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Introduction to Statistical Field Theory - Edouard Brezin

Introduction to Statistical Field Theory

Hardcover Published: 22nd July 2010
ISBN: 9780521193030
Number Of Pages: 176

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Knowledge of the renormalization group and field theory is a key part of physics, and is essential in condensed matter and particle physics. Written for advanced undergraduate and beginning graduate students, this textbook provides a concise introduction to this subject.

The textbook deals directly with the loop expansion of the free energy, also known as the background field method. This is a powerful method, especially when dealing with symmetries, and statistical mechanics. In focussing on free energy, the author avoids long developments on field theory techniques. The necessity of renormalization then follows.

'... a concise but clear introduction to this now-classic material. ... a magisterial introduction to the subject ... an excellent example of what such lectures should be: fast-paced, succinct and clear ... an excellent concise treatment of the modern theory of phase transitions and the renormalisation group by an expert in the field.' Contemporary Physics '... the author deals directly with the loop expansion of the free energy, also known as the background field method. This is a powerful method, especially when dealing with symmetries, and statistical mechanics.' Nenad Manojlovic, Zentralblatt MATH

Prefacep. ix
A few well-known basic resultsp. 1
The Boltzmann lawp. 1
The classical canonical ensemblep. 1
The quantum canonical ensemblep. 2
The grand canonical ensemblep. 3
Thermodynamics from statistical physicsp. 3
The thermodynamic limitp. 3
Gaussian integrals and Wick's theoremp. 4
Functional derivativesp. 6
d-dimensional integralsp. 6
Additional referencesp. 8
Introduction: order parameters, broken symmetriesp. 9
Can statistical mechanics be used to describe phase transitions?p. 9
The order-disorder competitionp. 10
Order parameter, symmetry and broken symmetryp. 12
More general symmetriesp. 16
Characterization of a phase transition through correlationsp. 18
Phase coexistence, critical points, critical exponentsp. 19
Examples of physical situations modelled by the Ising modelp. 22
Heisenberg's exchange forcesp. 22
Heisenberg and Ising Hamiltoniansp. 24
Lattice gasp. 26
More examplesp. 28
A first connection with field theoryp. 29
A few results for the Ising modelp. 32
One-dimensional Ising model: transfer matrixp. 32
One-dimensional Ising model: correlation functionsp. 35
Absence of phase transition in one dimensionp. 37
A glance at the two-dimensional Ising modelp. 38
Proof of broken symmetry in two dimensions (and more)p. 38
Correlation inequalitiesp. 42
Lower critical dimension: heuristic approachp. 44
Digression: Feynman path integrals, the transfer matrix and the Schrödinger equationp. 47
High-temperature and low-temperature expansionsp. 52
High-temperature expansion for the Ising modelp. 52
Continuous symmetryp. 55
Low-temperature expansionp. 56
Kramers-Wannier dualityp. 57
Low-temperature expansion for a continuous symmetry groupp. 58
Some geometric problems related to phase transitionsp. 60
Polymers and self-avoiding walksp. 60
Potts model and percolationp. 64
Phenomenological description of critical behaviourp. 68
Landau theoryp. 68
Landau theory near the critical point: homogeneous casep. 71
Landau theory and spatial correlationsp. 75
Transitions without symmetry breaking: the liquid-gas transitionp. 78
Thermodynamic meaning of (m)p. 79
Universalityp. 80
Scaling lawsp. 82
Mean field theoryp. 85
Weiss 'molecular field'p. 85
Mean field theory: the variational methodp. 87
A simpler alternative approachp. 92
Beyond the mean field theoryp. 95
The first correction to the mean-field free energyp. 95
Physical consequencesp. 97
Introduction to the renormalization groupp. 100
Renormalized theories and critical pointsp. 101
Kadanoff block spinsp. 101
Examples of real space renormalization groups: 'decimation'p. 103
Structure of the renormalization group equationsp. 109
Renormalization group for the 4 theoryp. 113
Renormalization group … without renormalizationp. 114
Study of the renormalization group flow in dimension fourp. 116
Critical behaviour of the susceptibility in dimension fourp. 118
Multi-component order parametersp. 120
Epsilon expansionp. 122
An exercise on the renormalization group: the cubic fixed pointp. 125
Renormalized theoryp. 128
The meaning of renormalizabilityp. 128
Renormalization of the massless theoryp. 132
The renormalized critical free energy (at one-loop order)p. 134
Away from Tcp. 136
Goldstone modesp. 138
Broken symmetries and massless modesp. 138
Linear and non-linear O(n) sigma modelsp. 142
Regularization and renormalization of the O(n) non-linear sigma model in two dimensionsp. 144
Regularizationp. 144
Perturbation expansion and renormalizationp. 148
Renormalization group equations for the O(n) non-linear sigma model and the (d - 2) expansionp. 150
Integration of RG equations and scalingp. 151
Extensions to other non-linear sigma modelsp. 153
Large np. 156
The linear O(n) modelp. 156
O(n) sigma modelp. 161
Indexp. 165
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780521193030
ISBN-10: 0521193036
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 176
Published: 22nd July 2010
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 24.7 x 17.4  x 1.3
Weight (kg): 0.5
Edition Number: 1

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