# Mathematics Colloquia and Seminars

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### The Quantum Hall Effect Revisited

**Mathematical Physics & Probability**

Speaker: | S. Michalakis, Los Alamos National Laboratory |

Location: | 2112 MSB |

Start time: | Wed, Feb 18 2009, 4:10PM |

Maxwell believed that a magnetic field through a conductor affected the conductor alone and not the distribution of its electrons. Edwin Hall, a student at the time at Johns Hopkins, found Maxwell's view peculiar so he set out to test the hypothesis by placing a thin gold plate in a transverse magnetic field. He observed that though a current was supplied in the x-direction of the plate, the magnetic field caused the roaming electrons to drift in the y-direction. That effect, now called the Quantum Hall Effect, earned Hall a position at Harvard 130 years ago. A hundred years later, in 1980, a German experimentalist observed that the the Hall conductance (the inverse of the resistance the Hall current felt) was quantized in integer multiples of e^2/h (e = electronic charge, h = plank's constant). Five years later he got the Nobel prize in physics. In the meantime, mathematical physicist Robert Laughlin gave an elegant heuristic/explanation for the integer quantum hall effect. He also got the Nobel prize for that, in 1998, along with Stormer and Tsui who observed that the Hall conductance could be fractionally quantized. Since then, one of the important open problems in Mathematical Physics has been the explanation of the quantization within a realistic framework of interacting electrons without a so-called "flux averaging" assumption. Using recently developed theoretical work on Lieb-Robinson bounds, I will present a proof that the Hall conductance of a tight-binding 2-D model of hopping electrons with finite range interactions is quantized exactly in the thermodynamic limit. More importantly, the proof avoids the "flux averaging" assumption that has plagued all previous attempts.