Interactions and non-commutativity in quantum Hall systems

Abstract

We discuss the role that interactions play in the non-commutative structure that arises when the relative coordinates of two interacting particles are projected onto the lowest Landau level. It is shown that the interactions in general renormalize the non-commutative parameter away from the non-interacting value 1 B. The effective non-commutative parameter is in general also angular momentum dependent. An heuristic argument, based on the non-commutative coordinates, is given to find the filling fractions at incompressibilty, which are in general renormalized by the interactions, and the results are consistent with known results in the case of singular magnetic fields.

Pair production of Dirac particles in a $$d+1$$ d + 1 -dimensional noncommutative space–time

“…The same observation appears in the framework of string theory [17,18]. Also, the quantum Hall effect well illustrates the NC quantum mechanics of space-time [19,20] (and references therein).…”

Topics in Noncommutative Geometry Inspired Physics

“…On the other hand, interparticle interactions can renormalize the noncommutative parameter away from ( 1 B ) and thus can have affect on the filling fraction in a Quantum Hall system. Indeed, it has been demonstrated that Jain fraction for Fractional Quantum Hall Effect can be obtained in a heuristic treatment, where the electrons are attached to appropriate magnetic flux tubes [92]. Furthermore, there are certain situations where interactions can be traded with noncommutativity within a certain approximation, as can be seen by constructing a dual families of noncommutative quantum systems [129].…”

“…An example is in Condensed Matter Physics where planar systems involving a perpendicular magnetic field becomes effectively noncommutative in the lowest Landau level [53,83] (see Szabo in [1,2]) the NC parameter being identified with inverse of the magnetic field or in Anyon models [84][85][86][87][88][89][90][91]. The Landau levels also get renormalized by interparticle interactions [92] that can have nontrivial impact in fractional Quantum Hall Effect. In another development it has been shown that NC phase space algebra can influence particle dynamics directly via induced Berry curvature effects [93][94][95] in studying many Condensed Matter phenomena such as Anomalous Hall effect [53,83,[96][97][98][99], Spin Hall effect [100][101][102], models of Graphene [103] among others.…”

“…1, we will not dwell too much on the topics we have covered in the present review simply because the individual chapters are reasonably self-contained. Section 2 introduces the star (Moyal) product used to define products of fields in NC spaces discusses effects of NC space on nonrelativistic many (Bose/Fermi) particle systems through twisted statistics [92,[107][108][109]. In Sect.…”

Aharonov-Bohm Phase for an Electric Dipole on a Noncommutative Space

“…And in Mirza and Zarei [16] the limit for the space noncommutativity parameter is 1/ √ θ ≥ 10 −7 GeV. The authors of Harms and Micu [23], Dayi and Jellal [24] and Chakraborty et al [25][26][27][28][29][30] have studied the noncommutative quantum Hall effect, and in Harms and Micu [23] a limit of 1/ √ θ ≥ 10GeV is given. The noncommutative spin Hall effect (SHE) is discussed through a semiclassical constrained Hamiltonian and interesting results are obtained in Dayi and Elbistan [31].…”