Possible permutation symmetry in two dimensional Heisenberg Model

S.B. Khokhlachev

Wang An-Min, Ren-Gui Zhu

We extend the BCS paring model with equally spaced energy levels to a general one-dimensional spin-1/2 Heisenberg model. The two well-known symmetries of the Heisenberg model, i.e. permutational and spin-inversion symmetries, no longer exist. However, when jointing these two operations together, we find a new symmetry of energy spectrum between its subspace n and subspace L − n of the Fock space. A rigorous proof is presented.

Véronique Hussin

We consider classical and quantum one and two-dimensional systems with ladder operators that satisfy generalized Heisenberg algebras.In the classical case, this construction is related to the existence of closed trajectories.In particular, we apply these results to the infinite well and Morse potentials.We discuss how the degeneracies of the permutation symmetry of quantum two-dimensional systems can be explained using products of ladder operators.These products satisfy interesting commutation relations.The two-dimensional Morse quantum system is also related to a generalized two-dimensional Morse supersymmetric model.Arithmetical or accidental degeneracies of such system are shown to be associated to additional supersymmetry.

Robert T. Powers

S. Moukouri

J. Žák

The known symmetries in one-dimensional systems are inversion and translations. These symmetries persist in finite phase plane, but a novel symmetry arises in view of the discrete nature of the coordinate x i and the momentum p i : x i and p i can undergo permutations. Thus, if x i assumes M discrete values, i = 0, 1,2,..., M – 1, a permutation will change the order of the set x 0 , x 1 ,..., x M −1 into a new ordered set. Such a symmetry element does not exist for a continuous x -coordinate in an infinite phase plane. Thus, in a finite phase plane, translations can be replaced by permutations. This is also true for the inversion operator. The new permutation symmetry has been used for the construction of conjugate representations and for the splitting of the M -dimensional vector space into independent subspaces. This splitting is exhaustive in the sense that if M = i M i with M i being prime numbers, the M -dimensional space splits into M 1 , M 2 ,... M n -dimensional independent subspaces. It is shown that following this splitting one can design new potentials with appropriate constants of motion. A related problem is the Weyl-Heisenberg group in the M-dimensional space which turns into a direct product of its subgroups in the M i -dimensional subspaces. As an example we consider the case of M = 8.