Many-Body Physics, Topology and Geometry

The following sections are included:

• Dedication
• Preface
• Contents

The challenge of condensed matter physics is to use non relativistic quantum ideas to explain and predict the observed macroscopic properties of matter. To do this great ingenuity and imagination is required. The Hamiltonian H of a many-body system can be written down schematically as

The following sections are included:

• Introduction
  • The Spectrum of Hydrogen
  • The Power of Qualitative Reasoning
  • Estimate of Speed of Electrons and Size of Their Orbits
  • Charge Distribution in an Atom
  • Estimating Lifetimes of Excited States
• The Lamb Shift
  • The Casimir Effect
  • Zero Point Effects for Nanoscale Structures
  • Estimating the Diffusion Coefficient
  • Kolmogorov's Law for Turbulence
• Turbulence in Graphene
• Gamow's Estimate of the Temperature of the Universe
• Quantum Field Theory
• Quasiparticles
  • Quasiparticles of Superfluid Helium
  • Quasiparticles of Superconductivity
  • Thermal Averages
• The Bogoliubov–de Gennes Equations
• Topology and Fermion Zero Energy Modes
  • The Proximity Approximation and Majorana Fermions

In this chapter we discuss topological ideas and methods. There is now growing awareness that such methods are useful for understanding and predicting the behaviour of condensed matter systems…

We are familiar with the idea that in order to obtain solutions of differential equations, we need to impose boundary conditions and that the nature of the solutions depend on which boundary conditions we impose. For example, the spectrum of bosons and fermions in quantum mechanics obtained by solving Schroedinger's equation are different as the boundary conditions in these two cases are different. So the natural question that immediately arises is what are the possible choices of boundary conditions for a given operator representing an observable in quantum mechanics. In order to address this issue we must remember that observables in quantum mechanics should have real eigenvalues, which is necessary for a unitary time evolution. These operators are conventionally called Hermetian or self-adjoint, although, as we shall see below, these two concepts are not identical. In view of this, we can thus ask the question what are the possible boundary conditions that can be imposed on an operator in quantum mechanics such that it is self-adjoint? The answer to this question can be obtained from the pioneering work of von Neumann on self-adjoint extensions of operators in quantum mechanics. Self-adjoint extensions are known to play important roles in a variety of physical contexts including Aharonov-Bohm effect, two and three dimensional delta function potentials, anyons, anomalies, ζ-function renormalization, particle statistics in one dimension, black holes and integrable models. In this Chapter we shall first explain the basic ideas of self-adjoint extension and their relation to boundary conditions. We shall then explain the method of von Neumann through various examples. Finally we shall apply the method of self-adjoint extensions to some interesting physics problems.

The following sections are included:

• Introduction
• Tight-Binding Model and the Dirac Equation
• Gapless Graphene with Coulomb Charge Impurities
  • Boundary Conditions and Self-Adjoint Extensions
  • Scattering Matrix for Gapless Graphene with Coulomb Charge
  • Gapless Graphene with Supercritical Coulomb Charge
• Gapped Graphene with Coulomb Charge Impurity
  • Boundary Conditions for Gapped Graphene with a Charge Impurity
  • Scattering Matrix for Gapped Graphene with Coulomb Impurity
• Gapped Graphene with a Supercritical Coulomb Charge
• Graphene with Charge Impurity and Topological Defects

The following section is included:

• Index