Mapping of Parent Hamiltonians: from Abelian and non-Abelian quantum hall states to exact models of critical spin chains
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2011
|
| Series: | Springer tracts in modern physics
244 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XIV, 194 S. 235 mm x 155 mm |
| ISBN: | 9783642243837 9783642243844 |
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Record in the Search Index
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IMAGE 1
CONTENTS
1 INTRODUCTION AND SUMMARY 1
REFERENCES 8
2 THREE MODELS AND A GROUND STATE 11
2.1 THE LAUGHLIN STATE AND ITS PARENT HAMILTONIAN 11
2.1.1 LANDAU LEVEL QUANTIZATION IN THE PLANAR GEOMETRY . . . 11 2.1.2
THE LAUGHLIN STATE 16
2.1.3 FRACTIONALLY CHARGED QUASIPARTICLE EXCITATIONS 18 2.1.4 FRACTIONAL
STATISTICS 20
2.1.5 LANDAU LEVEL QUANTIZATION IN THE SPHERICAL GEOMETRY. . 22 2.1.6
THE LAUGHLIN STATE AND ITS PARENT HAMILTONIAN ON THE SPHERE 27
2.2 THE HALDANE-SHASTRY MODEL 29
2.2.1 THE 1/R 2 MODEL OF HALDANE AND SHASTRY 29
2.2.2 SYMMETRIES AND INTEGRABILITY 30
2.2.3 GROUND STATE PROPERTIES 31
2.2.4 EXPLICT SOLUTION 34
2.2.5 FACTORIZATION OF THE HAMILTONIAN 38
2.2.6 SPINON EXCITATIONS AND FRACTIONAL STATISTICS 41
2.2.7 YOUNG TABLEAUX AND MANY SPINON STATES 45
2.3 THE MOORE-READ STATE AND ITS PARENT HAMILTONIAN 47
2.3.1 THE PFAFFIAN STATE AND ITS PARENT HAMILTONIAN 47 2.3.2
QUASIPARTICLE EXCITATIONS AND THE INTERNAL HUBERT SPACE 50
2.3.3 MAJORANA FERMIONS AND NON-ABELIAN STATISTICS 51 2.3.4 THE PFAFFIAN
STATE AND ITS PARENT HAMILTONIAN ON THE SPHERE 56
2.4 AN S= 1 SPIN LIQUID STATE DESCRIBED BY A PFAFFIAN 57
2.4.1 THE GROUND STATE 57
2.4.2 SYMMETRIES 58
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1014764181
DIGITALISIERT DURCH
IMAGE 2
CONTENTS
2.4.3 SCHWINGER BOSONS 61
2.4.4 GENERATION BY PROJECTION FROM GUTZWILLER STATES 62 2.4.5
TOPOLOGICAL DEGENERACIES AND NON-ABELIAN STATISTICS . . 64 2.4.6
GENERALIZATION TO ARBITRARY SPIN S 67
2.4.7 MOMENTUM SPACINGS AND TOPOLOGICAL DEGENERACIES FOR ARBITRARY SPIN
S 68
REFERENCES 72
FROM A LAUGHLIN STATE TO THE HALDANE-SHASTRY MODEL 79 3.1 GENERAL
CONSIDERATIONS 79
3.1.1 COMPARISON OF THE MODELS 79
3.1.2 A HOLE AT A POLE 81
3.2 HILBERT SPACE RENORMALIZATION 82
3.3 FOURIER TRANSFORMATION 86
3.3.1 PARTICLE CREATION AND ANNIHILATION OPERATORS 86
3.3.2 RENORMALIZED MATRIX ELEMENTS 87
3.3.3 AN ALTERNATIVE DERIVATION 90
3.4 THE DEFINING CONDITION FOR THE GUTZWILLER STATE 91
3.4.1 ANNIHILATION OPERATORS 91
3.4.2 DIRECT VERIFICATION 92
3.4.3 THE ROLE OF THE HOLE 93
3.5 ROTATIONS AND SPHERICAL TENSOR OPERATORS 94
3.5.1 REPRESENTATIONS OF ROTATIONS 94
3.5.2 TENSOR OPERATORS 95
3.5.3 PRODUCTS OF TENSOR OPERATORS 96
3.6 CONSTRUCTION OF A PARENT HAMILTONIAN FOR THE GUTZWILLER STATE 98
3.6.1 TRANSLATIONAL, TIME REVERSAL, AND PARITY SYMMETRY . . . 98 3.6.2
SPIN ROTATION SYMMETRY 100
3.6.3 AN ALTERNATIVE DERIVATION 101
3.7 THE RAPIDITY OPERATOR AND MORE 102
3.7.1 ANNIHILATION OPERATORS WHICH TRANSFORM EVEN UNDER T 102
3.7.2 ANNIHILATION OPERATORS WHICH TRANSFORM ODD UNDER T 104
3.8 CONCLUDING REMARKS 106
REFERENCES 106
FROM A BOSOM'C PFAFFIAN STATE TO AN S = 1 SPIN CHAIN 109 4.1 GENERAL
CONSIDERATIONS 109
4.1.1 A MODEL AND A GROUND STATE 109
4.1.2 CREATION OF A QUASIHOLE I LL
4.2 HILBERT SPACE RENORMALIZATION 112
IMAGE 3
CONTENTS
4.3 FOURIER TRANSFORMATION 113
4.3.1 PARTICLE CREATION AND ANNIHILATION OPERATORS 113 4.3.2
SUBSTITUTION OF SPIN FLIP OPERATORS FOR BOSON OPERATORS 114
4.3.3 MANY BODY ANNIHILATION OPERATORS 115
4.3.4 EVALUATION OF B*. AI AIAI 116
4.3.5 EVALUATION OF B=. AUAI 119
4.4 THE DEFINING CONDITION FOR THE S = 1 PFAFFIAN CHAIN 122
4.4.1 DERIVATION 122
4.4.2 A SECOND CONDITION 124
4.4.3 DIRECT VERIFICATION 125
4.5 CONSTRUCTION OF A PARENT HAMILTONIAN 127
4.5.1 TRANSLATIONAL, TIME REVERSAL, AND PARITY SYMMETRY . . . 127 4.5.2
SPIN ROTATION SYMMETRY 128
4.5.3 EVALUATION OF {T% A T I } 0 129
4.5.4 WRITING OUT THE HAMILTONIAN 132
4.6 VECTOR ANNIHILATION OPERATORS 133
4.6.1 ANNIHILATION OPERATORS WHICH TRANSFORM EVEN UNDER T 133
4.6.2 ANNIHILATION OPERATORS WHICH TRANSFORM ODD UNDER T 136
4.7 CONCLUDING REMARKS 138
REFERENCES 139
GENERALIZATION TO ARBITRARY SPIN S 141
5.1 A CRITICAL SPIN LIQUID STATE WITH SPIN 5 141
5.1.1 GENERATION THROUGH PROJECTION OF GUTZWILLER STATES . . . 141
5.1.2 DIRECT VERIFICATION OF THE SINGLET PROPERTY 142
5.2 THE DEFINING CONDITION FOR THE SPIN S CHAIN 144
5.2.1 STATEMENT 144
5.2.2 DIRECT VERIFICATION 144
5.3 CONSTRUCTION OF A PARENT HAMILTONIAN 146
5.3.1 TRANSLATIONAL SYMMETRY 146
5.3.2 TENSOR DECOMPOSITION OF (S*-) 2S {S-) 2 ' 146
5.3.3 TIME REVERSAL AND PARITY SYMMETRY 151
5.3.4 SPIN ROTATION SYMMETRY 152
5.4 VECTOR ANNIHILATION OPERATORS 155
5.4.1 ANNIHILATION OPERATORS WHICH TRANSFORM EVEN UNDER T 155
5.4.2 ANNIHILATION OPERATORS WHICH TRANSFORM ODD UNDER T 157
5.4.3 EVALUATION OF {W% AA T^ Y } L 159
5.4.4 ANNIHILATION OPERATORS WHICH TRANSFORM ODD UNDER T (CONTINUED) 161
IMAGE 4
XIV CONTENTS
5.5 SCALAR OPERATORS CONSTRUCTED FROM VECTORS 164
5.5.1 FACTORIZATION OF THE HAMILTONIAN 165
5.5.2 A VARIATION OF THE MODEL 167
5.5.3 THE THIRD DERIVATION 169
5.6 THE CASE S = \ ONCE MORE 171
REFERENCES 172
6 CONCLUSIONS AND UNRESOLVED ISSUES 173
REFERENCES 177
APPENDIX A: SPHERICAL COORDINATES 179
APPENDIX B: FOURIER SUMS FOR ONE-DIMENSIONAL LATTICES 181
APPENDIX C: ANGULAR MOMENTUM ALGEBRA 187
APPENDIX D: TENSOR DECOMPOSITIONS OF SPIN OPERATORS 189
REFERENCES * 194 |
| any_adam_object | 1 |
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| dewey-ones | 530 - Physics |
| dewey-raw | 530.12 |
| dewey-search | 530.12 |
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| dewey-tens | 530 - Physics |
| discipline | Physik |
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| spelling | Greiter, Martin Verfasser aut Mapping of Parent Hamiltonians from Abelian and non-Abelian quantum hall states to exact models of critical spin chains Martin Greiter Berlin [u.a.] Springer 2011 XIV, 194 S. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Springer tracts in modern physics 244 Kritisches Phänomen (DE-588)4165788-3 gnd rswk-swf Quanten-Hall-Effekt (DE-588)4124013-3 gnd rswk-swf Spinkette (DE-588)4263231-6 gnd rswk-swf Quantenflüssigkeit (DE-588)4176598-9 gnd rswk-swf Quantenflüssigkeit (DE-588)4176598-9 s Quanten-Hall-Effekt (DE-588)4124013-3 s Spinkette (DE-588)4263231-6 s Kritisches Phänomen (DE-588)4165788-3 s DE-604 Springer tracts in modern physics 244 (DE-604)BV000000153 244 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3873350&prov=M&dok%5Fvar=1&dok%5Fext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024650196&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Greiter, Martin Mapping of Parent Hamiltonians from Abelian and non-Abelian quantum hall states to exact models of critical spin chains Springer tracts in modern physics Kritisches Phänomen (DE-588)4165788-3 gnd Quanten-Hall-Effekt (DE-588)4124013-3 gnd Spinkette (DE-588)4263231-6 gnd Quantenflüssigkeit (DE-588)4176598-9 gnd |
| subject_GND | (DE-588)4165788-3 (DE-588)4124013-3 (DE-588)4263231-6 (DE-588)4176598-9 |
| title | Mapping of Parent Hamiltonians from Abelian and non-Abelian quantum hall states to exact models of critical spin chains |
| title_auth | Mapping of Parent Hamiltonians from Abelian and non-Abelian quantum hall states to exact models of critical spin chains |
| title_exact_search | Mapping of Parent Hamiltonians from Abelian and non-Abelian quantum hall states to exact models of critical spin chains |
| title_full | Mapping of Parent Hamiltonians from Abelian and non-Abelian quantum hall states to exact models of critical spin chains Martin Greiter |
| title_fullStr | Mapping of Parent Hamiltonians from Abelian and non-Abelian quantum hall states to exact models of critical spin chains Martin Greiter |
| title_full_unstemmed | Mapping of Parent Hamiltonians from Abelian and non-Abelian quantum hall states to exact models of critical spin chains Martin Greiter |
| title_short | Mapping of Parent Hamiltonians |
| title_sort | mapping of parent hamiltonians from abelian and non abelian quantum hall states to exact models of critical spin chains |
| title_sub | from Abelian and non-Abelian quantum hall states to exact models of critical spin chains |
| topic | Kritisches Phänomen (DE-588)4165788-3 gnd Quanten-Hall-Effekt (DE-588)4124013-3 gnd Spinkette (DE-588)4263231-6 gnd Quantenflüssigkeit (DE-588)4176598-9 gnd |
| topic_facet | Kritisches Phänomen Quanten-Hall-Effekt Spinkette Quantenflüssigkeit |
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| volume_link | (DE-604)BV000000153 |
| work_keys_str_mv | AT greitermartin mappingofparenthamiltoniansfromabelianandnonabelianquantumhallstatestoexactmodelsofcriticalspinchains |