Rigorous results on unexpected conductance of certain low-dimensional materials
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Rigorous results on unexpected conductance of certain low-dimensional materials

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In this thesis, we will study the conductance of several models originating from condensed matter physics, including the Anderson model for random systems and the Bistritzer-MacDonald (BM) model for twisted bilayer graphene (TBG). In fact, in their time, both models exhibited unexpected conductance properties which bewildered mathematicians and physicists. The Anderson model, developed in 1958 by physicist P.W. Anderson, exhibited unexpected localization/insulating phenomena in the 1D and 2D cases, while TBG was discovered experimentally in 2018 \cite{C18} to have unconventional superconductance at certain ``magic angles'' with relatively flat bands. This thesis has two primary parts. In the first part, we prove different types of localization results in the Anderson model and other related models. In the second part, we study the BM model in various magnetic fields from spectral, semi-classical and physical perspectives; in particular, we focus on the existence and persistence of flat bands which, though mysterious, is believed to be related to the superconductance of TBG \cite{LKV21}. More specifically, for the first part, we initially provide a short non-perturbative proof of Anderson localization and dynamical localization for the 1D Anderson model with arbitrary disorder (e.g. including Bernoulli potential). After that, we derive the dynamical localization in expectation in a related random CMV model with arbitrary disorder. Finally, we work with 2D Anderson model with Bernoulli potential and prove strong dynamical localization in expectation in this setting.

We start the second part by first discussing the influence of different magnetic and electric potentials on the existence/persistence of flat bands for TBG. After the general discussion, we divert our attention the strong constant magnetic fields and provide the explicit asymptotic expansion of the density of states (DOS). In particular, we point out the intrinsically different roles that chiral and anti-chiral potentials play in the magnetic response of TBG. Finally, from the expansion of the DOS, we are able to study the physical phenomena, including magnetic oscillations and quantum Hall effect of the TBG. We find that the chiral potential enhances these phenomena, while the anti-chiral potential diminishes them.

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