Noncommutative gauge symmetry in the fractional quantum Hall effect

Abstract

We show that a system of particles on the lowest Landau level can be coupled to a probe U(1) gauge field ${\mathcal{A}}_{\mu }$ in such a way that the theory is invariant under a noncommutative U(1) gauge symmetry. While the temporal component ${\mathcal{A}}_0$ of the probe field is coupled to the projected density operator, the spatial components ${\mathcal{A}}_i$ are best interpreted as quantum displacements, which distort the interaction potential between the particles. We develop a Seiberg-Witten-type map from the noncommutative U(1) gauge symmetry to a simpler version, which we call "baby noncommutative" gauge symmetry, where the Moyal brackets are replaced by the Poisson brackets. The latter symmetry group is isomorphic to the group of volume preserving diffeomorphisms. By using this map, we resolve the apparent contradiction between the noncommutative gauge symmetry, on the one hand, and the particle-hole symmetry of the half-filled Landau level and the presence of the mixed Chern-Simons terms in the effective Lagrangian of the fractional quantum Hall states, on the other hand. We outline the general procedure which can be used to write down effective field theories which respect the noncommutative U(1) symmetry.

arXiv comment14 pages

References in corpus (13)

• String Theory and Noncommutative Geometry
• Is the Composite Fermion a Dirac Particle?
• Quantum Hall states as matrix Chern-Simons theory
• Lowest-Landau-level theory of the quantum Hall effect: the Fermi-liquid-like state
• A dipole interpretation of the $\nu =1/2$ state
• Quantum Hall Physics = Noncommutative Field Theory
• Volume-preserving diffeomorphism as nonabelian higher-rank gauge symmetry
• Geometry of Compressible and Incompressible Quantum Hall States: Application to Anisotropic Composite Fermion Liquids
• Non-commutative Chern-Simons for the Quantum Hall System and Duality
• Non-commutative field theory and composite Fermi Liquids in some quantum Hall systems
• Microscopic derivation of Dirac composite fermion theory: Aspects of noncommutativity and pairing instabilities
• Lowest Landau level theory of the bosonic Jain states
• Matrix Effective Theories of the Fractional Quantum Hall effect

Cited by in corpus (2)

• Fermion liquids as quantum Hall liquids in phase space: A unified approach for anomalies and responses
• Quadrupole-conserving dynamics in the non-commutative plane