As an introductory account of the theory of phase transitions and critical phenomena, this book reflects lectures given by the authors to graduate students at their departments and is thus classroom-tested to help beginners enter the field. Most parts are written as self-contained units and
every new concept or calculation is explained in detail without assuming prior knowledge of the subject. The book significantly enhances and revises a Japanese version which is a bestseller in the Japenese market and is considered a standard textbook in the field. It contains new pedagogical
presentations of field theory methods, including a chapter on conformal field theory, and various modern developments hard to find in a single textbook on phase transitions. Exercises are presented as the topics develop, with solutions found at the end of the book, making the usefil for
self-teaching, as well as for classroom learning.
Author(s): Hidetoshi Nishimori, Gerardo Ortiz
Edition: 1
Publisher: Oxford University Press
Year: 2011
Language: English
Commentary: True PDF - Fully Bookmarked
Pages: 384
Tags: Phase transitions; Statistical physics
Contents
1 Phase transitions and critical phenomena
1.1 Phase and phase diagram
1.2 Phase transitions
1.3 Critical phenomena
1.4 Scale transformation and renormalization group
1.5 Ising model and related systems
2 Mean-field theories
2.1 Mean-field approximation
2.2 Critical exponents of the mean-field theory
2.3 Landau theory
2.4 Landau theory of the tricritical point
2.5 Infinite-range model
2.6 Variational method
2.7 Antiferromagnetic Ising model
2.8 Bethe approximation
2.9 Correlation function
2.10 Limit of applicability of the mean-field approximation
2.11 Dynamic critical phenomena
3 Renormalization group and scaling
3.1 Coarse-graining and scale transformations
3.2 Parameter space and renormalization group equation
3.3 Renormalization group flow near a fixed point and universality
3.4 Scaling law and critical exponents
3.5 Scaling law for correlation functions and hyperscaling
3.6 A simple example: One-dimensional Ising model
3.7 Mean-field theory and scaling law
3.8 Scaling dimension and scaling law
3.9 Scaling and anomalous dimensions
3.10 Data analysis by scaling law and finite-size scaling
3.11 Crossover phenomena
3.12 Dynamic scaling law
4 Implementation of the renormalization group
4.1 Real-space renormalization group for arbitrary dimensions
4.2 Momentum-space renormalization group: ε = 4 – d expansion
4.3 Real-space renormalization group for a quantum system
5 Statistical field theory
5.1 From bits to fields
5.2 Continuum limit and field theory
5.3 Hubbard–Stratonovich transformation
5.4 Integrating out degrees of freedom: Coarse graining
5.5 Phenomenological Landau–Ginzburg approach
5.6 Symmetry and its breakdown
5.7 Nambu–Goldstone modes
5.8 Topological defects
6 Conformal field theory
6.1 From scale invariance to conformal symmetry
6.2 Conformal transformation
6.3 Primary and quasi-primary operators
6.4 Energy–momentum tensor and the Ward identity
6.5 Virasoro algebra
6.6 Gaussian theory
6.7 Operator formalism
6.8 Unitary representation of the Virasoro algebra
6.9 Ising model
6.10 Finite-size effects
7 Kosterlitz–Thouless transition
7.1 Peierls argument
7.2 Lower critical dimension of the XY model
7.3 Mermin–Wagner theorem: Absence of spontaneous magnetization
7.4 Kosterlitz–Thouless transition
7.5 Interaction energy of vortex pairs
7.6 Renormalization group analysis
7.7 Lattice gauge theory and Elitzur’s theorem
8 Random systems
8.1 Random fields
8.2 Spin glass
8.3 Diluted ferromagnet and percolation
9 Exact solutions and related topics
9.1 One-dimensional Ising model
9.2 One-dimensional n-vector model
9.3 Spherical model
9.4 One-dimensional quantum XY model
9.5 Two-dimensional Ising model
9.6 Zeros of the partition function
10 Duality
10.1 Classical duality
10.2 High- and low-temperature series expansions
10.3 Duality by Fourier transformation
10.4 Quantum duality
11 Numerical methods
11.1 Master equation
11.2 Monte Carlo simulation
11.3 Numerical transfer matrix method
For further reading
Appendix A
A.1 Saddle-point method
A.2 Expressing the susceptibility in terms of correlation functions
A.3 Rushbrooke’s inequality
A.4 Cumulants
A.5 Renormalization group equations from the ε expansion
A.6 Symmetry and Noether’s theorem
A.7 Basics of group theory and Lie algebras
A.8 Basics of homotopy theory
A.9 Restrictions on the type of conformal mappings
A.10 Properties of the energy–momentum tensor
A.11 Energy–momentum tensor of the Gaussian theory
A.12 Existence of spontaneous magnetization in the two-dimensional Ising model
A.13 Quantum version of the Mermin–Wagner theorem
A.14 Replica symmetric solution of the SK model
A.15 Integral for the partition function of the n-vector model
A.16 Multiple Gaussian integral and lattice Green function
A.17 Jordan–Wigner transformation
A.18 Proof of Theorem 9.1
A.19 Poisson summation formula
A.20 Sample codes for Monte Carlo simulation of the Ising model
Appendix B: Solutions to exercises
Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
Z
Y
