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arXiv:0706.0310 · Journal of Physics A Mathematical and Theoretical

Quantum superintegrable systems for arbitrary spin

G P Pronko

Published Oct 16, 2007• 20 citations
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Abstract

In Pronko and Stroganov (1977 Zh. Eksp. Teor. Fiz. 72 2048, 1997 Sov. Phys.—JETP 45 1072) the superintegrable system which describes the magnetic dipole with spin (neutron) in the field of linear current was considered. Here we present its generalization for any spin which preserves superintegrability. The dynamical symmetry stays the same as it is for spin .

DOI
10.1088/1751-8113/40/44/013

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In Pronko and Stroganov (1977 Zh. Eksp. Teor. Fiz. 72 2048, 1997 Sov. Phys.—JETP 45 1072) the superintegrable system which describes the magnetic dipole with spin (neutron) in the field of linear current was considered. Here we present its generalization for any spin which preserves superintegrability. The dynamical symmetry stays the same as it is for spin .
We develop a dynamical approach based on the Schwinger-Keldysh formalism to derive a field-theoretic description of disordered and interacting electron systems. Within this formalism we calculate the perturbative renormalization-group (RG) equations for interacting electrons expanded around a diffusive Fermi liquid fixed point, as obtained originally by Finkelstein (Zh. \'Eksp. Teor. Fiz. 84, 168 (1983) [Sov. Phys. JETP 57, 97 (1983)]) using replicas. The major simplifying feature of this approach, as compared to Finkelstein's, is that instead of $\stackrel{\ensuremath{\rightarrow}}{N}0$ replicas, we only need to consider N=2N=2N=2 species. We compare the dynamical Schwinger-Keldysh approach and the replica methods, and we present a simple and pedagogical RG procedure to obtain Finkelstein's RG equations.
The study of spin-polarized atomic hydrogen in the three-dimensional gas phase has been extended into a new regime of magnetic field strength. Analysis of decay data for B=10--20 T reveal for the first time the characteristic peak in the three-body dipolar-decay rate constant as predicted by Kagan, Vartanyants, and Shlyapnikov (Zh. Eksp. Teor. Fiz. 81, 1113 (1981) [Sov. Phys.---JETP 54, 590 (1981)]). Beyond 17 T we observe a rapid falloff of the three-body-decay rate constant. We also determine a value for the three-body-exchange rate constant which is in accord with the calculation of Kagan et al. In contrast to the picture which was emerging from previous studies at more moderate fields, the data presented here suggest that existing theories are capable of describing the decay of the system in the very-high-field limit. The new picture which is emerging suggests that Bose-Einstein condensation may be achievable at somewhat higher fields which are currently available at high-magnetic-field facilities.
We consider the Bose-Einstein condensation of atoms in a trap where the density of particles is so high that the low-density approach of Gross [J. Math. Phys. (N.Y.) 4, 195 (1963)] and Pitaevskii (((V. L. Ginsburg and L. P. Pitaevskii, Zh. \'Eksp. Teor. Fiz. 34, 1240 (1958) [Sov. Phys. JETP 7, 1858 (1958)]))) will not be applicable. For this purpose we use the slave boson representation which is valid for hard-core bosons at any density. This description leads to the same results as the Gross-Pitaevskii approach in the low-density limit, but for higher densities, it predicts the depletion of the condensate in the regions where the density of the atomic cloud is high.