Harper-Hofstader Model: Butterfly & Landau Levels Relationship

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Discussion Overview

The discussion centers on the relationship between Hofstadter's Butterfly and Landau Levels in two-dimensional systems, particularly in the context of the Quantum Hall Effect (QHE). Participants explore theoretical connections and implications of these concepts.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant questions the relationship between Hofstadter's Butterfly and Landau Levels, seeking clarification on any connections.
  • Another participant suggests that Hofstadter's Butterfly arises from the solution to Harper's equation, linking it to the Almost Mathieu Operator.
  • A different participant expresses skepticism about the adequacy of Landau Levels in explaining the Quantum Hall Effect, proposing that the butterfly structure is only remotely related and emphasizing the mathematical nature of the QHE.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between Hofstadter's Butterfly and Landau Levels, with no consensus reached on their connection or the implications for the Quantum Hall Effect.

Contextual Notes

Some claims depend on specific mathematical formulations and interpretations, which remain unresolved in the discussion.

VishalSharma
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How are Hofstader's Butterfly and Landau Levels in 2-Dimesnions related to each other, if at all ?
 
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VishalSharma said:
Thanks for your help!
I'm currently writing a paper about Quantum Hall Effect, and I've run into these two concepts during my research. And as far as I am concerned, Landau level doesn't fully explain QHE and the "butterfly" is only remotely related. Quantum Hall effect is more of the result of mathematics than physics. The formulation of the resulting current in y direction under a weak electric filed in x direction, just so happens to consists of an integration that is mathematically proven to be an integer.
 
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Yes, I read that Hofstadter's butterfly is the result of the solution of the Harper's equation, which is just a particular case of a mathematical tool called Almost Mathieu Operator (i.e. λ = 1).
 

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