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
• Banach algebras
• $K$-theory
• Noncommutative Geometry
• Bloch Theory
• Algebra of observables of the solid body
• Symmetries
• Observables of the solid body in terms of $K$-theory
• Algebra of boundary observables
• $BB$-Correspondence
• E.A. Teplyakov, Mathematical methods in solids. Additional 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.

Lecturer
Sergeev Armen Glebovich

Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center