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Hofstadter’s Butterfly and interaction driven quantum Hall ferromagnetism in graphene

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Electrons moving in a periodic electric potential form Bloch energy
bands where the mass of electrons are effectively changed. In a strong
magnetic field, the cyclotron orbits of free electrons are quantized and
Landau levels forms with a massive degeneracy within. In 1976,
Hofstadter showed that for 2-dimensional electronic system, the
intriguing interplay between these two quantization effects can lead
into a self-similar fractal set of energy spectrum known as
“Hofstadter’s Butterfly.” Experimental efforts to demonstrate this
fascinating electron energy spectrum have continued ever since. Recent
advent of graphene, where its Bloch electrons can be described by Dirac
feremions, provides a new opportunity to investigate this half century
old problem experimentally. In this presentation, I will discuss the
experimental realization Hofstadter’s Butterfly via substrate engineered
graphene under extremely high magnetic fields controlling two competing
length scales governing Dirac-Bloch states and Landau orbits,
respectively. In addition, the strong Coulomb interactions and
approximate spin-pseudo spin symmetry are predicted to lead to a variety
of integer quantum Hall ferromagnetic and fractional quantum Hall
states and the quantum phase transition between them in graphene. I will
discuss several recent experimental evidences to demonstrate the role
of the electron interaction in single and bilayer graphene.