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[资源] Symmetry and Condensed Matter Physics. A Computational Approach

Contents
Preface page xi
1 Symmetry and physics 1
1.1 Introduction 1
1.2 Hamiltonians, eigenfunctions, and eigenvalues 5
1.3 Symmetry operators and operator algebra 9
1.4 Point-symmetry operations 14
1.5 Applications to quantum mechanics 17
Exercises 19
2 Symmetry and group theory 21
2.1 Groups and their realizations 21
2.2 The symmetric group 25
2.3 Computational aspects 30
2.4 Classes 34
2.5 Homomorphism, isomorphism, and automorphism 41
2.6 Direct- or outer-product groups 42
Exercises 43
3 Group representations: concepts 51
3.1 Representations and realizations 52
3.2 Generation of representations on a set of basis functions 62
Exercises 67
4 Group representations: formalism and methodology 69
4.1 Matrix representations 69
4.2 Character of a matrix representation 78
4.3 Burnside’s method 85
Exercises 94
Computational projects 95
5 Dixon’s method for computing group characters 96
5.1 The eigenvalue equation modulo p 96
5.2 Dixon’s method for irreducible characters 99
vii
viii Contents
5.3 Computer codes for Dixon’s method 107
Appendix 1 Finding eigenvalues and eigenvectors 116
Exercises 128
Appendix 2 128
Computation project 133
6 Group action and symmetry projection operators 134
6.1 Group action 134
6.2 Symmetry projection operators 138
6.3 The regular projection matrices: the simple characteristic 157
Exercises 162
7 Construction of the irreducible representations 164
7.1 Eigenvectors of the regular Rep 164
7.2 The symmetry structure of the regular Rep eigenvectors 168
7.3 Symmetry projection on regular Rep eigenvectors 170
7.4 Computer construction of Irreps with d α >1 172
7.5 Summary of the method 176
Exercise 178
8 Product groups and product representations 179
8.1 Introduction 179
8.2 Subgroups and cosets 179
8.3 Direct outer-product groups 185
8.4 Semidirect product groups 190
8.5 Direct inner-product groups and their representations 191
8.6 Product representations and the Clebsch–Gordan series 192
8.7 Computer codes 208
8.8 Summary 214
Exercises 215
9 Induced representations 217
9.1 Introduction 217
9.2 Subduced Reps and compatibility relations 217
9.3 Induction of group Reps from the Irreps of its subgroups 219
9.4 Irreps induced from invariant subgroups 230
9.5 Examples of Irrep induction using the method of little-groups 247
Appendix Frobenius reciprocity theorem and other useful theorems 257
Exercises 261
10 Crystallographic symmetry and space-groups 263
10.1 Euclidean space 263
10.2 Crystallography 272
10.3 The perfect crystal 273
10.4 Space-group operations: the Seitz operators 294
10.5 Symmorphic and nonsymmorphic space-groups 297
10.6 Site-symmetries and the Wyckoff notation 323
Contents ix
10.7 Fourier space crystallography 347
Exercises 357
11 Space-groups: Irreps 362
11.1 Irreps of the translation group 362
11.2 Induction of Irreps of space-groups 377
Exercises 407
12 Time-reversal symmetry: color groups and the Onsager relations 409
12.1 Introduction 409
12.2 The time-reversal operator in quantum mechanics 409
12.3 Spin-1/2 and double-groups 419
12.4 Magnetic and color groups 429
12.5 The time-reversed representation: theory of corepresentations 442
12.6 Theory of crystal fields 458
12.7 Onsager reciprocity theorem (Onsager relations)
and transport properties 464
Exercises 472
13 Tensors and tensor fields 474
13.1 Tensors and their space-time symmetries 474
13.2 Construction of symmetry-adapted tensors 487
13.3 Description and classification of matter tensors 498
13.4 Tensor field representations 536
Exercises 551
14 Electronic properties of solids 552
14.1 Introduction 552
14.2 The one-electron approximations and self-consistent-field theories 552
14.3 Methods and techniques for band structure calculations 566
14.4 Electronic structure of magnetically ordered systems 618
Appendix 1 Derivation of the Hartree–Fock equations 632
Appendix 2 Holstein–Primakoff (HP) operators 633
Exercises 636
15 Dynamical properties of molecules, solids, and surfaces 638
15.1 Introduction 638
15.2 Dynamical properties of molecules 638
15.3 Dynamical properties of solids 668
15.4 Dynamical properties of surfaces 702
Appendix 1 Coulomb interactions and the method of Ewald summations 704
Appendix 2 Electronic effects on phonons in insulators and
semiconductors 708
Exercises 713
16 Experimental measurements and selection rules 716
16.1 Introduction 716
16.2 Selection rules 717
16.3 Differential scattering cross-sections in the Born approximation 721
x Contents
16.4 Light scattering spectroscopies 727
16.5 Photoemission and dipole selection rules 739
16.6 Neutron and atom scattering spectroscopies 748
Exercises 775
17.1 Phase transitions and their classification 777
17.2 Landau theory of phase transitions: principles 781
17.3 Construction and minimization techniques for ∆Φ 830
Exercises 856
18 Incommensurate systems and quasi-crystals 858
18.1 Introduction 858
18.2 The concept of higher-dimensional spaces: superspaces and
superlattices 865
18.3 Quasi-crystal symmetry: the notion of indistinguishability and the
classification of space-groups 879
18.4 Two-dimensional lattices, cyclotomic integers, and axial stacking 887
Bibliography 901
References 903
Index 912

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2016-09-16 01:47:31, 6.26 M