Explicit Solutions of the Bethe Ansatz Equations for Bloch Electrons in a Magnetic Field

Abstract

For Bloch electrons in a magnetic field, explicit solutions are obtained at the center of the spectrum for the Bethe ansatz equations recently proposed by Wiegmann and Zabrodin. When the magnetic flux per plaquette is 1/Q where Q is an odd integer, distribution of the roots is uniform on the unit circle in the complex plane. For the semi-classical limit, Q → ∞, the wavefunction obeys the power low and is given by |ψ(x)| 2 = (2/ sin πx) which is critical and unnormalizable. For the golden mean flux, the distribution of roots has the exact self-similarity and the distribution function is nowhere differentiable.The corresponding wavefunction also shows a clear self-similar structure.

Self-similar occurrence of massless Dirac particles in graphene under a magnetic field

“…[6][7][8][9] In addition to numerical studies solving Harper's equation, there have been extensive efforts to obtain analytic solutions. [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] The reason for such efforts is multifaceted. For one, many researchers have been curious about the very origin of the self-similar fractal structure seen in the Hofstadter butterfly and tried to make a connection to other known systems exhibiting similar fractal structures.…”

Quantum group, Bethe ansatz equations, and Bloch wave functions in magnetic fields

“…Indeed, we have turned the dream into reality: We have found explicit analytical solutions to this problem at the first time in the literature. [14] The algebraic structure of U q (sl 2 ) and the associated Bethe Ansatz equations also gives us a new method to solve the problem numerically. Especially when we study the irrational limit of a well-organized sequence of rational fluxes, the new method is convenient and beneficial in revealing the multifractal behavior, as we briefly reported also in ref.…”

“…Especially when we study the irrational limit of a well-organized sequence of rational fluxes, the new method is convenient and beneficial in revealing the multifractal behavior, as we briefly reported also in ref. [14].…”

Conductivity of 2D Lattice Electrons in an Incommensurate Magnetic Field

“…Even for non-interacting electrons, the various physical quantities (for example, the wavefunctions, and the energy spectra) exhibit extremely rich behaviors [1][2][3][4][5][6]8,15,17] and it has been attracted great attentions in relation to the quantum Hall effect [7][8][9][10][11][12], one-dimensional quasiperiodic systems [13][14][15][16][17], flux states for the high-T c superconductivity [18][19][20][21]. The algebraic structure of this problem has also been revealed recently [22][23][24].…”