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arXiv:cond-mat/9908254 · Physical review. B, Condensed matter

Landau levels from the Bethe Ansatz equations

K. Hoshi, Yasuhiro Hatsugai

Published Feb 15, 2000• 9 citations
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Abstract

The Bethe ansatz (BA) equations for the two-dimensional Bloch electrons in a uniform magnetic field are treated in the weak-field limit. We have calculated energies near the lower boundary of the energy spectrum up to the first nontrivial order. It corresponds to calculating a finite size correction for the excitation energies of the BA solvable lattice models and gives the Landau levels in the present problem.

DOI
10.1103/physrevb.61.4409

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The Bethe ansatz (BA) equations for the two-dimensional Bloch electrons in a uniform magnetic field are treated in the weak-field limit. We have calculated energies near the lower boundary of the energy spectrum up to the first nontrivial order. It corresponds to calculating a finite size correction for the excitation energies of the BA solvable lattice models and gives the Landau levels in the present problem.
For Bloch electrons in a magnetic field, explicit solutions are obtained at the center of the spectrum for the Bethe ansatz equations of Wiegmann and Zabrodin. When the magnetic flux per plaquette is 1Q\frac{1}{Q}Q1​ with QQQ an odd integer, distribution of the roots of the Bethe ansatz equation is uniform except at two points on the unit circle in the complex plane. For the semiclassical limit $Q\ensuremath{\rightarrow}\ensuremath{\infty}$, the wave function is ${|\ensuremath{\psi}(x)|}^{2}=(\frac{2}{sin \ensuremath{\pi}x})$, which is critical and unnormalizable. For the golden-mean flux, the distribution of roots has exact self-similarity and the distribution function is nowhere differentiable. The corresponding wave function also shows a clear self-similar structure.
The wave functions for a two-dimensional Bloch electron in uniform magnetic fields at the mid-band points are studied by exploiting a connection to the quantum group Uq{\mathit{U}}_{\mathit{q}}Uq​(sl2{\mathit{sl}}_{2}sl2​): A linear combination of its generators gives the Hamiltonian. We apply both analytic and numerical methods to obtain and analyze the wave functions, by solving the functional Bethe ansatz equations first proposed by Wiegmann and Zabrodin on the basis of the above observation. The semiclassical case with the flux per plaquette \ensuremath{\varphi}=1/Q is analyzed in detail, by exploring a structure of the Bethe ansatz equations. We also reveal the multifractal structure of the solutions to Bethe ansatz equations and corresponding wave functions when \ensuremath{\varphi} is irrational, such as the golden or silver mean. \textcopyright{} 1996 The American Physical Society.
We present a new approach to the problem of Bloch electrons in magnetic field, by making explicit a natural relation between magnetic translations and the quantum group Uq{\mathit{U}}_{\mathit{q}}Uq​(sl2{\mathit{sl}}_{2}sl2​). The approach allows us to express the spectrum and the Bloch function as solutions of the Bethe-ansatz equations typical for completely integrable quantum systems.
Building on the fundamentals introduced in part I, we employ the Bethe ansatz to study some ground-state properties (energy, magnetization, susceptibility) of the one-dimensional s=1/2 Heisenberg antiferromagnet in zero and nonzero magnetic field. The 2-spinon triplet and singlet excitations from the zero-field ground state are discussed in detail, and their energies are calculated for finite and infinite chains. Procedures for the numerical calculation of real and complex solutions of the Bethe ansatz equations are discussed and applied. The paper is designed as a tutorial for beginning graduate students. It includes 10 problems for further study.
We discuss the size dependence of small-system periodic-boundary-condition calculations of ground-state energies and excitation energies for two-dimensional electron systems in a strong magnetic field. The discussion is based on an examination of the Hartree-Fock approximation for a finite-size system. We find that finite-size-system calculations overestimate the magnitude of the ground-state energy and underestimate excitation energies, especially in the case of higher-orbital Landau levels. Numerical estimates of the finite-size corrections are given for 4-, 9-, and 16-electron systems.