Field Theories in Condensed Matter Physics

Sumathi Rao

• Publisher: CRC Press
• Year: 2019
• Print ISBN:
  9780750308762
• eBook ISBN:
  9780429530395
• Topic: Physical Sciences
• Subtopic: Condensed Matter

The application of field theoretic techniques to problems in condensed matter physics has generated an array of concepts and mathematical techniques to attack a range of problems such as the theory of quantum phase transitions, the quantum Hall effect, and quantum wires. While concepts such as the renormalization group, topology, and bosonization h

Chapter 1

Quantum Many Particle Physics

Pinaki Majumdar

Harish-Chandra Research Institute

Chhatnag Road, Jhusi

Allahabad – 211 019

This set of lectures focuses on some problems in itinerant Bose and Fermi systems. The first lecture will be an introduction to many-particle physics. In the second we will discuss phase transitions and symmetry breaking and the mean field approach to condensation in Bose systems. In the third we will look at strong interactions in ‘normal’ Fermi systems. The final lecture will be on strong coupling superconductivity which involves both strong interactions and symmetry breaking.

1.1 Preamble

‘Statistical field theory’ encompasses all the applications of quantum field theory to statistical physics. In the context of these notes, it is interpreted as the study of ‘statistical mechanics of quantum many particle systems, employing the methods of field theory’. In this interpretation we are closer to the classic text of Abrikosov, Gorkov and Dzyaloshinski [1], than to the more modern approaches of Parisi [2] or Itzykson and Drouffe [3]. These lectures cover only quantum systems, fermions and bosons; other lectures in the school covered applications of field theory to problems in classical statistical mechanics and spin systems.

These lectures are meant to provide an introduction to itinerant quantum systems and discuss a few model problems. There is no claim at originality! The material is readily available in the literature [4, 5, 6, 7] and anyone interested should consult the references for a more comprehensive and rigorous coverage.

1.2 Introduction

We begin our survey of methods available for handling interacting many particle systems. There is no single method that is valid in all cases. We may tentatively classify the approaches as:

1. Exact solution!

2. ‘Mean field’ theory for symmetry broken states.

3. Perturbation theory; weak coupling.

4. Strong coupling expansion (if/when possible).

5. Renormalisation group calculations.

6. Exact numerical calculations.

7. Large dimension expansion.

This is not an exhaustive list. Special techniques exist, for instance in one dimensional problems, as discussed elsewhere in the text. The emphasis here is on general applicability.

Of these methods ‘exact solutions’ are not generic [8]. For itinerant fermions only the simplest interacting model (the Hubbard model) can be solved in one dimension (Id). There are no exactly solvable models in two or higher dimensions. Renormalisation group calculations are discussed elsewhere, in the context of classical systems and for Id fermions and quantum impurity models. Exact numerical calculations, for instance Quantum Monte Carlo (QMC) or exact diagonalisation methods, are powerful and quite generally applicable but lie beyond the scope of our discussion.

The large dimension, d → ∞, approach to lattice models is a promising new meth...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Introduction
• 1 Quantum Many Particle Physics
• 2 Critical Phenomena
• 3 Phase Transitions and Critical Phenomena
• 4 Topological Defects
• 5 Introduction to Bosonization
• 6 Quantum Hall Effect
• 7 Low-dimensional Quantum Spin Systems