The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics

699 Pages·1998·14.118 MB·\699

by  A.N. Vasil`ev (Author)

The Field Theoretic
Renormalization Group in
Critical Behavior Theory
and Stochastic Dynamics

The Field Theoretic
Renormalization Group in
Critical Behavior Theory
and Stochastic Dynamics
A. N. Vasil’ev
St. Petersburg State University
Russia
Translated by
Patricia A. de Forcrand-Millard
CHAPMAN & HALL/CRC
A CRC Press Company
Boca Raton London New York Washington, D.C.
Originally published in Russian in 1998 as
Квантовополевая ренормгруппа в теории критического поведения
и стохастической динамике
by St. Petersburg Institute of Nuclear Physics Press, St. Petersburg
© 1998 A.N. Vasil’ev
Library of Congress Cataloging-in-Publication Data
Vasil’ev, A.N. (Aleksandr Nikolaevich)
The field theoretic renormalization group in critical behavior theory and stochastic
dynamics / by A.N. Vasil’ev.
p. cm.
Includes bibliographical references and index.
ISBN 0-415-31002-4 (alk. paper)
1. Renormalization group. 2. Critical phenomena (Physics). 3. Stochastic processes. 4.
Statistical physics. I. Title.
QC20.7.R43V37 2004
530.13'3—dc22 2004043573
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Contents
PREFACE xiii
CHAPTER 1 Foundations of the Theory of Critical
Phenomena 1
1.1 Historical review . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Generalized homogeneity. . . . . . . . . . . . . . . . . . . . . . 13
1.3 The scaling hypothesis (critical scaling) in thermodynamics . . 15
1.4 The Ising model and thermodynamics of a ferromagnet. . . . . 17
1.5 The scaling hypothesis for the uniaxial ferromagnet . . . . . . . 19
1.6 The On-symmetric classical Heisenberg ferromagnet . . . . . . 22
1.7 The classical nonideal gas: the model and thermodynamics . . 23
1.8 The thermodynamical scaling hypothesis for the critical point
of the liquid–gas transition . . . . . . . . . . . . . . . . . . . . 27
1.9 The scaling hypothesis for the correlation functions . . . . . . . 31
1.10 The functional formulation . . . . . . . . . . . . . . . . . . . . 35
1.11 Exact variational principle for the mean field . . . . . . . . . . 37
1.12 The Landau theory . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.13 The fluctuation theory of critical behavior . . . . . . . . . . . . 41
1.14 Examples of specific models . . . . . . . . . . . . . . . . . . . . 44
1.15 Canonical dimensions and canonical scale invariance . . . . . . 47
1.16 Relevant and irrelevant interactions. The logarithmic
dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.17 An example of a two-scale model: the uniaxial ferroelectric . . 52
1.18 Ultraviolet multiplicative renormalization . . . . . . . . . . . . 54
1.19 Dimensional regularization. Relation between the exact and
formal expressions for one-loop integrals . . . . . . . . . . . . . 58
1.20 The renormalization problem in dimensional regularization . . 62
1.21 Explicit renormalization formulas . . . . . . . . . . . . . . . . . 66
1.22 The constants Z in the minimal subtraction scheme . . . . . . 68
1.23 The relation between the IR and UV problems . . . . . . . . . 69
1.24 The differential RG equations . . . . . . . . . . . . . . . . . . . 70
1.25 The RG functions in terms of the renormalization constants . . 72
1.26 Relations between the residues of poles in Z of various order
in ε. Representation of Z in terms of RG functions . . . . . . . 74
1.27 Relation between the renormalized and bare charges . . . . . . 75
1.28 Renormalization and RG equations for T <Tc . . . . . . . . . 77
1.29 Solution of the linear partial differential equations . . . . . . . 78
v
vi CONTENTS
1.30 The RG equation for the correlator of the ϕ4 model in zero
field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
1.31 Fixed points and their classification . . . . . . . . . . . . . . . 82
1.32 Invariant charge of the RG equation for the correlator . . . . . 84
1.33 Criticalscaling,anomalouscriticaldimensions,scalingfunction
of the correlator . . . . . . . . . . . . . . . . . . . . . . . . . . 86
1.34 Conditions for reaching the critical regime. Corrections to
scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
1.35 What is summed in the solution of the RG equation? . . . . . . 90
1.36 Algorithm for calculating the coefficients of ε expansions of
critical exponents and scaling functions . . . . . . . . . . . . . 92
1.37 Results of calculating the ε expansions of the exponents of the
On ϕ4 model in dimension d=4−2ε . . . . . . . . . . . . . . 93
1.38 Summation of the ε expansions. Results . . . . . . . . . . . . . 96
1.39 The RG equation for Γ(α) (the equation of state) . . . . . . . . 100
1.40 Subtraction-scheme independence of the critical exponents
and normalized scaling functions . . . . . . . . . . . . . . . . . 101
1.41 The renormalization group in real dimension . . . . . . . . . . 104
1.42 Multicharge theories . . . . . . . . . . . . . . . . . . . . . . . . 107
1.43 Logarithmic corrections for ε=0 . . . . . . . . . . . . . . . . . 109
1.44 Summation of the glns contributions at ε=0 using the RG
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
CHAPTER 2 Functional and Diagrammatic Technique
of Quantum Field Theory 115
2.1 Basic formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
2.2 The universal graph technique. . . . . . . . . . . . . . . . . . . 119
2.3 Graph representations of Green functions . . . . . . . . . . . . 124
2.4 Graph technique for spontaneous symmetry breaking (τ <0) . 127
2.5 One-irreducible Green functions . . . . . . . . . . . . . . . . . . 129
2.6 Graph representations of Γ(α) and the functions Γn . . . . . . 131
2.7 Passage to momentum space. . . . . . . . . . . . . . . . . . . . 134
2.8 The saddle-point method. Loop expansion of W(A) . . . . . . 137
2.9 Loop expansion of Γ(α) . . . . . . . . . . . . . . . . . . . . . . 139
2.10 Loop calculation of Γ(α) in the On ϕ4 model . . . . . . . . . . 141
2.11 The Schwinger equations . . . . . . . . . . . . . . . . . . . . . . 145
2.12 Solutions of the equations of motion . . . . . . . . . . . . . . . 147
2.13 Green functions with insertion of composite operators . . . . . 149
2.14 Summary of definitions of various Green functions . . . . . . . 152
2.15 Symmetries, currents, and the energy–momentum tensor . . . . 153
2.16 Ward identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
2.17 The relation between scale and conformal invariance . . . . . . 164
2.18 Conformal structures for dressed propagators and triple
vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
2.19 The large-n expansion in the On ϕ4 model for T ≥Tc . . . . . 168
CONTENTS vii
2.20 A simple method of constructing the large-n expansion . . . . . 174
2.21 The large-n expansion of the functionals W and Γ for
A∼α∼n1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
2.22 The solution for arbitrary A, T in leading order in 1/n . . . . . 178
2.23 The A→0 asymptote. Singularity of the longitudinal
susceptibility for T <Tc . . . . . . . . . . . . . . . . . . . . . . 181
2.24 Critical behavior in leading order in 1/n . . . . . . . . . . . . . 182
2.25 A simplified field model for calculating the large-n
expansions of critical exponents . . . . . . . . . . . . . . . . . . 184
2.26 The classical Heisenberg magnet and the nonlinear σ model . . 187
2.27 The large-n expansion in the nonlinear σ model . . . . . . . . . 189
2.28 Generalizations: the CPn−1 and matrix σ models. . . . . . . . 191
2.29 The large-n expansion for (ϕ2)3-type interactions . . . . . . . . 192
2.30 Systems with random admixtures . . . . . . . . . . . . . . . . . 193
2.31 The replica method for a system with frozen admixtures . . . . 196
CHAPTER 3 Ultraviolet Renormalization 199
3.1 Preliminary remarks . . . . . . . . . . . . . . . . . . . . . . . . 199
3.2 Superficially divergent graphs. Classification of theories
according to their renormalizability . . . . . . . . . . . . . . . . 201
3.3 Primitive and superficial divergences . . . . . . . . . . . . . . . 202
3.4 Renormalization of the parameters τ and g in the one-loop
approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
3.5 Various subtraction schemes. The physical meaning of the
parameter τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
3.6 The two-loop approximation . . . . . . . . . . . . . . . . . . . . 208
3.7 The basic action and counterterms . . . . . . . . . . . . . . . . 210
3.8 The operators L, R, and R(cid:1) . . . . . . . . . . . . . . . . . . . . 212
3.9 The Bogolyubov–Parasyuk R operation . . . . . . . . . . . . . 215
3.10 Recursive construction of L in terms of the subtraction
operator K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
3.11 The commutativity of L, R(cid:1), and R with ∂τ-type operators . . 219
3.12 The basic statements of renormalization theory . . . . . . . . . 220
3.13 Remarks about the basic statements . . . . . . . . . . . . . . . 222
3.14 Proof of the basic combinatorial formula for the R operation . 225
3.15 Graph calculations in arbitrary dimension . . . . . . . . . . . . 232
3.16 Dimensional regularization and minimal subtractions . . . . . . 236
3.17 Normalized functions . . . . . . . . . . . . . . . . . . . . . . . . 238
3.18 The renormalization constants in terms of counterterms in
the MS scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
3.19 The passage to massless graphs . . . . . . . . . . . . . . . . . . 243
3.20 The constants Z in three-loop order in the MS scheme for
the On ϕ4 model . . . . . . . . . . . . . . . . . . . . . . . . . . 247
3.21 Technique for calculating the ξγ . . . . . . . . . . . . . . . . . . 250
viii CONTENTS
3.22 Nonmultiplicativity of the renormalization in analytic
regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
3.23 The inclusion of composite operators . . . . . . . . . . . . . . . 262
3.24 The renormalized composite operator. . . . . . . . . . . . . . . 264
3.25 Renormalization of the action and Green functions of the
extended model . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
3.26 Structure of the operator counterterms . . . . . . . . . . . . . . 266
3.27 An example of calculating operator counterterms . . . . . . . . 269
3.28 Matrix multiplicative renormalization of families of operators . 273
3.29 UV finiteness of operators associated with the renormalized
action and conserved currents . . . . . . . . . . . . . . . . . . . 275
3.30 The On ϕ4 model: renormalization of scalar operators with
d∗ =2,3,4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
F
3.31 Renormalization of conserved currents . . . . . . . . . . . . . . 280
3.32 Renormalization of tensor operators with d∗F =4 in the On ϕ4
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
3.33 The Wilson operator expansion for short distances . . . . . . . 283
3.34 Calculation of the Wilson coefficients in the one-loop
approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
3.35 Expandability of multiloop counterterms Lξγ in p and τ . . . . 291
3.36 Renormalization in the case of spontaneous symmetry
breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
CHAPTER 4 Critical Statics 299
4.1 General scheme for the RG analysis of an arbitrary model . . . 299
4.2 The On ϕ4 model: the constants Z, RG functions, and
4−ε expansion of the exponents . . . . . . . . . . . . . . . . . 301
4.3 Renormalization and the RG equations for the renormalized
functional WR(A) including vacuum loops . . . . . . . . . . . . 305
4.4 The On ϕ4 model: renormalization and the RG equation for
the free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
4.5 General solution of the inhomogeneous RG equation for the
freeenergyoftheϕ4 modelandtheamplituderatioA+/A− in
the specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . 309
4.6 RG equations for composite operators and coefficients of the
Wilson operator expansion. . . . . . . . . . . . . . . . . . . . . 312
4.7 Critical dimensions of composite operators . . . . . . . . . . . . 314
4.8 Correction exponents ω associated with IR-irrelevant
composite operators . . . . . . . . . . . . . . . . . . . . . . . . 319
4.9 Example: the system F ={1,ϕ2} in the On ϕ4 model . . . . . 319
4.10 Second example: scalar operators with d∗ =4. . . . . . . . . . 321
F
4.11 Determinationofthecriticaldimensionsofcompositeoperators
following Sec. 3.29 . . . . . . . . . . . . . . . . . . . . . . . . . 324
4.12 The On ϕ4 model: calculation of the 1- and 2-loop graphs of
the renormalized correlator in the symmetric phase . . . . . . . 325