Dissipative quantum mechanics and Kondo-like impurities on noncommutative two-tori

TL;DR

This paper explores the interplay between noncommutative geometry, quantum Hall bilayers, and dissipative quantum mechanics, highlighting the role of impurities and generalized magnetic translations in such systems.

Contribution

It develops a framework connecting noncommutative field theories, quantum Hall bilayers, and dissipative quantum mechanics, emphasizing impurity effects and magnetic translation symmetries.

Findings

Established a correspondence between noncommutative field theories and quantum Hall fluids.

Analyzed the role of impurities in quantum Hall bilayers using generalized magnetic translations.

Developed a tensor product structure for magnetic translation operators acting on system and defect spaces.

Abstract

In a recent paper, by exploiting the notion of Morita equivalence for field theories on noncommutative tori and choosing rational values of the noncommutativity parameter $\theta $ (in appropriate units), a general one-to-one correspondence between the $m$-reduced conformal field theory (CFT) describing a quantum Hall fluid (QHF) at paired states fillings $\nu =% \frac{m}{pm+2}$ and an Abelian noncommutative field theory (NCFT) has been established . That allowed us to add new evidence to the relationship between noncommutativity and quantum Hall fluids\cite% {ncmanybody}. On the other hand, the $m$-reduced CFT is equivalent to a system of two massless scalar bosons with a magnetic boundary interaction as introduced by Callan et al., at the so called ``magic''\ points. We are then able to describe, within such a framework, the dissipative quantum mechanics of a particle confined to a plane and subject to an external magnetic field normal to it. Here we develop such a point of view by focusing on the case $m=2$ which corresponds to a quantum Hall bilayer. The key role of a localized impurity which couples the two layers is emphasized and the effect of noncommutativity in terms of generalized magnetic translations (GMT) is fully exploited. As a result, general GMT operators are introduced, in the form of a tensor product, which act on the QHF and defect space respectively, and a comprehensive study of their rich structure is performed.