"Topological" (Chern-Simons) quantum mechanics

Testing quantum duality using cold Rydberg atoms

“…Using the complex coordinates z = x + iy and z † = x − iy, the Lagrangian L ν CO is written (apart from a total time derivative) as L ν CO = − g 2 z † i d dt + a z + νa and its effective quantum action is Γ ν CO (a) = −i ln det i d dt + a + ν adt. The determinant is not invariant against gauge transformations with nontrivial winding number [3,5,15] because under this "large" gauge transformation it acquires the sign (−1) N , what is usually called a Z 2 -anomaly [6]. Hence, to ensure the mathematical consistency of the L ν CO , a "large" (N = 0) gauge invariance of e ī h Γ ν CO (a) requires the half-integer quantization of ν.…”

“…It is clear that both the LM and the CO are invariant under "global" rotations of the kind δx i (t) = −λǫ ij x j (t), λ being time independent, and that their respective angular momentum are the generators of these transformations. To promote this "global" symmetry to a "local" gauge symmetry with time dependent λ, we follow [3,5] and introduce a "gauge potential" a(t) which produces the covariant time derivative Dx i ≡ẋ i + aǫ ij x j and the local rotation δx i (t) = −λ(t)ǫ ij x j (t), where δa(t) =λ. In this dimensionality, only a further true CS term can be added to the Lagrangian and it must be linear in a.…”

“…The issue of duality at the quantum level is however more subtle. It was shown in [3,5] that the CO has a Z 2 -anomaly that forces its orbital angular momentum to be quantized in half-integer units ofh, either positive or negative, letting it with an uncommon behavior through rotations. This anomaly has a great resemblance with the Z 2 -anomaly of a single Weyl fermion coupled to a SU(2) Yang-Mills field discovered by Witten [6].…”

“…Both models describe the classical trajectories of a particle in the x-y plane subject to a constant magnetic field B = B z [2,3,4], and therefore, the classical duality would be already expected. The issue of duality at the quantum level is however more subtle.…”

“…determined by its symplectic structure [3,11], and Hamiltonian H + = k 2 x + i x + i , one recognizes that this model has the structure of an one-dimensional harmonic oscillator [12]. It should be noticed that the coordinates of the CO system are noncommutative like the coordinates of the electrons in the lowest Landau level [13].…”