A Modern Approach to Critical Phenomena
Igor Herbut
The studies of the critical behaviors for a system near its phase
transition point constitutes an important subfield of the
researches in statistical mechanics and condensed matter physics.
It has acquired a renewed interest due to the discovery of quantum
critical behaviors in correlated electron systems. The development
of the theory of critical phenomena has a tremendous impact on
both statistical mechanics and quantum field theory, and the
associated renormalization group idea has now become a basic
language in thinking about many fundamental problems in condensed
matter physics. No wonder that there exist many books in the
market which address this issue. Among them, I have to mention the
classic book by S. K. Ma and the more recent little, but
refreshingly clear, book by John Cardy. There is also an
overwhelming monograph by J. Zinn-Justin. In my opinion, Igor
Herbut's book stands out from its pedagogy and its modern flavors.
To be more precise, although the contents of this book is, by its
very nature, challenging, this book is accessible to any motivated
graduate students with a solid background in quantum mechanics and
statistical mechanics. Moreover, unlike some old monographs, this
book not only teach you the phi-4 theory, but also contains nice
discussions of the superconducting transition, the nonliear-sigma
model, the KT transition, and the charge-vortex duality. In the
end of the book, it gives the readers a brief but useful
discussion of the quantum phase transition. On the technical side,
this book is devoted almost exclusively to the momentum-shell RG
approach which is a powerful tool and it is still widely used in
the research literature. However, I would like to recommend the
uninitiated readers to read this book in companion with a standard
field theory textbook, such as chapter 10-13 of Peskin's book, to
learn some different perspectives on the renormalization group, in
particular, the Callen-Symanzik equation and the related stuff.
As far as I can tell, the only missing "standard" topic in this
book about the critical phenomena is a systematic discussion of
the large-N calculation of the critical exponent. In the mean
time, since most of the book is devoted to the classical phase
transitions, this book deals exclusively with bosonic field
theories. The author may like to consider to add more discussions
about critical properties of fermonic systems or even the
fermion-gauge coupled systems in the chapter about quantum phase
transition (in the 2nd edition?) of this book. Of course, nobody
promise you a rose garden. As a whole, this book serves as a
useful reference to bridge the gap between the usual graduate
course and the research literature. After reading this book, the
student can go on studying the more specialized monograph, such as
Sachdev's book, and more importantly, starting doing their own
research works. I highly recommend this book to anyone interested
in the field theoretical approach to condensed matter physics.
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