Physical Review Letters

Bethe-ansatz for the Bloch electron in magnetic field

Paul B. Wiegmann, Anton Vladimirovich Zabrodin

Published Mar 21, 1994• 137 citations

Abstract

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.

DOI: 10.1103/physrevlett.72.1890

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Physical Review Letters

Bethe-ansatz for the Bloch electron in magnetic field

Paul B. Wiegmann, Anton Vladimirovich Zabrodin

Published Mar 21, 1994• 137 citations

Publisher

Abstract

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.

DOI: 10.1103/physrevlett.72.1890

Related papers

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.

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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.

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Paul Zinn-Justin

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Bethe-ansatz equations for quantum Heisenberg chains with elliptic exchange

V. I. Inozemtsev

The eigenvectors of the Hamiltonian ${\cal H}_{N}$ of $N$ -sites quantum spin chains with elliptic exchange are connected with the double Bloch meromorphic solutions of the quantum continuous elliptic Calogero-Moser problem. This fact allows one to find the eigenvectors via the solutions to the system of highly transcendental equations of Bethe-ansatz type which is presented in explicit form.The eigenvectors of the Hamiltonian ${\cal H}_{N}$ of $N$-sites quantum spin chains with elliptic exchange are connected with the double Bloch meromorphic solutions of the quantum continuous elliptic Calogero-Moser problem. This fact allows one to find the eigenvectors via the solutions to the system of highly transcendental equations of Bethe-ansatz type which is presented in explicit form.

arXiv:math-ph/9911022Nov 17, 19992 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.

The eigenvectors of the Hamiltonian

{\cal H}_{N}

N

-sites quantum spin chains with elliptic exchange are connected with the double Bloch meromorphic solutions of the quantum continuous elliptic Calogero-Moser problem. This fact allows one to find the eigenvectors via the solutions to the system of highly transcendental equations of Bethe-ansatz type which is presented in explicit form.