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The goal of the course is to acquaint the listeners with the mathematical,
first of all topological, methods used in the solid state theory. The role of
topology in the solid state physics revealed on full scale in the investigation of
the quantum Hall effect. After its discovery by von Klitzing in 1980 appeared the
publications of Loughlin and Thouless et al. in which it was proposed the
topological explanation of this effect.
The key role in the investigation of topological properties of solid bodies is
played by the study of their symmetry groups. The description of the possible
symmetry types goes back to Kitaev who proposed a classification of topological
objects based on the representation theory of Clifford algebras. The Clifford
algebras were followed by $K$-theory in which terms it is natural to formulate the
topological properties of solid bodies.
In our course we shall present applications of the mentioned mathematical
theories in the solid state physics. We start by recalling the basic notions of
Bloch theory describing the properties of the solid bodies having the crystal
lattice. Then we construct the algebras of observables of topological objects and
arising symmetry classes.
We give next the description of the algebra of observables in terms of $K$-theory of the graded $C*$-algebras and introduce the topological invariants of the
solid body. The algebra of boundary observables is also defined in terms of the $K$-theory proposed by Kasparov.
We conclude the course by construction of the $BB$-correspondence between
the topological invariants of the solid body and its boundary. This
correspondence admits a natural formulation in terms of $K$-theory. In the
particular case of the periodic unitary model it can be described in an explicit
way.
 Course materials::
INTRODUCTION
I. $C*$- ALGEBRAS
1.1. $C*$-algebras.
1.2. $C*$-modules .
1.3. Tensor products.
1.4. Operators of $A$-finite rank and $A$-compact operators.
1.5. Projections and unitary operators.
II. $K$-THEORY
2.1. $K_0$-group.
2.2. Grothendieck construction.
2.3. $K_1$-group.
III. NONCOMMUTATIVE GEOMETRY
3.1. Spectral triples.
3.2. Fredholm modules.
3.3. Theory of index.
3.4. $K$-theory.
IV. BLOCH THEORY
4.1. Oneparticle Schrödinger operator.
4.2. Fermionic Fock space.
4.3. Fermionic Fock space of the solid body.
4.4. Tight-binding approximation.
V. ALGEBRA OF OBSERVABLES OF THE SOLID BODY
5.1. Real $C*$-algebras.
5.2. Local observables.
5.3. Algebra of observables of the solid body.
5.4. Crossed products.
5.5. Graded $C*$-algebras
VI. SYMMETRIES
6.1. Clifford algebras.
6.2. Symmetry classes.
6.3. Pseudosymmetries.
VII. ALGEBRA OF OBSERVABLES OF THE SOLID BODY IN TERMS
OF $K$-THEORY
7.1. $K$-theory.
7.2. Topological invariants of the solid body.
VIII. ALGEBRA OF BOUNDARY OBSERVABLES
8.1. Boundary observables.
8.2. Fredholm $K$-theory.
8.3. Construction of the boundary classes.
IX. $BB$-CORRESPONDENCE
9.1. Main theorem and its corollaries.
9.2. $BB$-correspondence in the unitary class.
9.3. $BB$-correspondence for the periodic model.
 Program
Lecturer
Sergeev Armen Glebovich
Financial support
The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no. 075-15-2022-265).
Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center |
