Abstract
Eigenstates of the planar magnetic Laplacian with a homogeneous magnetic field form degenerate energy bands, the Landau levels. We discuss the unitary correspondence between states in higher Landau levels and those in the lowest Landau level, where wave functions are holomorphic. We apply this correspondence to many-body systems; in particular, we represent effective Hamiltonians and particle densities in higher Landau levels by using corresponding quantities in the lowest Landau level.
Highlights
The state space of a charged particle moving in a homogeneous magnetic field in a plane orthogonal to the field decomposes into Landau levels, differing in energy by integral multiples of the magnetic field strength
As noted by many people for a long time, and emphasized, in particular, in Refs. 1–3, a holomorphic representation of states is not limited to the lowest Landau level, where it has proved to be important for deriving some basic properties, e.g., Refs. 4–8
We discuss several ways to arrive at the holomorphic representations and derive explicit formulas for particle densities and effective Hamiltonians in higher Landau levels, expressed in terms of corresponding quantities in the lowest Landau level
Summary
The state space of a charged particle moving in a homogeneous magnetic field in a plane orthogonal to the field decomposes into Landau levels, differing in energy by integral multiples of the magnetic field strength. A formally simpler and more direct approach is to use the creation and annihilation operators of the cyclotron oscillator to define unitary mappings between different Landau levels.12 This gives explicit formulas for particle densities of many-body states in one Landau level in terms of polynomials in the Laplacian applied to the corresponding densities in the lowest Landau level. The physics boils down to the motion of the electrons in the last, partially filled, Landau level In both cases, only one Landau level has to be taken into account, and an effective model of widespread use in the literature is given in terms of a Hamiltonian acting on holomorphic functions. VI, we apply our formulas to Laughlin states in an arbitrary Landau level, computing their density profiles and extending rigidity results from Refs. 4–6 and 8
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