Symmetry and Condensed Matter Physics: A Computational Approach

Cover......Page 1
Contents......Page 9
Preface......Page 13
1.1 Introduction......Page 17
1.2 Hamiltonians, eigenfunctions, and eigenvalues......Page 21
1.2.1 Examples of symmetry and conservation laws......Page 24
1.3.1 Configuration-space operators......Page 25
1.3.2 Function-space operations......Page 26
1.3.3 Operator algebra......Page 29
1.4 Point-symmetry operations......Page 30
1.5 Applications to quantum mechanics......Page 33
Exercises......Page 35
2.1 Groups and their realizations......Page 37
2.2 The symmetric group......Page 41
2.2.1 The permutation group......Page 44
2.3.1 Generation of the Cayley (multiplication) table......Page 46
2.3.2 Computer generation of group elements......Page 47
2.4 Classes......Page 50
2.4.1 Computer generation of classes and class arrays......Page 52
2.4.2 Class multiplication......Page 54
2.4.3 Computer generation of class multiplication matrices......Page 56
2.5 Homomorphism, isomorphism, and automorphism......Page 57
2.6 Direct- or outer-product groups......Page 58
Exercises......Page 59
3 Group representations: concepts......Page 67
3.1 Representations and realizations......Page 68
3.1.1 Transformation of coordinates......Page 70
3.1.2 Transformation of functions......Page 71
3.2 Generation of representations on a set of basis functions......Page 78
Exercises......Page 83
4.1 Matrix representations......Page 85
4.1.1 Diagonal matrix representatives......Page 87
4.1.2 Reducible representations......Page 88
4.1.3 The regular representation......Page 89
4.1.4 The great orthogonality theorem......Page 91
4.2.1 Character orthogonality relations......Page 94
4.2.2 Character decomposition......Page 97
4.2.3 Class matrices and Dirac characters......Page 98
4.3 Burnside’s method......Page 101
Exercises......Page 110
5.1 The eigenvalue equation modulo p......Page 112
5.2 Dixon’s method for irreducible characters......Page 115
5.2.1 Integer polynomials modulo p......Page 118
5.2.2 Multiplicities and the character......Page 121
5.3 Computer codes for Dixon’s method......Page 123
Appendix 1 Finding eigenvalues and eigenvectors......Page 132
Appendix 2......Page 144
6.1 Group action......Page 150
6.2 Symmetry projection operators......Page 154
6.2.1 Construction of the symmetry transfer operators......Page 155
6.2.2 The Wigner projection operator......Page 157
6.3 The regular projection matrices: the simple characteristic......Page 173
Exercises......Page 178
7.1 Eigenvectors of the regular Rep......Page 180
7.1.1 Computer generation of eigenvectors of the regular Rep......Page 182
7.2 The symmetry structure of the regular Rep eigenvectors......Page 184
7.3 Symmetry projection on regular Rep eigenvectors......Page 186
7.4 Computer construction of Irreps with dα >1......Page 188
7.5 Summary of the method......Page 192
Exercise......Page 194
8.2 Subgroups and cosets......Page 195
8.2.1 Conjugate subgroups......Page 197
8.2.2 Self-conjugate (invariant or normal) subgroups and their quotient groups......Page 199
8.3 Direct outer-product groups......Page 201
8.3.1 Representations of direct outer-product groups: Kronecker-product representations......Page 203
8.4 Semidirect product groups......Page 206
8.5 Direct inner-product groups and their representations......Page 207
8.6 Product representations and the Clebsch–Gordan series......Page 208
8.6.1 Reduction of Kronecker products: the Clebsch–Gordan series......Page 209
8.6.2 Symmetrizing Kronecker products of the same Irrep......Page 210
8.6.3 Symmetrized pth power of an Irrep and Molien functions2......Page 215
8.6.4 Reduction of product basis sets: Clebsch–Gordan or Wigner coefficients......Page 218
8.7 Computer codes......Page 224
8.8 Summary......Page 230
Exercises......Page 231
9.2 Subduced Reps and compatibility relations......Page 233
9.3.1 Inducing group Reps from Irreps of subgroups......Page 235
9.3.2 The ground Rep......Page 239
9.3.3 Inducing Reps of C3v from Irreps of H = {E,σ1}......Page 244
9.4.1 Conjugate Irreps of normal subgroups......Page 246
9.4.2 Group action on Irreps of normal subgroups......Page 249
9.4.3 Establishing the road map to induction of Irreps......Page 255
9.4.4 Irrep induction procedures based on the method of little-groups......Page 261
9.5.1 Example 1: The induced Irrep for C3v......Page 263
9.5.2 Example 2: induced Irreps for C4v......Page 268
Appendix Frobenius reciprocity theorem and other useful theorems......Page 273
Exercises......Page 277
10.1.1 Rotations and angular momentum......Page 279
10.1.2 The Euclidean group in n-dimensional space: E(n)......Page 286
10.2 Crystallography......Page 288
10.3.1 The translation group......Page 289
10.3.2 The Holohedry group and classification of Bravais lattices......Page 290
10.3.3 Lattice metric, arithmetic holohedries, and Bravais classes......Page 294
10.3.4 Lattice systems and Bravais classes in two and three dimensions......Page 299
10.3.5 Unit, primitive, and Wigner–Seitz cells......Page 305
10.3.6 The crystal......Page 307
10.4 Space-group operations: the Seitz operators......Page 310
10.4.1 Important properties of the Seitz operator......Page 311
10.5.1 Symmorphic space-groups......Page 313
10.5.2 Nonsymmorphic space-groups......Page 316
10.5.3 Point-group of the crystal revisited......Page 320
10.5.4 Classification of space-groups......Page 322
10.5.5 Computer generation of space-group matrices......Page 327
10.5.6 Subgroups and supergroups of space-groups......Page 332
10.6.1 Space-group action and crystallographic orbits......Page 339
10.6.3 Wyckoff positions and the Wyckoff notation......Page 341
10.6.4 Wyckoff sets: Euclidean and affine normalizers......Page 345
10.6.5 Examples of two- and three-dimensional crystals......Page 356
10.7.1 Reciprocal space, reciprocal lattice, and diffraction patterns......Page 363
10.7.2 Action of space-group operators in Λ∗......Page 365
10.7.3 Equivalent space-groups, gauge transformations......Page 367
10.7.4 Extinctions in Fourier space......Page 372
Exercises......Page 373
11.1 Irreps of the translation group......Page 378
11.1.1 Brillouin zones and the reciprocal lattice......Page 380
11.1.2 Symmetry projection operators of T and Bloch functions......Page 384
11.1.3 Point-groups and conjugate Irreps of the translation group......Page 386
11.1.4 Orbits and little-groups......Page 387
11.1.5 Rectangular Brillouin zone......Page 390
11.2 Induction of Irreps of space-groups......Page 393
11.2.1 The method of the little-group of the first kind: projective Irreps and multiplier factor systems......Page 394
11.2.2 Examples of symmorphic space-groups......Page 396
11.2.3 Program for constructing subgroups of the wavevectors......Page 406
11.2.4 Nonsymmorphic space-groups: Herring’s method of kernel and quotient subgroups......Page 409
11.2.5 Examples of nonsymmorphic space-groups......Page 414
Exercises......Page 423
12.2 The time-reversal operator in quantum mechanics......Page 425
12.2.1 The conjugation operator and time-reversal......Page 427
12.2.2 Transformation of operators and wavefunctions under time-reversal......Page 428
12.2.3 Time-reversal and spin-1/2 systems......Page 430
12.2.4 Time-reversal in external fields......Page 434
12.3.1 Representations of rotation operators for spin angular momentum 1/2......Page 435
12.3.2 Double-groups......Page 437
12.4 Magnetic and color groups......Page 445
12.4.1 Classification of magnetic or Shubnikov groups......Page 446
12.4.2 Polychromatic groups......Page 456
12.5.1 Construction of corepresentations (CoRep)......Page 458
12.5.2 Reality of Irreps......Page 460
12.5.4 Reducibility of Γ......Page 463
12.5.5 Double-valued corepresentations and Kramers’ theorem......Page 467
12.6.1 Conventional theory of crystal fields......Page 474
12.6.2 Extension of crystal-field theory to Shubnikov point-groups......Page 477
12.7.1 Irreversible thermodynamics and time-reversal symmetry......Page 480
12.7.2 The Onsager reciprocity theorem and time-reversal......Page 487
Exercises......Page 488
13.1.1 Neumann’s principle......Page 490
13.1.2 Tensor transformation and invariance under O(3) operations......Page 491
13.1.3 Tensors and time-reversal symmetry......Page 498
13.1.4 Intrinsic symmetry......Page 500
13.2 Construction of symmetry-adapted tensors......Page 503
13.2.1 Method using Mathematica......Page 504
13.2.2 The method of Erd¨os......Page 508
13.2.3 The method of Bradley and Davies for magnetic crystals......Page 510
13.3 Description and classification of matter tensors......Page 514
13.3.1 Symmetric second-rank matter tensors......Page 516
13.3.2 Elasticity tensors......Page 518
13.3.3 Tensors associated with electric and magnetic fields......Page 527
13.4 Tensor field representations......Page 552
13.4.1 The tensor representation ΓT......Page 553
13.4.3 Simple system decomposition......Page 554
13.4.4 Simple system tensor field Rep......Page 555
Exercises......Page 567
14.2.1 The many-body problem......Page 568
14.2.2 The Hartree–Fock approach......Page 570
14.2.3 Density functional formalisms......Page 575
14.3.1 Bloch and Wannier bases......Page 582
14.3.2 Symmetries of the electron dispersion curves (bands) E(k)......Page 583
14.3.3 Plane-wave methods......Page 585
14.3.4 The tight-binding method......Page 615
14.3.5 d-Bands and resonances......Page 632
14.4 Electronic structure of magnetically ordered systems......Page 634
14.4.1 The space-group and Brillouin zone of a magnetic metal......Page 636
14.4.2 Labeling of spin-wave dispersion curves......Page 638
Appendix 1 Derivation of the Hartree–Fock equations......Page 648
Appendix 2 Holstein–Primakoff (HP) operators......Page 649
Exercises......Page 652
15.2.1 The adiabatic approximation......Page 654
15.2.2 The equations of motion in the harmonic approximation......Page 655
15.2.3 Displacement tensor field and its tensor field Rep......Page 663
15.2.4 Computer programs......Page 673
15.3.1 The harmonic approximation revisited......Page 684
15.3.2 The dynamical matrix and its symmetries......Page 687
15.3.3 Displacement tensor field and its symmetry decomposition......Page 695
15.3.4 Elastic equations of motion......Page 712
15.4.1 Surface modes of an elastic medium......Page 718
15.4.2 Surface modes of a crystalline solid......Page 719
Appendix 1 Coulomb interactions and the method of Ewald summations......Page 720
Appendix 2 Electronic effects on phonons in insulators and semiconductors......Page 724
Exercises......Page 729
16.1 Introduction......Page 732
16.2 Selection rules......Page 733
16.3.1 Scattering states......Page 737
16.3.3 The differential scattering cross-section......Page 741
16.4.1 Interaction Hamiltonian......Page 743
16.4.2 Phonons......Page 748
16.5 Photoemission and dipole selection rules......Page 755
16.5.1 Wavevector selection rules......Page 757
16.5.2 Photoexcitation selection rules......Page 758
16.6.1 Neutron scattering spectroscopy......Page 764
16.6.2 Atom scattering spectroscopy......Page 784
Exercises......Page 791
17.1.1 Classification of phase transitions......Page 793
17.1.2 The Ehrenfest definition of second-order phase transition......Page 794
17.1.3 Characteristics of phase transitions......Page 796
17.2 Landau theory of phase transitions: principles......Page 797
17.2.1 The density function......Page 798
17.2.2 Symmetry properties of the density increment......Page 801
17.2.3 The Landau free energy......Page 804
17.2.4 The low-temperature, low-symmetry phase......Page 808
17.2.5 Instability in the presence of F(3)......Page 810
17.2.6 Fourth-degree term......Page 815
17.2.7 Group–subgroup relations: isotropy subgroups......Page 816
17.2.8 Homogeneity of the order-parameter......Page 828
17.3.1 Construction of the Landau free energy: invariant polynomials......Page 846
17.3.2 Group action and OP image groups......Page 857
17.3.3 Finding the minima of the Landau free energy......Page 865
Exercises......Page 872
18.1 Introduction......Page 874
18.1.1 Diffraction patterns of QP systems: rank and indexing dimension......Page 875
18.1.2 Describing the symmetry of QP systems......Page 876
18.1.3 Types of quasi-periodic systems......Page 877
18.2.1 Projection from higher dimensions: pedagogy......Page 881
18.2.2 Symmetry and superspace groups......Page 889
18.3.1 Quasi-periodic systems and the concept of indistinguishability......Page 895
18.4 Two-dimensional lattices, cyclotomic integers, and axial stacking......Page 903
18.4.1 The standard lattices: classification of 2D generalized lattices......Page 904
18.4.2 Standard lattices and cyclotomic integers......Page 905
18.4.3 Classification of two-dimensional N-lattices......Page 906
18.4.4 The action of point-groups on cyclotomic integers in the complex plane......Page 907
18.4.5 Scale invariance of standard lattices: cyclotomic units......Page 909
18.4.6 Quasi-crystallographic space-groups in two dimensions......Page 912
18.4.7 Cubic lattices in n-dimensions......Page 916
Bibliography......Page 917
References......Page 919
Index......Page 928