Interacting Quantum Topologies and the Quantum Hall Effect

Abstract

The algebra of observables of planar electrons subject to a constant background magnetic field B is given by A θ (R 2 ) ⊗ A θ (R 2 ) (θ = − 4 eB ), the product of two mutually commuting Moyal algebras. It describes the free Hamiltonian and the guiding centre coordinates. We argue that A θ (R 2 ) itself furnishes a representation space for the actions of these two Moyal algebras, and suggest physical arguments for this choice of the representation space. We give the proper setup to couple the matter fields based on A θ (R 2 ) to electromagnetic fields which are described by the abelian commutative gauge group G c (U (1)), i.e. gauge fields based on A 0 (R 2 ). This enables us to give a manifestly gauge covariant formulation of integer quantum Hall effect (IQHE). Thus, we can view IQHE as an elementary example of interacting quantum topologies, where matter and gauge fields based on algebras A θ ′ with different θ ′ appear. Two-particle wave functions in this approach are based on A θ (R 2 ) ⊗ A θ (R 2 ). We find that the full symmetry group in IQHE, which is the semi-direct product SO(2) ⋉ G c (U (1)) acts on this tensor product using the twisted coproduct ∆ θ . Consequently, as we show, many particle sectors of each Landau level have twisted statistics. As an example, we find the twisted two particle Laughlin wave functions.

Quantum entanglement in a non-commutative system

“…Though θ ij = θǫ ij turns out to be an SO(2) scalar as mentioned earlier and the technique of twist symmetrization is motivated by its generalizability to higher dimensions. Indeed, general arguments coming from diffeomorphism invariance [32] and quantum integrability [33] support this.…”

Noncommutative electrodynamics from $$SO(2,3)_\star $$ S O ( 2 , 3 ) ⋆

“…Especially, corrections to the relativistic Landau levels of an electron in a constant magnetic field are given by (5.17) and their non-relativistic limit is (5.19). Motion of an electron in a constant background magnetic field and NC corrections to Landau levels were investigated in the case of canonical noncommutaivity in [37][38][39] and for other types of NC space-times in [40,41]. It can be seen both from (5.17) and (5.19) that NC correction to (non)-relativistic Landau levels depends on the mass m, the principal quantum number n and the spin s. In particular, the NC correction to energy levels will be different for different levels.…”

Bogoliubov's Vision: Quasiaverages and Broken Symmetry to Quantum Protectorate and Emergence

“…62 Other systems of interest are dissipative quantum systems, Bose-Einstein condensation, symmetry breaking and gapless excitations, phase transitions, Fermi liquids, spin density wave states, Fermi and fractional statistics, quantum Hall effects, topological/quantum order, spin liquid and string condensation. 208 The typical example of emergent phenomena is in fractional quantum Hall systems 209 -two dimensional systems of electrons at low temperature and in high magnetic fields. In this case, the underlying degrees of freedom are the electron, but the emergent quasiparticles have charge which is only a fraction of that of the electron.…”