Bloch and Bethe Ansatze for the Harper model: A butterfly with a boundary

Article

关键词: QUANTIZED HALL CONDUCTANCE;    EDGE STATES;    ENERGY-SPECTRUM;    DIRAC FERMIONS;    ELECTRONS;   

DOI  :  10.1103/PhysRevB.104.165140

来源: SCIE

【 摘 要 】

Based on a recent generalization of Bloch's theorem, we present a Bloch Ansatz for the Harper model with an arbitrary rational magnetic flux in various geometries, and solve the associated Ansatz equations analytically. In the case of a cylinder and a particular boundary condition, the energy spectrum of edge states has no dependence on the length of the cylinder, which allows us to construct a quasi-one-dimensional edge theory that is exact and describes two edges simultaneously. We prove that energies of bulk states, generating the so-called Hofstadter's butterfly, depend on a single geometry-dependent spectral parameter and have exactly the same functional form for the cylinder and the torus with general twisted boundary conditions, and argue that the (edge) bulk spectrum of a semi-infinite cylinder in an irrational magnetic field is (the complement of) a Cantor set. Finally, realizing that the bulk projection of the Harper Hamiltonian is a linear form over a deformed Weyl algebra, we introduce a Bethe Ansatz valid for both cylinder and torus geometries.

【 授权许可】

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