The geometry of algebraic Fermi curves
This book outlines a mathematical model for electronic motion at low temperature in a finite, pure sample of a d-dimensional crystal. The authors present current research using the machinery of algebraic geometry and topological methods to determine the entire independent electron approximation.
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The Periodic Schrodinger Operator and Electrons in a Crystal
OneDimensional Algebraic Bloch Varieties
13 other sections not shown
0)-vanishing cycles algebraic analytic continuation arithmetic genus assume B(q)comp band function Bloch variety branch Chapter coefficient compactification components compute consists coordinates Corollary curve F cusp vanishing cycle defined deformation denote density determinant divisor eigenvalues equation Euler characteristic Fermi curve fia x m fiber Figure finite fixed points Furthermore germ Hodge structures holomorphic Hove vanishing cycle hyperelliptic curves implies independent electron approximation involution ip(m irreducible isomorphism matrix monodromy group morphism neighborhood Observation one-dimensional ordinary double points P1 x P1 parallel transport parametrized path Picard-Lefschetz formulas point of B(0 points q polynomial potential proof of Lemma proof of Proposition proof of Theorem Proposition 7.2 prove Qe'f quadrics Remark resp Sing(B singular point smooth points spectral quadruple point spectral van Hove spectral vanishing cycle strict transform sufficiently small tacnode tacnode point Theorem 6.1 transversal Zariski open zero