Employing the method of a symmetry lowering (form the permutation group Pn to the dihedral group Cnv), the eigenstates of the ring Heisenberg Hamiltonian H=-2J ∑ni=1Si·Si+1 are classified not only by total spin S and its z-component M, but also by wave number k=2 πL/n, where L= ±0, ±1, ±2, ∼, ±p for n=2p+1 or 2p. By the characters of irreducible representation of Pn and Cnv, we express the sum of the energy eigenvalues at fixed S, M and L, together with the number of the eigenstates contained in this sum. With respect to the spin configurations in the ring, it is revealed that there exist invariant subspaces under all symmetry operations of Cnv in the space of the spin-functions specified by M. By connecting this fact with the Wigner coefficients, the eigenfunctions are easily obtained.
With a view to obtaining further insight into the nature of eigenvalues and eigenfunctions of a stationary state one-dimensional Schrödinger equation corresponding to a non-Hermitian Hamiltonian H ( x , p ) we investigate the ground-state solutions for a variety of potentials within the framework of an extended complex phase space characterized by x = x 1 + i p 2 , p = p 1 + i x 2 , where ( x 1 , p 1 ) and ( x 2 , p 2 ) are real and considered as canonical pairs. The analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalues and eigenfunctions for different systems. It is noted that the imaginary part of the eigenvalue, E i , turns out to be zero for all potentials V ( x ) with real couplings whereas it turns out to be nonzero for the case when the couplings are complex. The prescription is also extended to study the excited states. The problems related to the normalization of the eigenfunction and the boundary conditions to be used within this framework are also discussed.
As an illustration of a quasiclassical spin formalism developed in a recent paper a new method for solving the one-dimensional Ising model is presented. The method used is an extension to spin systems of the Wigner method familiar in computations involving position and momentum. The partition function, pair correlation function and the zero-field susceptibility for the one-dimensional Ising model with arbitrary spin are shown to be expressed in terms of the eigenfunctions and eigenvalues of a certain integral equation whose kernel approaches the classical kernel as the spin approaches infinity.
The eigenvalue problem in a one-dimensional system of many fermions interacting via a delta-function potential is solved exactly in a field theoretical framework. Simultaneous eigenstates of spin and energy are represented in a systematic and compact form, that is, in such a form that the orbital and spin eigenfunctions are unified in harmony without the aid of Young’s tableau. The eigenstate and energy eigenvalue are specified by two sets of quantum numbers, particle momenta and spin momenta, besides two eigenvalues of a total spin angular momentum and its z-component. It is shown that for given particle momenta and spin momenta the energy eigenvalue is given by the solutions of coupled equations.
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.