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arXiv:cond-mat/9509062 · Physical review. B, Condensed matter

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

Yasuhiro Hatsugai, Mahito Kohmoto, Yong-Shi Wu

Published Apr 15, 1996• 27 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.53.9697

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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.
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.
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.
The Bethe Ansatz is a method that is used in quantum integrable models in order to solve them explicitly. This method is explained here in a general framework, which applies to 1D quantum spin chains, 2D statistical lattice models (vertex models) and relativistic field theories with 1 space dimension and 1 time dimension. The connection with quantum groups is expounded. Several applications are then presented. Finite size corrections are calculated via two methods: The Non-Linear Integral Equations, which are applied to the study of the states of the affine Toda model with imaginary coupling, and their interpolation between the high energy (ultra-violet) and low energy (infra-red) regions; and the Thermodynamic Bethe Ansatz Equations, along with the associated Fusion Equations, which are used to determine the thermodynamic properties of the generalized multi-channel Kondo model. The latter is then studied in more detail, still using the Bethe Ansatz and quantum groups, so as to characterize the spectrum of the low energy excitations.
For the case of uniform magnetic field and a potential quadratic in its coordinates an electron state originally represented by a Gaussian wave packet remains Gaussian in its subsequent evolution. Under these conditions the state is completely characterized by the first-and second-order quantum means, that is, quantities such as 〈qi〉〈{q}_{i}〉〈qi​〉, 〈pi〉〈{p}_{i}〉〈pi​〉 and 〈qipj〉〈{q}_{i}{p}_{j}〉〈qi​pj​〉, 〈pipj〉〈{p}_{i}{p}_{j}〉〈pi​pj​〉, respectively. A tensor-product relationship is shown to exist between the problem of the determination of the second- and first-order means which facilitates the solution. Two problems are treated in detail: (i) an isotropic wave packet in an isotropic potential well and (ii) the dynamics of a wave packet on a quadratic saddle point. In the former case, the wave packet "spins" and its radius pulsates as its mean follows the classical trajectory. In the second the magnetic field effect on tunneling through such barriers is studied and found to be small at attainable field strengths unless the effective electronic mass is very small. A method for the numerical solution for the time-dependent Schr\"odinger equation, termed the particle method and based on the hydrodynamic analogy to that equation, is here extended to include magnetic fields. It is applied to the two problems just described and found to agree well with the analytical solutions.