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arXiv:cond-mat/9405028 · Physical Review Letters

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

Yasuhiro Hatsugai, Mahito Kohmoto, Yong-Shi Wu

Published Aug 22, 1994• 44 citations

Abstract

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 $\frac{1}{Q}$ with $Q$ 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.

DOI: 10.1103/physrevlett.73.1134

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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

\frac{1}{Q}

with

Q

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

{\mathit{U}}_{\mathit{q}}

(

{\mathit{sl}}_{2}

): 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

{\mathit{U}}_{\mathit{q}}

(

{\mathit{sl}}_{2}

). 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.

The bare existence of this book is due to the amazing fact that the solution of the stationary Schrödinger equation for the one-dimensional Hubbard model can be reduced to a set of algebraic equations, which is tractable in the thermodynamic limit. These equations will be derived in this chapter. We will call them the Lieb-Wu equations to honour E. H. Lieb and F. Y. Wu, who first obtained them [298]. The derivation is based on a method, called the nested (coordinate) Bethe ansatz, which goes back to the seminal articles of C. N. Yang [493] and M. Gaudin [154], who generalized earlier work [60, 296] on exactly solvable models to models with internal degrees of freedom. The roots of the Lieb-Wu equations parameterize the eigenvalues and the eigenstates of the Hamiltonian of the one-dimensional Hubbard model. They encode the complete information about the model. These roots are not explicitly known in the general N -particle case. In the thermodynamic limit ( N →∞), however, only the distributions of the roots in the complex plane matter, and many physical quantities can be calculated as solutions of linear or non-linear Fredholm type integral equations. Moreover, it is sometimes possible to use the Lieb-Wu equations in an implicit way even for finite N , e.g., in the proof of the SO(4) highest weight properties of the eigenstates in Sections 3.D and 3.F of the appendix or in the calculation of their norm in Section 3.5.

An exactly solvable strongly correlated electron model with two independent parameters is constructed in the frame of the quantum inverse scattering method, which can be seen as a generalization of the Bariev model. Through the Bethe ansatz method, a set of Bethe ansatz equations is derived. In the thermodynamic limit, to study the ground state of the model, we obtain the integral equations for the density of Bethe roots. Numerical validation is done to confirm the accuracy of our analytic results.