\ensuremath{\Vert} for l odd and l even, with, surprisingly, a broad maximum attained within the odd-l values. The bounds obtained from the chain results were essential to seeing this maximum (because of the larger l values available for given N). The quantity l\ensuremath{\Vert}${\ensuremath{\omega}}_{l}$(N)\ensuremath{\Vert} for chains with fixed N also shows such a maximum, and in addition shows a similar maximum for even l's. If the indicated trends for large l and N continue in ${\ensuremath{\omega}}_{l}$(\ensuremath{\infty}) and in
SN, the structure factor at wave vector \ensuremath{\pi}, then finite-size contributions to ${\ensuremath{\omega}}_{l}$(N) will have to contribute to the (seemingly) logarithmic divergence of
SN as N\ensuremath{\rightarrow}\ensuremath{\infty}. We are not aware of any models where a similarly weak divergence shows such a finite-size contribution.
The solution of the Bloch equation for many-particle systems on a finite lattice is given in terms of the “path summation”. The boundary conditions we consider are “zero on the boundary” and “the periodic boundary condition”. The obtained expression under the periodic boundary condition is appropriate for the Monte Carlo calculation of the density matrix and the ground state energy of Bose lattice systems and the Heisenberg magnet with a ferromagnetic interaction. For the antiferromagnetic Heisenberg magnet, the present formalism is useful only for the lattices in which each of the closed paths has an even number of edges.
Thermodynamic properties of S =1 antiferromagnetic Heisenberg chains with free and periodic boundaries are investigated by means of a quantum Monte Carlo method. The specific heat is shown to have a peak at a temperature T peak ∼2Δ, where Δ is the energy gap of the present model, although global shape of the temperature dependence is very similar to the Schottky type specific heat. In open chains, due to the four-fold degeneracy of the ground state, the magnetic susceptibility shows a Curie-like divergence regardless of the number of spins. The extracted diverging part is consistent with the edge moment S =1/2.