Quantum Field Theory in Condensed Matter Physics

Rating:
Summary: physical review
Review: diffusion in the semiconducto

Rating:
Summary: Modern overview that emphasizes the physics
Review: Quantum field theory has been applied to many different areas of physics, and has done a fairly good job of explaining the phenomena in these areas. When applying quantum field theory to a physical problem one usually takes a pragmatic attitude, and ignores the many existing difficulties in its formalism. Quantum field theory has yet to be put on a rigorous mathematical foundation, but this has not deterred its use in a myriad of applications, with condensed matter physics, the subject of this book, being one of them. The author has done a superb job here, since he emphasizes the physics behind quantum field theory, and not just the formalism. Anyone interested in quantum field theory, and especially those outside the "oral tradition", will definitely benefit from its perusal.

That quantum field theory is similar to statistical mechanics is used extensively in this book. Loosely speaking, one can view the quantum field theory of a system in a certain dimension as a statistical mechanical system in one dimension more. This simplifies calculations considerably, and in condensed matter physics things get even easier since a lattice is present, thus allowing one to deal more transparently with the problems with infinities that will always appear in quantum field theory.

The author gives an overview of quantum field theory in the first part of the book, it being assumed that the reader already has a strong background in it. The calculation of correlation functions is the main goal of the book, and to facilitate this, the author introduces the path integral formalism. Wick's theorem, the tour-de-force of calculations in quantum field theory is then derived. Explicit calculations are done for a bosonic field in an external field using the now ubiquitous mathematical identity that "the determinant of an operator is the exponential of the trace of the logarithm of the operator. One should remember when reading these pages that the considerations are purely formal since no mathematical justification has been given for the path integral measure. Perturbation theory and Feynman diagrams are discussed (of course) and the infinities that arise in perturbation series are dealt with using regularization procedures. Since the author is dealing with problems in condensed matter, where a lattice is present, he labels quantum field theories as "universal" if there is no dependence of the correlation functions in the lattice. Since regularization procedures are obviously dependent on the lattice spacing (the "ultraviolet" and "infrared" divergences), physical quantities that depend on this are called "non-universal" by the author. The standard characterization of a theory as being "renormalizable" is reserved for those where the perturbation expansion can be reformulated so that non-universal quantities appear as a finite number of parameters. This leads to a formulation of the "universality hypothesis" and the renormalization group. The author states the Gellman-Low equation, and shows that the behavior of the Gellman-Low function graphically. The properties of this function in predicting asymptotic freedom and phase transitions are discussed in detail. The O(N) model is used to illustrate some of the phenomena exhibited by quantum field theories, such as symmetry breaking and the origin of Goldstone bosons.

All of these considerations involve only bosonic quantum field theories, but the inclusion of fermions is done in the second part of the book. The discussion here is also more physical, as the author discusses electrodynamics in metals, the treatment however being non-relativistic. This is remedied though later as the author treats quantum electrodynamics. The Schwinger model, and the origin of anomalies as a screening of the electromagnetic field is discussed, and this discussion is more physically motivated and better appreciated intuitively than the one based on path integral measures. The famous Boson-Fermion equivalence in (2+1) dimensions is discussed in terms of the Aharonov-Bohm effect. This is an interesting discussion and one that is somewhat unorhodox, as it is usually not presented in this way. It clearly shows the physical meaning of adding the Chern-Simons term to the Lagrangian, presented in most books as being merely a mathematical device.

Spin systems are the subject of part 3 of the book, with the author noting at the beginning that such systems are complicated to study due to the commutation relations of the spin operators. The emphasis is on disordered magnetic systems, and the presentation is crystal clear from a physical standpoint. The role of continuous symmetry in the nonlinear sigma model, and the breaking of discrete symmetry by short range quantum fluctuations is discussed in detail. The reader is also briefly introduced to the physics of doped antiferromagnets.

The last part of the book is the most exotic, and one that is better understood from a mathematical standpoint. The physics of (1+1)-dimensional quantum systems has turned out to be more of a mathematical playground however, as it turns out to have many experimental manifestations, as the author points out many times. In addition, his treatment of the quantum field theory of the free massless bosonic scalar field shows that even a seemingly trivial action can have non-trivial properties in terms of its correlation functions. Perturbing this action by a cosine term gives the sine-Gordon model, which is exactly solvable, and its connection with conformal field theory is shown by the author. The famous Kosterlitz-Thouless transition is also treated in fair detail. The Ising and spin 1/2 Heisenberg models are discussed in terms of conformal field theories and bosonization. The reader thus gets a physical motivation for the consideration of conformal field theories that have resulted in an enormous amount of research in the past decade. And, also, the reader can see clearly the origin of Kac-Moody algebras and non-Abelian bosonization in these and latter discussions on current operators. The Kondo problem, dealing with a magnetic impurity in a metal, and one of the most difficult problems in condensed matter physics, is treated here in detail in one dimension at half-filling.

Rating:
Summary: The Best Introduction to CFT for CMT
Review: This book gives a fairly comprehensive and very comprehendible introduction to conformal field theory as seen from a condensed-matter theorist's point of view. The review of the book from Book News given above actually describes it very well. The only things that could be added are a warning that there are typos in the text and a praise to the author for the superb (as always) illustrations, which he drew himself.

The author gives a fuller treatment of the subject in his new book "Bosonisation and Strongly Correlated Systems", which, quite unfortunately, has not yet come out in paperback and is prohibitively expensive in hardcover.