Quantum ising phases and transitions in transverse ising models:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Heidelberg [u.a.]
Springer
2013
|
| Edition: | 2. ed. |
| Series: | Lecture Notes in Physics
862 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XI, 403 S. graph. Darst. |
| ISBN: | 364233038X 9783642330384 |
Staff View
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| 245 | 1 | 0 | |a Quantum ising phases and transitions in transverse ising models |c Sei Suzuki ; Jun-ichi Inoue ; Bikas K. Chakrabarti |
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Record in the Search Index
| _version_ | 1807954783574687744 |
|---|---|
| adam_text |
CONTENTS
1 INTRODUCTION 1
1.1 THE TRANSVERSE ISING MODELS 1
1.2 A SIMPLE VERSION OF THE MODEL AND MEAN FIELD PHASE DIAGRAM . . 2
1.3 PROPERTIES OF ISING MODELS IN A TRANSVERSE FIELD: A SUMMARY . 5
2 TRANSVERSE ISING CHAIN (PURE SYSTEM) 13
2.1 SYMMETRIES AND THE CRITICAL POINT 13
2.1.1 DUALITY SYMMETRY OF THE TRANSVERSE ISING MODEL 13
2.1.2 PERTURBATIVE APPROACH 15
2.2 EIGENVALUE SPECTRUM: FERMIONIC REPRESENTATION 17
2.2.1 THE GROUND STATE ENERGY, CORRELATIONS AND EXPONENTS . 22
2.3 DIAGONALISATION TECHNIQUES FOR FINITE TRANSVERSE ISING CHAIN . 26
2.3.1 FINITE-SIZE SCALING 27
2.3.2 THE DIAGONALISATION TECHNIQUES 28
2.4 REAL-SPACE RENORMALISATION 30
2.4.1 BLOCK RENORMALISATION GROUP METHOD 31
2.5 FINITE TEMPERATURE BEHAVIOUR OF THE TRANSVERSE ISING CHAIN . 35
2.6 EXPERIMENTAL STUDIES OF THE TRANSVERSE ISING CHAIN 36
APPENDIX 2.A 38
2. A. 1 JORDAN-WIGNER FERMIONS 38
2.A.2 TO DIAGONALISE A GENERAL HAMILTONIAN QUADRATIC IN
FERMIONS 39
2. A.3 CALCULATION OF CORRELATION FUNCTIONS 42
3 TRANSVERSE ISING SYSTEM IN HIGHER DIMENSIONS (PURE SYSTEMS) . 47
3.1 MAPPING TO THE EFFECTIVE CLASSICAL HAMILTONIAN: SUZUKI-TROTTER
FORMALISM 47
3.2 THE QUANTUM MONTE CARLO METHOD 49
3.2.1 INFINITE
M
METHOD 50
3.3 DISCRETISED PATH INTEGRAL TECHNIQUE FOR A TRANSVERSE ISING SYSTEM .
54
3.4 INFINITE-RANGE MODELS 55
VII
HTTP://D-NB.INFO/1024629333
VIII
CONTENTS
3.4.1 HUSIMI-TEMPERLEY-CURIE-WEISS MODEL IN A TRANSVERSE FIELD 56
3.4.2 FULLY CONNECTED/?-BODY MODEL IN A TRANSVERSE FIELD . 60
3.5 SCALING PROPERTIES CLOSE TO THE CRITICAL POINT 63
3.6 REAL-SPACE AND FIELD-THEORETIC RENORMALISATION GROUP 66
3.6.1 REAL-SPACE RENORMALISATION GROUP 66
3.6.2 FIELD-THEORETIC RENORMALISATION GROUP 67
APPENDIX 3.A 68
3.A.1 EFFECTIVE CLASSICAL HAMILTONIAN OF THE TRANSVERSE ISING
MODEL 68
3.A.2 DERIVATION OF THE EQUIVALENT QUANTUM HAMILTONIAN OF A
CLASSICAL SPIN SYSTEM 70
4 ANNNI MODEL IN TRANSVERSE FIELD 73
4.1 INTRODUCTION 73
4.2 CLASSICAL ANNNI MODEL 74
4.3 ANNNI CHAIN IN A TRANSVERSE FIELD 75
4.3.1 SOME RESULTS IN THE HAMILTONIAN LIMIT: THE
PESCHEL-EMERY LINE 77
4.3.2 INTERACTING FERMION PICTURE 79
4.3.3 REAL-SPACE RENORMALISATION GROUP CALCULATIONS 81
4.3.4 FIELD-THEORETIC RENORMALISATION GROUP 83
4.3.5 NUMERICAL METHODS 83
4.3.6 MONTE CARLO STUDY 86
4.3.7 RECENT WORKS 87
4.4 LARGE
S
ANALYSIS 89
4.5 RESULTS IN HIGHER DIMENSIONS 91
4.6 NEAREST NEIGHBOUR CORRELATIONS IN THE GROUND STATE 95
APPENDIX 4. A 96
4.A.1 HARTREE-FOCK METHOD: MATHEMATICAL DETAILS 96
4. A.2 LARGE
S
ANALYSIS: DIAGONALISATION OF THE HAMILTONIAN IN
SPIN WAVE ANALYSIS 98
4. A.3 PERTURBATIVE ANALYSIS 99
5 DILUTE AND RANDOM TRANSVERSE ISING SYSTEMS 105
5.1 INTRODUCTION 105
5.2 DILUTE ISING SYSTEM IN A TRANSVERSE FIELD 105
5.2.1 MAPPING TO THE EFFECTIVE CLASSICAL HAMILTONIAN: HARRIS
CRITERION 107
5.2.2 DISCONTINUOUS JUMP IN
R
C
(P,
T =
0) AT THE PERCOLATION
THRESHOLD 108
5.2.3 REAL-SPACE RENORMALISATION GROUP STUDIES AND SCALING . . 109
5.3 CRITICAL BEHAVIOUR OF RANDOM TRANSVERSE FIELD ISING MODELS . 114
5.3.1 ANALYTICAL RESULTS IN ONE DIMENSION 114
5.3.2 MAPPING TO FREE FERMIONS 117
5.3.3 NUMERICAL RESULTS IN TWO AND HIGHER DIMENSIONS 119
CONTENTS IX
6 TRANSVERSE ISING SPIN GLASS AND RANDOM FIELD SYSTEMS 123
6.1 CLASSICAL ISING SPIN GLASSES: A SUMMARY 123
6.2 QUANTUM SPIN GLASSES 125
6.2.1 EXPERIMENTAL REALISATIONS OF QUANTUM SPIN GLASSES . 127
6.3 SHERRINGTON-KIRKPATRICK (SK) MODEL IN A TRANSVERSE FIELD 127
6.3.1 PHASE DIAGRAM 128
6.3.2 SUSCEPTIBILITY AND ENERGY GAP DISTRIBUTION 136
6.3.3 SK MODEL WITH ANTIFERROMAGNETIC BIAS 140
6.4 EDWARDS-ANDERSON MODEL IN A TRANSVERSE FIELD 142
6.4.1 QUANTUM MONTE CARLO RESULTS 143
6.5 A GENERAL DISCUSSION ON TRANSVERSE ISING SPIN GLASSES 148
6.5.1 THE POSSIBILITY OF REPLICA SYMMETRIC GROUND STATES IN
QUANTUM GLASSES 149
6.6 ISING SPIN GLASS WITH /?-SPIN INTERACTIONS IN A TRANSVERSE FIELD . .
152
6.6.1 P-BODY SPIN GLASS WITH FERROMAGNETIC BIAS 155
6.7 RANDOM FIELDS 161
6.7.1 CLASSICAL RANDOM FIELD ISING MODELS 161
6.7.2 RANDOM FIELD TRANSVERSE ISING MODELS (RFTIM) 162
6.7.3 CONCLUDING REMARKS ON THE RANDOM FIELD TRANSVERSE
ISING MODEL 167
6.8 MATTIS MODEL IN A TRANSVERSE FIELD 167
APPENDIX 6. A 169
6.A.1 THE VECTOR SPIN GLASS MODEL 169
6. A.2 THE EFFECTIVE CLASSICAL HAMILTONIAN OF A TRANSVERSE ISING
SPIN GLASS 170
6.A.3 EFFECTIVE SINGLE-SITE HAMILTONIAN FOR LONG-RANGE
INTERACTING RFTIM 171
6.A.4 MAPPING OF RANDOM ISING ANTIFERROMAGNET IN UNIFORM
LONGITUDINAL AND TRANSVERSE FIELDS TO RFTIM 173
6.A.5 DERIVATION OF FREE ENERGY FOR THE SK MODEL WITH
ANTIFERROMAGNETIC BIAS IN A TRANSVERSE FIELD 175
7 DYNAMICS OF QUANTUM ISING SYSTEMS 179
7.1 TUNNELLING DYNAMICS FOR HAMILTONIANS WITHOUT EXPLICIT TIME
DEPENDENCE 179
7.1.1 DYNAMICS IN ISING SYSTEMS: RANDOM PHASE APPROXIMATION 179
7.1.2 DYNAMICS IN DILUTE ISING SPIN SYSTEMS 180
7.1.3 DYNAMICS IN QUANTUM ISING GLASSES 182
7.2 NON-EQUILIBRIUM DYNAMICS IN PRESENCE OF TIME-DEPENDENT FIELDS 185
7.2.1 TIME-DEPENDENT BOGOLIUBOV-DE GENNES FORMALISM . 185
7.2.2 QUANTUM QUENCHES 189
7.2.3 OSCILLATING FIELDS: QUANTUM HYSTERESIS 203
7.2.4 RESPONSE DUE TO A PULSED TRANSVERSE FIELD IN ABSENCE OF
A LONGITUDINAL FIELD 213
X CONTENTS
APPENDIX 7.A 215
7.A.1 MEAN FIELD EQUATION OF MOTION 215
7.A.2 LANDAU-ZENER PROBLEM AND PARABOLIC CYLINDER FUNCTIONS . 217
7.A.3 MICROSCOPIC EQUATION OF MOTION FOR OSCILLATORY
TRANSVERSE FIELD 220
8 QUANTUM ANNEALING 225
8.1 INTRODUCTION 225
8.2 COMBINATORIAL OPTIMISATION PROBLEMS 227
8.3 OPTIMISATION BY A QUANTUM ADIABATIC EVOLUTION 229
8.3.1 NON-CROSSING RULE 229
8.3.2 QUANTUM ADIABATIC THEOREM 232
8.4 IMPLEMENTATION OF QUANTUM ANNEALING 237
8.4.1 NUMERICAL EXPERIMENTS 237
8.4.2 EXPERIMENTS 242
8.5 SIZE SCALING OF ENERGY GAPS 243
8.5.1 SIMPLE CASE 243
8.5.2 ANNEALING OVER AN INFINITE RANDOMNESS FIXED POINT . 244
8.5.3 ANNEALING OVER A FIRST ORDER QUANTUM PHASE TRANSITION . . 246
8.5.4 ANDERSON LOCALISATION 253
8.6 SCALING OF ERRORS 255
8.7 CONVERGENCE THEOREMS 258
8.7.1 SUFFICIENT CONDITION OF THE SCHEDULE 258
8.7.2 CONVERGENCE CONDITION OF QUANTUM ANNEALING WITH
QUANTUM MONTE CARLO DYNAMICS 262
8.8 CONCLUSION 270
APPENDIX 8.A 272
8.A.1 HOPF'S THEOREM 272
8.A.2 PERRON-FROBENIUS THEOREM 276
8. A.3 THEORY OF THE MARKOV CHAIN 277
9 APPLICATIONS 291
9.1 HOPFIELD MODEL IN A TRANSVERSE FIELD 291
9.1.1 STATICS AND PHASE DIAGRAMS 292
9.1.2 PATTERN-RECALLING PROCESSES 294
9.2 STATISTICAL MECHANICS OF INFORMATION 302
9.2.1 BAYESIAN STATISTICS AND INFORMATION PROCESSING 305
9.2.2 THE PRIORS AND CORRESPONDING SPIN SYSTEMS 309
9.2.3 QUANTUM VERSION OF MODELS 311
9.2.4 ANALYSIS OF THE INFINITE RANGE MODEL 312
9.2.5 MEAN FIELD ALGORITHMS 334
9.2.6 QUANTUM MONTE CARLO METHOD FOR INFORMATION PROCESSING . 340
APPENDIX 9. A DERIVATION OF SADDLE POINT EQUATIONS FOR THE QUANTUM
HOPFIELD MODEL 348
9.A.1 REPLICA SYMMETRIC AND STATIC APPROXIMATION 352
9. A.2 ZERO TEMPERATURE LIMIT 354
CONTENTS XI
10 RELATED MODELS 355
10.1 XY MODEL IN A TRANSVERSE FIELD 355
10.1.1 MEAN FIELD THEORY AND BCS EQUATIONS 355
10.1.2 EXACT SOLUTION OF TRANSVERSE XY CHAIN 357
10.1.3 TRANSVERSE XY CHAIN AND HARPER MODEL 363
10.1.4 INFINITE RANGE XY SPIN GLASS IN A TRANSVERSE FIELD . 364
10.2 KITAEV MODEL 367
10.2.1 FERMION REPRESENTATION AND DIAGONALISATION 367
10.2.2 CORRELATION FUNCTIONS 373
10.2.3 SLOW QUENCH DYNAMICS 373
11 BRIEF SUMMARY AND OUTLOOK 377
REFERENCES 381
INDEX 401 |
| any_adam_object | 1 |
| author | Suzuki, Sei Inoue, Jun-ichi Chakrabarti, Bikas K. 1952- |
| author_GND | (DE-588)1031570608 (DE-588)11423244X |
| author_facet | Suzuki, Sei Inoue, Jun-ichi Chakrabarti, Bikas K. 1952- |
| author_role | aut aut aut |
| author_sort | Suzuki, Sei |
| author_variant | s s ss j i i jii b k c bk bkc |
| building | Verbundindex |
| bvnumber | BV040669288 |
| classification_rvk | UD 8220 UG 3100 |
| classification_tum | PHY 721f PHY 602f |
| ctrlnum | (OCoLC)823890568 (DE-599)DNB1024629333 |
| dewey-full | 530.474 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.474 |
| dewey-search | 530.474 |
| dewey-sort | 3530.474 |
| dewey-tens | 530 - Physics |
| discipline | Physik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV040669288 |
| illustrated | Illustrated |
| indexdate | 2024-08-21T00:27:21Z |
| institution | BVB |
| isbn | 364233038X 9783642330384 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025495888 |
| oclc_num | 823890568 |
| open_access_boolean | |
| owner | DE-11 DE-703 DE-19 DE-BY-UBM DE-91G DE-BY-TUM DE-188 DE-29T DE-20 DE-384 |
| owner_facet | DE-11 DE-703 DE-19 DE-BY-UBM DE-91G DE-BY-TUM DE-188 DE-29T DE-20 DE-384 |
| physical | XI, 403 S. graph. Darst. |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | Springer |
| record_format | marc |
| series | Lecture Notes in Physics |
| series2 | Lecture Notes in Physics |
| spelling | Suzuki, Sei Verfasser (DE-588)1031570608 aut Quantum ising phases and transitions in transverse ising models Sei Suzuki ; Jun-ichi Inoue ; Bikas K. Chakrabarti 2. ed. Heidelberg [u.a.] Springer 2013 XI, 403 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture Notes in Physics 862 Quantenphasenübergang (DE-588)4793582-0 gnd rswk-swf Querfeld (DE-588)4338647-7 gnd rswk-swf Ising-Modell (DE-588)4127615-2 gnd rswk-swf Quantenmechanisches System (DE-588)4300046-0 gnd rswk-swf Ising-Modell (DE-588)4127615-2 s Quantenmechanisches System (DE-588)4300046-0 s Quantenphasenübergang (DE-588)4793582-0 s DE-604 Querfeld (DE-588)4338647-7 s 1\p DE-604 Inoue, Jun-ichi Verfasser aut Chakrabarti, Bikas K. 1952- Verfasser (DE-588)11423244X aut Erscheint auch als Online-Ausgabe 978-3-642-33039-1 Lecture Notes in Physics 862 (DE-604)BV000003166 862 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4093488&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025495888&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Suzuki, Sei Inoue, Jun-ichi Chakrabarti, Bikas K. 1952- Quantum ising phases and transitions in transverse ising models Lecture Notes in Physics Quantenphasenübergang (DE-588)4793582-0 gnd Querfeld (DE-588)4338647-7 gnd Ising-Modell (DE-588)4127615-2 gnd Quantenmechanisches System (DE-588)4300046-0 gnd |
| subject_GND | (DE-588)4793582-0 (DE-588)4338647-7 (DE-588)4127615-2 (DE-588)4300046-0 |
| title | Quantum ising phases and transitions in transverse ising models |
| title_auth | Quantum ising phases and transitions in transverse ising models |
| title_exact_search | Quantum ising phases and transitions in transverse ising models |
| title_full | Quantum ising phases and transitions in transverse ising models Sei Suzuki ; Jun-ichi Inoue ; Bikas K. Chakrabarti |
| title_fullStr | Quantum ising phases and transitions in transverse ising models Sei Suzuki ; Jun-ichi Inoue ; Bikas K. Chakrabarti |
| title_full_unstemmed | Quantum ising phases and transitions in transverse ising models Sei Suzuki ; Jun-ichi Inoue ; Bikas K. Chakrabarti |
| title_short | Quantum ising phases and transitions in transverse ising models |
| title_sort | quantum ising phases and transitions in transverse ising models |
| topic | Quantenphasenübergang (DE-588)4793582-0 gnd Querfeld (DE-588)4338647-7 gnd Ising-Modell (DE-588)4127615-2 gnd Quantenmechanisches System (DE-588)4300046-0 gnd |
| topic_facet | Quantenphasenübergang Querfeld Ising-Modell Quantenmechanisches System |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=4093488&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025495888&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000003166 |
| work_keys_str_mv | AT suzukisei quantumisingphasesandtransitionsintransverseisingmodels AT inouejunichi quantumisingphasesandtransitionsintransverseisingmodels AT chakrabartibikask quantumisingphasesandtransitionsintransverseisingmodels |