Introduction to the theory of critical phenomena: mean field, fluctuations and renormalization
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| Language: | English |
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World Scientific
2010
|
| Edition: | 2. ed. |
| Subjects: | |
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| Physical Description: | XXV, 673 S. |
| ISBN: | 9789814299497 9814299499 9789814299480 9814299480 |
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| 100 | 1 | |a Uzunov, Dimo I. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Introduction to the theory of critical phenomena |b mean field, fluctuations and renormalization |c Dimo I. Uzunov |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New Jersey [u.a.] |b World Scientific |c 2010 | |
| 300 | |a XXV, 673 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Critical phenomena (Physics) | |
| 650 | 4 | |a Phase transformations (Statistical physics) | |
| 650 | 0 | 7 | |a Kritisches Phänomen |0 (DE-588)4165788-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Phasenumwandlung |0 (DE-588)4132140-6 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Kritisches Phänomen |0 (DE-588)4165788-3 |D s |
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| 689 | 1 | 0 | |a Phasenumwandlung |0 (DE-588)4132140-6 |D s |
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| adam_text | Contents
Foreword
vii
Preface to the Second Edition
ix
Preface to the First Edition
xi
1.
Introduction
1
1.1
Thermodynamic Description of Phase Transitions
..... 1
1.2
Statistical Description of Phase Transitions
........ 3
1.3
Scaling and Critical Exponents
............... 8
1.4
Dynamics
........................... 12
1.5
Experiment
.......................... 14
1.6
Scaling and Homogeneity. Scaling Relations
....... 17
1.7
Length-Scale
Invariance
and Renormalization
....... 20
2.
Equilibrium States and Phases
25
2.1
Phases
............................. 25
2.2
Fundamental Relations
.................... 26
2.2.1
Fundamental equations
............... 27
2.2.2
Properties of entropy and energy
.......... 28
2.2.3
Notations
....................... 30
2.2.4
Macroscopic scales
.................. 31
2.3
Entropy Increase
....................... 33
2.4
Conditions of Equilibrium and Stability
.......... 36
2.4.1
Equilibrium conditions
............... 36
2.4.2
Stability conditions
................. 39
2.5
Classification of Equilibria
.................. 45
xvi
Introduction to the Theory of Critical Phenomena
2.6
Thermodynamic Potentials
................. 48
2.6.1
Energy minimum
.................. 48
2.6.2
Legendre transformations and
extrémům
conditions
...................... 50
2.6.3
Legendre transformations in entropy
representation
.................... 53
2.6.4
Application
...................... 55
2.7
Coexistence of Phases
.................... 57
2.7.1
Equilibrium conditions
............... 58
2.7.2
Gibbs phase rule
................... 60
2.7.3
Stability conditions
................. 61
2.7.4
Limitation of the thermodynamic theory of
stability
........................ 64
3.
Thermodynamic Theory of Phase Transitions
67
3.1
Order Parameter
....................... 67
3.2
Coexistence of Two Phases
................. 69
3.2.1
Phase diagram
.................... 70
3.2.2
Equilibrium isotherms
................ 71
3.2.3
Relations on the coexistence line
.......... 73
3.3
Metastability
......................... 75
3.4
Critical Point
......................... 78
3.5
Classifications of Phase Transitions
............. 82
3.6
Compressible Systems
.................... 85
3.6.1
Gas-liquid-solid diagram
.............. 86
3.6.2
The solid phase
................... 87
3.6.3
The liquid state
................... 91
3.7
Phase Separation in Mixtures
................ 94
3.7.1
Binary mixtures
................... 95
3.7.2
Stability conditions and critical states in
binary mixtures
................... 97
3.7.3
Coexistence surfaces, critical lines, and second
Gibbs rule
...................... 99
3.7.4
Tricritical and critical-end points.
Multicritical phenomena
.............. 100
3.8
Other Systems
........................ 103
3.8.1
General notations
.................. 103
3.8.2
Ferromagnets
.................... 106
3.8.3
Anisotropy in ferromagnets
............. 110
Contents xvii
3.8.4
Complex
magnetic
order
..............
Ill
3.8.5
Structural and ferroelectric phase transitions
... 115
3.8.6
Superfluidity and superconductivity
........ 118
3.8.7
Liquid crystals
.................... 121
3.9
Advantages and Disadvantages of the Thermodynamic
Theory
............................. 122
4.
Landau Expansion
125
4.1
Introductory Remarks
.................... 125
4.2
Appropriate Variables
.................... 126
4.3
Generalized Gibbs Potential
................. 129
4.3.1
Nonequilibrium potential
Fť
............ 129
4.3.2
Nonequilibrium potential
Φ
............. 130
4.3.3
Large reservoir and relation between
Fł
and
Φ
. . 134
4.4
Landau Potential
....................... 135
4.4.1
Taylor expansion
.................. 135
4.4.2
Equilibrium
..................... 136
4.4.3
Stability
....................... 137
4.4.4
Aims of the Landau theory
............. 138
4.4.5
Examples
....................... 139
4.5
Phase Transition Line
.................... 140
4.6
Second Order Phase Transitions
.............. 143
4.7
First Order Transitions and the Isolated Critical Point
. . 148
4.7.1
Phase diagram of the first order transition
.... 148
4.7.2
Metastability. Specific heat
............. 151
4.7.3
The isolated critical point
.............. 153
4.8
Extended Expansion:
ψ6-
and (/^-theories
......... 154
4.8.1
First order transition (u
< 0)............ 155
4.8.2
The tricritical point (u
= 0) ............ 160
4.8.3
Irrelevance of the
<¿>6-term
for
и
> 0........ 162
4.8.4
<¿>8-theory.......................
162
4.9
Multicomponent Order Parameter and Symmetry
Breaking
............................ 163
4.9.1
Thermodynamic functions
............. 164
4.9.2
Symmetry breaking and symmetry conserving
phase transitions. Landau criteria for second
order phase transitions
............... 164
4.10
Effect of External Field and Susceptibility
......... 167
4.10.1
One-component order parameter
.......... 167
xviii
Introduction
to the Theory of Critical Phenomena
4.10.2
Two-component order parameter
.......... 172
4.11
Miscellaneous Topics
..................... 173
4.11.1
Summary of critical exponents within the Landau
theory
........................ 173
4.11.2
Notion of crossover phenomena.
Critical-to-tricritical crossover
........... 174
4.11.3
Anisotropy
...................... 179
4.11.4
Coupling to strains
................. 180
4.11.5
Validity and further development of
the Landau theory
.................. 181
5.
Statistical Mechanics and Mean-Field Approximation
185
5.1
Partition Function
...................... 185
5.1.1
Microscopic variables
................ 186
5.1.2
Canonical distribution
................ 187
5.1.3
Thermodynamic relations
.............. 189
5.2
Fixed External Field and Order Parameter
Fluctuations
.......................... 189
5.2.1
Continuous variable
φ
................ 190
5.2.2
Discrete variable
φ
................. 193
5.3
Inhomogeneous Systems and Translational
Invariance
. . 194
5.4
Coarse-Graining and Effective Hamiltonian
........ 196
5.4.1
Coarse-graining
................... 196
5.4.2
Transformation of the canonical partition function
198
5.4.3
Quasimacroscopic separability
........... 200
5.4.4
Separability of the Hamiltonian
.......... 202
5.4.5
Effective Hamiltonian
................ 204
5.4.6
Remarks
....................... 205
5.5
Generating
Funcţionale
and Correlation Functions
.... 206
5.6
Ising Model
.......................... 214
5.6.1
Definition of the Ising model
............ 214
5.6.2
General exchange, general values of spin and
angular momentum
................. 217
5.6.3
Brief notes
...................... 220
5.7
Other Lattice Models
.................... 222
5.7.1 Heisenberg
and XY model
............. 222
5.7.2
Lattice gas model
.................. 224
5.7.3
n-component vector models
............. 226
5.7.4
The s-state Potts model
.............. 226
Contents xix
5.7.5
The spherical model
................. 226
5.7.6
Biquadratic exchange and single spin energy
. . . 227
5.7.7
About the exact solutions of lattice models
.... 228
5.8
Mean-Field Approximation
................. 228
5.8.1
Mean field
...................... 229
5.8.2
Thermodynamic potential in mean-field
approximation and symmetry breaking
...... 231
5.8.3
Self-consistency of the mean-field approximation
. 237
5.8.4
Free spins and Curie law
.............. 239
5.8.5
Uniform systems and equation of state
...... 240
5.8.6
Expansion of the equation of state and the critical
point
......................... 245
5.8.7
Helmholtz free energy and generalized potential
. 248
5.9
Real Space Condensation and Statistical Mechanics
.... 249
5.9.1
On the derivation of the Van
der Waals
equation
. 249
5.9.2
Corresponding states and universality
....... 252
5.9.3
Critical properties of Van
der Waals
fluids
.... 253
6.
Fluctuations and Fields
257
6.1
Description of Macroscopic Fluctuations
.......... 257
6.2
Spatially Independent Fluctuations and Anisotropy
.... 260
6.2.1
Gaussian fluctuations
................ 262
6.2.2
Gaussian-like and non-Gaussian fluctuations
. . . 262
6.3
Partition Function of Spatially Independent
Fluctuations
.......................... 264
6.3.1
Treatment of Gaussian-like fluctuations
...... 265
6.3.2
Beyond the Gaussian approximation.
Weak coupling
.................... 267
6.3.3
Strong coupling and closed-form expression of
the partition integral
................ 272
6.4
Fluctuation Field
....................... 275
6.5
From Lattice to Field Models and Vice Versa
....... 281
6.5.1
Continuum limit
................... 281
6.5.2
The origin of fields in the generalized
thermodynamics
................... 284
6.5.3
Discretization of field models
............ 285
6.5.4
Continuum limit in the momentum space
..... 286
6.5.5
Functional integral
................. 289
6.5.6
Truncated series
................... 290
xx
Introduction
to the Theory of Critical Phenomena
6.6
Integral Transformation of Microscopic Models
...... 291
6.6.1
Integral transformation of the Ising model
.... 293
6.6.2
Continuum limit for the Ising model and relation
to the mean-field theory
.............. 296
6.6.3
Transformation of other models
.......... 303
6.7
Short-Range and Long-Range Translationally Invariant
Interactions
.......................... 308
7.
Perturbation Theory of Fluctuation Fields
313
7.1
Basic Fluctuation Hamiltonian
............... 313
7.2
Free Field
........................... 317
7.2.1
Equation of state
.................. 318
7.2.2
Green function
.................... 320
7.3
Thermodynamic Properties of Free Fields
......... 324
7.4
Correlation Function of the Free Fields
........... 332
7.5
Gaussian Approximation
................... 340
7.5.1 Goldstone
modes
.................. 342
7.5.2
Ginzburg-Levanyuk criterion and critical region
. 343
7.6
Perturbation Expansion
................... 346
7.6.1
Perturbation expansion for the partition function
and the thermodynamic potential
......... 346
7.6.2
Perturbation expansion for the averages
...... 353
7.7
i/54-theory. Coordinate Representation
........... 354
7.7.1
First order perturbation contribution to
the thermodynamic potential
............ 355
7.7.2
Perturbation contribution to
the correlation functions
.............. 357
7.7.3
Remarks and calculation of the combinatorial
factors
........................ 363
7.8
¡¿^-theory. Momentum Representation
........... 368
7.8.1
First-order perturbation terms
........... 368
7.8.2
Second-order perturbation terms
.......... 371
7.9
Dyson Equation and Self-energy Function
......... 374
7.9.1
Dyson equation
................... 375
7.9.2
Expansion of the self-energy function to order
и$
376
7.9.3
Remarks on the reducibility of the diagrams
. . . 379
7.10
Notion of Renormalization
.................. 381
7.10.1
Mass renormalization from the perturbation series
383
7.10.2
Mass-renormalization counter-term
........ 385
Contents xxi
7.11
Properties of the
Standard
Perturbation
Series
...... 388
7.11.1
How to determine the critical region from
the perturbation expansion
............. 388
7.11.2
Effective expansion parameter
........... 391
7.11.3
Dependence on the symmetry index
η
and
the spatial dimension d. Universality classes
. . . 393
7.11.4
Borderline dimensions
................ 394
7.11.5
Breakdown of the standard perturbation expansion
396
Renormalization Group
399
8.1
Modified Perturbation Scheme
............... 399
8.2
Perturbation Expansion for the Effective Hamiltonian
. . 404
8.2.1
First order calculation
................ 405
8.2.2
Vertex functions
................... 407
8.2.3
Second order calculation
.............. 408
8.3
Loop Expansion
....................... 411
8.3.1
One-loop approximation
.............. 412
8.3.2
Two-loop approximation
.............. 413
8.3.3
Large-ò
limit and one-component field
....... 415
8.3.4
Remarks on the renormalization
.......... 417
8.4
Large
-η
Limit and
Hartree
Approximation
......... 418
8.4.1
Hartree
diagrams
.................. 418
8.4.2
Large-n critical behaviour
............. 420
8.4.3
Hartree
approximation
............... 423
8.5
Renormalization-group Recursion Relations
........ 424
8.6
Renormalization Group in the One-loop
Approximation
........................ 431
8.6.1
General features of the renormalization group
equations
...................... 431
8.6.2
Renormalization group at the critical point
.... 433
8.6.3
Critical Hamiltonian and fixed points
....... 437
8.7
Linearized Renormalization Group in the One-loop
Approximation
........................ 440
8.7.1
Recursion relations and fixed points
........ 441
8.7.2
Linearization and exponents
............ 443
8.7.3
Renormalization group flows and classification of
the scaling fields
................... 447
8.7.4
Remarks on the effect of the high-order interaction
terms
......................... 449
xxii
Introduction to the Theory of Critical Phenomena
8.7.5
Dimensional analysis
................ 451
8.7.6
Differential renormalization group relations
.... 452
8.8
Renormalization Group and Thermodynamics
....... 454
8.8.1
Scaling form of the thermodynamic potential and
classification of the scaling fields
.......... 456
8.8.2
Scaling relations
................... 461
8.8.3
Remarks on the crossover phenomena in
the critical region
.................. 464
8.8.4
Scaling and e-expansion
............... 466
8.9
Renormalization Group in the Two-loop
Approximation
........................ 467
8.9.1
Recursion relations to order e2
.......... 468
8.9.2
Fixed points
..................... 469
8.9.3
Exponents and stability properties
......... 473
8.10
Other Theoretical Schemes and Methods of
Calculation
.......................... 477
8.10.1
Direct calculation of the critical exponents
.... 477
8.10.2
Renormalization group in three dimensions
.... 478
8.10.3
(l/n)-expansion
................... 480
8.10.4
Renormalization group in the
х
-space and
references to other topics
.............. 481
8.11
Interrelations with other theories
.............. 481
9.
Some Applications
485
9.1
Preliminary Remarks
..................... 485
9.2
Ideal
Bose
Gas
........................ 486
9.2.1
Thermodynamics
.................. 487
9.2.2
Bose-Einstein condensation at finite temperatures
490
9.2.3
Zero temperature condensation
........... 491
9.3
Nonideał Bose
Gas...................... 491
9.3.1
Notion of universality of the critical behaviour at
finite critical temperatures
............. 491
9.3.2
Functional formulation
............... 492
9.3.3
Renormalization group results
........... 495
9.3.4
Notes
......................... 500
9.4
Quantum Critical Phenomena and
Classical-to-Quantum Crossover
............... 501
9.5
Disorder Effects
........................ 504
9.5.1
Random critical temperature
............ 505
Contents xxiii
9.5.2
Extended impurities and long-range random
correlations
...................... 508
9.5.3
Random field
..................... 510
9.5.4
Technical remarks
.................. 512
9.5.5
Ginzburg criterion
.................. 512
9.6
Anisotropy, Coupled Order Parameters
and Multicritical Behaviour
................. 513
9.6.1 Anisotropie
systems
................. 513
9.6.2
Mean-field analysis of anisotropic systems
..... 515
9.6.3
Cubic anisotropy in mean-field approximation
. . 517
9.6.4
Renormalization group studies of
anisotropic systems
................. 520
9.6.5
Tricritical points and renormalization group
. . . 521
9.6.6
Bicritical and tetracritical points
.......... 522
9.6.7
Lifshitz s and tricritical Lifshitz s points
...... 525
9.7
Critical Properties of Superconductors and
Liquid Crystals
........................ 526
9.7.1
Conventional superconductors
........... 526
9.7.2
Unconventional superconductors
.......... 533
9.7.3
Layered superconductors
.............. 536
9.7.4
Fluctuations of the superconducting order
parameter
...................... 537
9.7.5
Magnetic fluctuations in superconductors
..... 538
9.7.6
Weakly-first-order smectic A-nematic
phase transition
................... 541
9.7.7
Renormalization group investigations
....... 542
9.8
Magnetic Fluctuations in Thin Superconducting Films
. . 553
9.8.1
Preliminary notes
.................. 553
9.8.2
General form of the effective free energy
..... 556
9.8.3
Dimensional crossover
................ 559
9.8.4
Free energy for particular spatial dimensions
. . . 562
9.8.5
Limitations of the theory
.............. 565
9.8.6
Bulk superconductors
................ 566
9.8.7
Thermodynamics of quasi-2D films
........ 572
9.9
Phase Transitions in Unconventional Ferromagnetic
Superconductors
....................... 579
9.9.1
Coexistence of unconventional superconductivity
and itinerant ferromagnetism
............ 579
xxiv
Introduction to the Theory of Critical Phenomena
9.9.2
Free energy of ferromagnetic
superconductors with spin-triplet electron pairing
581
9.9.3
Phases
........................ 583
9.9.4
Phase diagram, quantum phase transitions, and
classification of ferromagnetic superconductors
. . 588
9.9.5
Peculiar renormalization group analysis and
unusual universality class of critical behaviour
. . 594
Appendix A Useful Mathematical Formulae
603
A.I Homogeneous Functions and
Euler
Theorem
........ 603
A.
2
Useful Formulae for the Gamma Function
......... 603
A.3 Logarithm of Matrices
.................... 604
Appendix
В
Gaussian Integrals and Transformations
605
B.I Gaussian Integrals
...................... 605
B.2 Multiple Gaussian Integrals
................. 606
Appendix
С
Some Functional
609
C.I Definition
........................... 609
C.2 Functional Power Series
................... 609
C.3 Functional Differentiation
.................. 611
C.4 Functional Integration
.................... 612
C.5 Functional Gaussian Integral
................ 612
Appendix
D D-dimensional
Integration and Some Fourier
Transformations
615
D.I .D-dimensional Angular Integration
............. 615
D.I.I Angular integration
................. 615
D.I.2
Relationship of Cartesian and spherical
coordinates in
đ-dimensional
space
......... 617
D.2 Some Fourier Transformations
................ 620
Appendix
E
Perturbation Integrals and Sums
623
E.I Simple Perturbation Integrals
................ 623
E.2 Treatment of Perturbation Integrals
............ 624
E.3
е
-Expansion of Perturbation Integrals
........... 628
E.4 Summation over Matsubara Frequencies
.......... 632
Contents
Appendix
F
Fourier
Amplitudes
of Lattice Exchange Interactions
635
Bibliography
639
Index
663
P
henomena
This book provides a comprehensive introduction
to the theory of phase transitions and critical
phenomena. The content covers a period of more
than
100
years of theoretical research of condensed
matter phases and phase transitions providing a
clear interrelationship with experimental problems.
It starts from certain basic University knowledge of
thermodynamics, statistical physics and quantum
mechanics. The text is illustrated with classic
examples of phase transitions. Various types of
phase transition and (multOcritical points are
introduced and explained. The classic aspects of
the theory are naturally related with the modern
developments. This interrelationship and the field-
theoretical renormalization group method are
presented in details. The main applications of the
renormalization group methods are presented.
Special attention is paid to the description of
quantum phase transitions. This edition contains a
more detailed presentation of the renormalization
group method and its applications to particular
systems.
Worid Scientific
|
| any_adam_object | 1 |
| author | Uzunov, Dimo I. |
| author_facet | Uzunov, Dimo I. |
| author_role | aut |
| author_sort | Uzunov, Dimo I. |
| author_variant | d i u di diu |
| building | Verbundindex |
| bvnumber | BV036620413 |
| classification_rvk | UG 3800 |
| ctrlnum | (OCoLC)699900673 (DE-599)BVBBV036620413 |
| discipline | Physik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV036620413 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T22:44:21Z |
| institution | BVB |
| isbn | 9789814299497 9814299499 9789814299480 9814299480 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020540444 |
| oclc_num | 699900673 |
| open_access_boolean | |
| owner | DE-703 DE-29T DE-19 DE-BY-UBM DE-20 |
| owner_facet | DE-703 DE-29T DE-19 DE-BY-UBM DE-20 |
| physical | XXV, 673 S. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | World Scientific |
| record_format | marc |
| spelling | Uzunov, Dimo I. Verfasser aut Introduction to the theory of critical phenomena mean field, fluctuations and renormalization Dimo I. Uzunov 2. ed. New Jersey [u.a.] World Scientific 2010 XXV, 673 S. txt rdacontent n rdamedia nc rdacarrier Critical phenomena (Physics) Phase transformations (Statistical physics) Kritisches Phänomen (DE-588)4165788-3 gnd rswk-swf Phasenumwandlung (DE-588)4132140-6 gnd rswk-swf Kritisches Phänomen (DE-588)4165788-3 s DE-604 Phasenumwandlung (DE-588)4132140-6 s Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020540444&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020540444&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Uzunov, Dimo I. Introduction to the theory of critical phenomena mean field, fluctuations and renormalization Critical phenomena (Physics) Phase transformations (Statistical physics) Kritisches Phänomen (DE-588)4165788-3 gnd Phasenumwandlung (DE-588)4132140-6 gnd |
| subject_GND | (DE-588)4165788-3 (DE-588)4132140-6 |
| title | Introduction to the theory of critical phenomena mean field, fluctuations and renormalization |
| title_auth | Introduction to the theory of critical phenomena mean field, fluctuations and renormalization |
| title_exact_search | Introduction to the theory of critical phenomena mean field, fluctuations and renormalization |
| title_full | Introduction to the theory of critical phenomena mean field, fluctuations and renormalization Dimo I. Uzunov |
| title_fullStr | Introduction to the theory of critical phenomena mean field, fluctuations and renormalization Dimo I. Uzunov |
| title_full_unstemmed | Introduction to the theory of critical phenomena mean field, fluctuations and renormalization Dimo I. Uzunov |
| title_short | Introduction to the theory of critical phenomena |
| title_sort | introduction to the theory of critical phenomena mean field fluctuations and renormalization |
| title_sub | mean field, fluctuations and renormalization |
| topic | Critical phenomena (Physics) Phase transformations (Statistical physics) Kritisches Phänomen (DE-588)4165788-3 gnd Phasenumwandlung (DE-588)4132140-6 gnd |
| topic_facet | Critical phenomena (Physics) Phase transformations (Statistical physics) Kritisches Phänomen Phasenumwandlung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020540444&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020540444&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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