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.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.

Physical Review LettersMar 21, 1994137 citations

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

Yasuhiro Hatsugai, Mahito Kohmoto, Yong-Shi Wu

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.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.

arXiv:cond-mat/9405028Physical Review LettersAug 22, 199444 citations

The Bethe Ansatz and One-Dimensional Magnetic Systems.

Michael Durnan Johnson

Jan 1, 19860 citations

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

Yasuhiro Hatsugai, Mahito Kohmoto, Yong-Shi Wu

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.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.

arXiv:cond-mat/9509062Physical review. B, Condensed matterApr 15, 199627 citations

Analytic Bethe Ansatz for the Hubbard model

哲生出口, 瑞宏岳

日本物理学会講演概要集. 秋の分科会Sep 13, 19960 citations

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.

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.