On the Hilbert space of supersymmetric quantum systems

A.I. Vainshtein, Andrei V. Smilga, Mikhail A Shifman

Maciej Trzetrzelewski, J. Wosiek

Standard quantum mechanics is viewed as a limit of a cut system with artificially restricted dimension of a Hilbert space. Exact spectrum of cut momentum and coordinate operators is derived and the limiting transition to the infinite dimensional Hilbert space is studied in detail. The difference between systems with discrete and continuous energy spectra is emphasized. In particular a new scaling law, characteristic for nonlocalized, states is found. Some applications for supersymmetric quantum mechanics are briefly outlined.

M. Kibler, Mohammed Daoud A

C Marques, G S Dias, Helder Hipólito Chávez Sánchez

Abstract Supersymmetry at a susy harmonic oscillator, H(ω 1, ω 1), can be broken or restored in certain conditions and parameters, that are linked with thermal interaction and with a polynomial interactions of creation and annihilation operators. All possibles supersymmetric harmonic oscillators represented by a point (ω 1, ω 1) in the frequency space of the system, are in a two dimensional surface parametrized by the (ω 2, α 2), which we call s-surface, where α 2 is the interaction parameter. The temperature in the s-surface are intended to be zero. Interaction with the thermal bath represented by the tilde Hilbert space from the doubling Hilbert space, establishes thermal oscillations that push the system from the s-surface. In such a way we can define the set of all supersymmetric harmonic oscillator or in a equivalent way the s-surface as a global standard reference for zero temperature.

Barry A. Freedman

Shan-Wen Tsai, John Bradley Marston

We use the density-matrix renormalization-group (DMRG) algorithm and finite-size scaling to study a supersymmetric (SUSY) spin chain that models plateau transitions in the integer quantum Hall effect. To illustrate the method, we first present results for the S=1 antiferromagnetic spin chain. Different scaling behavior of the dimerization operator is obtained for the Haldane phase, the dimerized phase, and at the critical point separating the two phases. We then investigate the supersymmetric spin chain. As the on-site Hilbert space has infinite dimension, it is non-compact. We present DMRG calculations for supersymmetric truncations of the on-site Hilbert space and also for non-supersymmetric truncations. These two cases scale differently, in accord with a Lieb-Schultz-Mattis type theorem.