We numerically caculate the exact ground state energy E 0 ( N , δ) and the first excited state energy E 1 ( N , δ) of the one-dimensional spin-
21 alternating antiferromagnetic Heisenberg model
The exact eigenvalue spectrum and thermodynamic properties of the spin Hamiltonian $\mathcal{H}=2|J|\ensuremath{\Sigma}\stackrel{\frac{N}{2}}{i=1}({\mathrm{S}}_{2i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{2i\ensuremath{-}1}+a{\mathrm{S}}_{2i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{2i+1})$ are calculated for short chains of
N=4,6,8,and10 spins, each of spin \textonehalf{}, for alternation parameter
a=0.2,0.4,0.6,and0.8, with ${\mathrm{S}}_{N+1}\ensuremath{\equiv}{\mathrm{S}}_{1}$. The behavior for $N=\ensuremath{\infty}$ is estimated by extrapolation. Comparison is made with the known results for
a=0 and 1. The ground-state energy, ground-state short-range order, energy gap, and triplet excitation spectrum are compared with various approximate theories. Zero-temperature infinite-chain magnetization curves are inferred from the finite-chain results. The energy, entropy, specific heat, and magnetic susceptibility for
N=10 are shown to approximate well the behavior for $N=\ensuremath{\infty}$ when
∣J∣kT>a. The magnetic-susceptibility data on the free radical 2,2-bis (
p-nitrophenyl)-1-picrylhydrazyl are shown to agree well with the theoretical results for an alternating chain with $a\ensuremath{\approx}0.6$.
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.
The generalized linear antiferromagnetic chain with two spins per unit cell
J>0, and
a,b,d,e> 0, ${\mathcal{H}}_{G}=\ensuremath{\Sigma}\stackrel{\frac{N}{2}}{j=1}J{a{{S}_{2j}}^{z}{{S}_{2j+1}}^{z}+b({{S}_{2j}}^{x}{{S}_{2j+1}}^{x}+{{S}_{2j}}^{y}{{S}_{2j+1}}^{y})+d{{S}_{2j}}^{z}{{S}_{2j\ensuremath{-}1}}^{z}+e({{S}_{2j}}^{x}{{S}_{2j\ensuremath{-}1}}^{x}+{{S}_{2j}}^{y}{{S}_{2j\ensuremath{-}1}}^{y})},$ is solved in the pseudospin approximation. The pseudospin method is compared with the exactly soluble special case of the Heisenberg-Ising antiferromagnet ($a=b=1+\ensuremath{\delta}$, $d=1\ensuremath{-}\ensuremath{\delta}$,
e=0) and good agreement is obtained even for a sensitive property like the long-range order. It is suggested that the pseudospin approach provides a single, systematic approximation scheme, not only for describing the magnetic properties of a class of solid free radicals, but also as a check for widely used many-body techniques.
We consider the problem of seeking the eigenvectors for a commuting family of quantum minors of the monodromy matrix for an SL(n,ℂ)-invariant inhomogeneous spin chain. The algebra generators and elements of the L-operator at each site of the chain are implemented as linear differential operators in the space of functions of n(n−1)/2 variables. In the general case, the representation of the sl n (ℂ) algebra at each site is infinite-dimensional and belongs to the principal unitary series. We solve this problem using a recursive procedure with respect to the rank n of the algebra. We obtain explicit expressions for the eigenvalues and eigenvectors of the commuting family. We consider the particular cases n = 2 and n = 3 and also the limit case of the one-site chain in detail.