Lecture Notes on Field Theory in Condensed Matter Physics

The following sections are included:

• Dedication
• Preface
• Acknowledgments
• Contents

The classical equations of motion for a finite chain of atoms are solved within the harmonic approximation. In the thermodynamic limit, an approximate hydrodynamical description, i.e., a one-dimensional classical field theory, is obtained. Quantization of the finite harmonic chain is undertaken. In the thermodynamic limit, phonons in a one-dimensional lattice are approximated by a quantum hydrodynamical theory, i.e., a one-dimensional quantum field theory.

Second quantization for bosons is reviewed. Bose-Einstein condensation for noninteracting bosons is interpreted as an example of spontaneous-symmetry breaking. The spectrum of a dilute Bose gas with hardcore repulsion is calculated within Bogoliubov mean-field theory using the operator formalism. It is shown that a Goldstone mode, an acoustic phonon, emerges in association with spontaneous-symmetry breaking. Landau criterion for superfluidity is presented. Bose-Einstein condensation as well as superfluidity at non-vanishing temperatures are treated using the path integral formalism.

The O(N) Non-Linear Sigma model (NLσM) is defined as an effective field theory that encodes the pattern of symmetry breaking of an O(N) classical Heisenberg model defined on a square lattice. A geometric and group-theoretical interpretation of the O(N) NLσM is given. The notions of fixed points and the notions of relevant, marginal, and irrelevant perturbations are introduced. The real-valued scalar (free) field theory in two-dimensional Euclidean space is taken as and example of a critical field theory and the Callan-Symanzik equations obeyed by (m + n) point correlation functions are derived for the two-dimensional O(2) NLσM. The Callan-Symanzik equations are generalized to include relevant coupling constants. The Callan-Symanzik equation obeyed by the spin-spin correlator in the two-dimensional O (N > 2) NLσM is derived. The beta function and the wave-function renormalization are computed up to second order in the coupling constant of the two-dimensional O(N > 2) NLσM. It is shown that the free-field fixed point is IR unstable in that the coupling constant flows to strong coupling upon coarse graining and that there exists a finite correlation length that increases exponentially fast upon approaching the free-field unstable fixed point. The beta function for the O(N) NLσM is also computed in more than two-dimensions, in which case it exhibits a non-trivial fixed point at a small but finite value of the coupling constant.

The classical two-dimensional XY (2d–XY) model is defined. The quasi-long-range ordered phase in the continuum spin-wave approximation derived in section 3.3.2 is shown to be unstable to the deconfining transition of topological defects, i.e., vortices, that drives the model into a high-temperature paramagnetic phase [40–42]. This transition, which we shall call the Kosterlitz-Thouless (KT) transition, although the terminology Berezinskii-Kosterlitz-Thouless (BKT) transition is also used in the literature, is studied within a perturbative renormalization-group analysis. To set up the renormalization-group analysis, the vortex sector of the 2d – XY is first shown to be equivalent to a 2d–Coulomb gas [40]. In turn the 2d–Coulomb gas is shown to be equivalent to the 2d–Sine-Gordon model. The renormalization-group analysis is made within the 2d–Sine-Gordon model.

The physics of non-interacting fermions is reviewed.

The jellium model for a three-dimensional Coulomb gas is defined. The path-integral representation of its grand-canonical partition function is presented. Collective degrees of freedom are introduced through a Hubbard-Stratonovich transformation. The low-energy and long-wavelength limit of the effective theory for the collective degrees of freedom that results from integration over the fermions is derived within the random-phase approximation (RPA). A diagrammatic interpretation to the RPA approximation is given. The ground-state energy in the RPA approximation is calculated. The dependence on momenta and frequencies of the RPA polarization function is studied. A qualitative argument is given for the existence of a particlehole continuum and for a branch of sharp excitations called plasmons. The quasistatic and dynamic limits of the polarization function are studied. The quasistatic limit is characterized by screening, Kohn effect, and Friedel oscillations. The dynamic limit is characterized by plasmons and Landau damping. The physical content of the RPA approximation for a repulsive short-range interaction is derived. The physics of zero-sound is discussed. The feedback effect of phonons on the RPA effective interaction between electrons is sketched.

The notion of a pairing order is introduced. A repulsive interaction is decoupled through the pairing-order parameter. It is shown that the coupling of a repulsive interaction decreases upon momentum shell integration, whereas the coupling of an attractive interaction increases. An effective action for a uniform and static pairing-order parameter is calculated and analyzed perturbatively as well as nonperturbatively. The BCS mean-field theory for a uniform and static pairing-order parameter is described. Effective theories for the superconducting order parameter are derived in the vicinity of T = 0 and T = T_c, respectively. First, an effective action is derived that describes in the vicinity of T = 0 the long-wavelength and low-frequency fluctuations of the phase of the superconducting order parameter. Second, the Gross-Pitaevskii non-linear Schrödinger equation, from which follows the Meissner effect, is derived at T = 0. Third, the polarization tensor in the pairing state is calculated to quadratic order in the expansion of the fermionic determinant and shown to encode the Anderson-Higgs mechanism by which the photon acquires an effective mass in a superconductor. Finally, the space-dependent Ginzburg-Landau functional is calculated in the vicinity of T_c.

A phenomenological model for a Josephson junction is presented. The DC- and AC-Josephson effects are described. A model for a dissipative Josephson junction is given both at the classical and quantum levels. At the quantum level, a dissipative Josephson junction is shown to realize the Caldeira-Leggett model of dissipative quantum mechanics. The method of instantons in quantum mechanics is reviewed. The phase diagram of a dissipative Josephson junction is discussed using renormalization-group methods and a duality transformation. The existence of lines of weak- and strong-coupling fixed points is established.

It is shown that the two-dimensional massive Thirring model is related to the two-dimensional Sine-Gordon model through Abelian bosonization. The bosonized solution to the quantum-xxz spin-1/2 chain and to the single-impurity problem in a Luttinger liquid are given.

The following sections are included:

• The harmonic-oscillator algebra and its coherent states
  • The harmonic-oscillator algebra and its coherent states
    • Bosonic algebra
    • Coherent states
  • Path-integral representation of the anharmonic oscillator
  • Higher dimensional generalizations
• Appendix B Some Gaussian integrals
  • Generating function
  • Bose-Einstein distribution and the residue theorem
• Appendix C Non-Linear Sigma Models (NLσM) on Riemannian manifolds
  • Introduction
  • A few preliminary definitions
  • Definition of a NLσM on a Riemannian manifold
  • Classical equations of motion for NLσM: Christoffel symbol and geodesics
  • Riemann, Ricci, and scalar curvature tensors
  • Normal coordinates and vielbeins for NLσM
    • The background-field method
    • A mathematical excursion
    • Normal coordinates for NLσM
    • Gaussian expansion of the action
    • Diagonalization of the metric tensor through vielbeins
    • Renormalization of the action after integration over the fast degrees of freedom
    • One-loop scaling flow obeyed by the metric tensor
  • How many couplings flow on a NLσM?
• Appendix D The Villain model
• Appendix E Coherent states for fermions, Jordan-Wigner fermions, and linear-response theory
  • Grassmann coherent states
  • Path-integral representation for fermions
  • Jordan-Wigner fermions
    • Introduction
    • Nearest-neighbor and quantum xy limit in one-dimensional position space
  • The ground state energy and the single-particle time-ordered Green function
  • Linear response
    • Introduction
    • The Kubo formula
    • Kubo formula for the conductivity
    • Kubo formula for the dc conductance
    • Kubo formula for the dielectric function
    • Fluctuation-dissipation theorem
• Appendix F Landau theory of Fermi liquids
  • Adiabatic continuity
  • Quasiparticles
  • Topological stability of the Fermi surface
    • The case of no many-body interactions
    • The case with many-body interactions
  • Quasiparticles in the Landau theory of Fermi liquids as poles of the two-point Green function
  • Breakdown of Landau Fermi liquid theory
    • Gapped phases
    • Luttinger liquids
• Appendix G First-order phase transitions induced by thermal fluctuations
  • Landau-Ginzburg theory and the mean-field theory of continuous phase transitions
  • Fluctuations induced by a local gauge symmetry
  • Applications
• Appendix H Useful identities
  • Proof of Equation (8.75)
    • Proof of (H.5)
    • Proof of (H.11)
    • Proof of (H.22)
    • Proof of (H.28)
    • Proof of (H.31)
    • Proof of (H.40)
• Appendix I Non-Abelian bosonization
  • Introduction
  • Minkowski versus Euclidean spaces
  • Free massless Dirac fermions and the Wess-Zumino-Witten theory
  • A quantum-mechanical example of a Wess-Zumino action
  • Wess-Zumino action in (1+1)-dimensional Minkowski space and time
  • Equations of motion for the WZNW action
  • One-loop RG flow for the WZNW theory
  • The Polyakov-Wiegmann identity
  • Integration of the anomaly in QCD₂
    • General symmetry considerations
    • The axial gauge anomaly for QCD₂
  • Bosonization of QCD₂ for infinitely strong gauge coupling

The following sections are included:

• Bibliography
• Index