title: Topological Insulators and Topological Superconductors
author:  B. Andrei Bernevig 
description: "The book . . . may be extremely useful to both graduate students and more senior researchers."<b>---Vicentiu D. Radulescu, <i>Zentralblatt MATH</i></b><br /><br />"Dr. Bernevig writes well and with insight. . . . with a beginning graduate student in mind who wants to enter quickly the research in this field."<b>---S.W. Lovesey, <i>Contemporary Physics</i></b><br /><br />"This book presents an array of increasingly complicated problems centered around the idea of the topology of a band in <i>k</i>-space and the theorem that the Chern number determines the Hall effect. It will be invaluable to those who are in tune with the conceptual structure of modern band theory as reconfigured by Haldane and his fellow travelers."<b>Philip W. Anderson, Nobel Laureate in Physics</b><br /><br />"One of the most exciting developments in condensed matter physics over the last seven or eight years has been the topic of topological insulators and superconductors. The present book, by one of the original pioneers in this area, is a very up-to-date and comprehensive introduction to the theory of these systems. It will be extremely useful to both graduate students and more senior researchers."<b>Anthony J. Leggett, University of Illinois, Urbana-Champaign</b><br /><br />"An authoritative and sophisticated introduction to the mathematics of topological insulation."<b>Robert B. Laughlin, Stanford University</b><br /><br />"This book gives the first comprehensive introduction to the theory of topological insulators and superconductorsan exciting new field in condensed matter physics. The authors contributed to the discovery of the first topological insulator HgTe, and now present a readable account accessible to most graduate students."<b>Shoucheng Zhang, Stanford University</b><br /><br />"This excellent book introduces a relatively new topic in condensed matter physics. The material is well developed and sufficient detail is given for students to follow arguments and derivations. With a hands-on, no-nonsense approach, <i>Topological Insulators and Topological Superconductors</i> will be a mainstay in the field for years to come."<b>Marcel Franz, University of British Columbia</b><br /><br />"<i>Topological Insulators and Topological Superconductors</i> deals with a very exciting subject that has become the focus of research in recent years. Bernevig and Hughes have made some of the most important theoretical contributions to this young field and this timely volume will have significant staying power. It will be of great interest to condensed matter physicists, high energy and string theorists, and mathematicians."<b>Eduardo Fradkin, University of Illinois, Urbana-Champaign</b>
zlib_text: TOPOLOGICAL
  INSULATORS AND
  TOPOLOGICAL
  SUPERCONDUCTORS
  
  Contents
  1
  Introduction
  1
  2
  Berry Phase
  6
  2.1 General Formalism
  6
  2.2 Gauge-Independent Computation of the Berry Phase
  8
  2.3 Degeneracies and Level Crossing
  10
  2.3.1 Two-Level System Using the Berry Curvature
  10
  2.3.2 Two-Level System Using the Hamiltonian Approach
  11
  2.4 Spin in a Magnetic Field
  13
  2.5 Can the Berry Phase Be Measured?
  14
  2.6 Problems
  14
  3
  Hall Conductance and Chern Numbers
  15
  3.1 Current Operators
  15
  3.1.1 Current Operators from the Continuity Equation
  16
  3.1.2 Current Operators from Peierls Substitution
  17
  3.2 Linear Response to an Applied External Electric Field
  18
  3.2.1 The Fluctuation Dissipation Theorem
  20
  3.2.2 Finite-Temperature Green’s Function
  22
  3.3 Current-Current Correlation Function and Electrical Conductivity
  23
  3.4 Computing the Hall Conductance
  24
  3.4.1 Diagonalizing the Hamiltonian and the Flat-Band Basis
  25
  3.5 Alternative Form of the Hall Response
  29
  3.6 Chern Number as an Obstruction to Stokes