Replica symmetry breaking at low temperatures [Elektronische Ressource] / vorgelegt von Manuel J. Schmidt

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• Physik

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• Publié par: julius-maximilians-universitat_wurzburg
• Publié le: 01 janvier 2008
• Nombre de lectures: 26
• Poids de l'ouvrage: 4 Mo

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Replica Symmetry Breaking at Low Temperatures
Dissertation
zur Erlangung des
naturwissenschaftlichen Doktorgrades
der Julius-Maximilians-Universit¨at Wurzburg¨
vorgelegt von
Manuel J. Schmidt
aus Schwemmelsbach in Bayern
Wurzburg,¨ 2008Eingereicht am 27. Juni 2008
bei der Fakult¨at fur¨ Physik und Astronomie
Gutachter der Dissertation:
1. Gutachter: Prof. Dr. Reinhold Oppermann
2. Gutachter: Prof. Dr. Bj¨orn Trauzettel
3. Gutachter: Prof. Dr. Cirano de Dominicis
Prufer¨ im Promotionskolloquium:
1. Prufer:¨ Prof. Dr. Reinhold Oppermann
2. Prufer:¨ Prof. Dr. Bj¨orn Trauzettel
3. Prufer:¨ Prof. Dr. Laurens W. Molenkamp
Tag des Promotionskolloquiums: 01.12.2008
Doktorurkunde ausgeh¨andigt am: ....to ElisabethContents
1 Introduction 1
1.1 Historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Modern applications in computer sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Computational complexity in optimization problems . . . . . . . . . . . . . . . . . . . 3
1.2.2 Errororrecting codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Neural networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Ergodicity breaking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Quenched disorder vs. annealed disorder . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.2 Replica trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Meaneld theory: from ferromagnetism to RSB . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Formalism at ite RSB 11
2.1 The free energy in replica formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.2 Special case: Ising spin glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.3 Special case: (Quasiisotropic nomponent spin glass . . . . . . . . . . . . . . . . . . 15
2.2 The replica symmetric approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Special case: Ising spin glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2 Special case: Quasisotropic nomponent spin glass . . . . . . . . . . . . . . . . . . . 18
2.3 Parisi RSB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.1 Ising spin glass (n=1) with ite J and ite external ld . . . . . . . . . . . . . . . 220
2.3.2 The quasisotropic nomponent spin glass . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.3 held integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Low temperature formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.1 Parameter rescaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.2 Asymptotic renormalization group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.3 The kernel correction function for an Ising spin glass . . . . . . . . . . . . . . . . . . 30
2.4.4 The kernel at T = 0 for general n . . . . . . . . . . . . . . . . . . . 31
2.5 Observables in the replica formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5.1 Free/internal energy and entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5.2 Magnetic observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.5.3 Zero temperature and zero external ld . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3 Analysis of the RS saddle point 39
3.1 The Ising spin glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Isotropic nomponent spin glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Results and discussion at ite order of RSB 45
4.1 Analysis of thermodynamic observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1.1 Free energy, internal energy and entropy . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.1.2 Magnetic susceptibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1.3 Connection to systems of ite size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 The order function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.1 q(a) at zero temperature and zero ld . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.2 q(a) at ite temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
iii CONTENTS
4.2.3 q(a) at ite external lds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3 Discreteness at zero temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.1 Discrete spectra in the block size ratios . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3.2 Two scales for the order function at zero temperature . . . . . . . . . . . . . . . . . . 64
4.3.3 Criticality of the block size ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4 Scaling analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4.1 PaTcaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4.2 Scaling in the continuous limit near h = 0, T = 0 . . . . . . . . . . . . . . . . . . . . . 68
4.4.3 Unid scaling at ite RSB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Continuous RSB 73
5.1 The continuous RSB transition of kerC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Continuous RSB at ite temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 The proper zero temperature continuous theory . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3.1 Replacement of the initial condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3.2 Solving the dirential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3.3 The ed point at a =∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4 Sommersupont Ansatz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4.1 The method of functional Lagrangeultipliers . . . . . . . . . . . . . . . . . . . . . . 82
5.4.2 The continuous selfonsistency equations . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4.3 Selfonsistence at a =∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.4.4 in the full continuous a interval . . . . . . . . . . . . . . . . . . . . . . 85
5.4.5 Relation to Pankov scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6 Conclusion and outlook 87
Acknowledgments 91
A Proofs and derivations 93
A.1 Steepest descent and replica limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
A.2 First integration in the recursion sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
A.3 Irrelevance of the kernel normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
A.4 Asymptotic regime of the recursion relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
A.5 Evaluation of replica sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
A.6 The trace term in the entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
B The temperature dependence of A(T) 99
C Deitions and conventions 103
C.1 Gaussian integral operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
C.2 Parisi block size parameters and matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . 104
D Numerics of the ite RSB formalism 105
D.1 The program suite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
D.2 High precision arithmetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Bibliography 107Chapter 1
Introduction
Spin glass science is a relatively mature ld of research and much work has been invested to experimentally
investigate, as well as to theoretically model and understand the intriguing properties of this magnetically
1(dis)ordered phase at low temperatures. However, up to now, many fundamental questions, e.g. what are
the critical dimensions and what happens in iteimensional spin glasses, remain unanswered. Already at
thebeginningofthetheoreticalinvestigationsitturnedoutthateventheclassicalmeaneldtheoryofaspin
glass is technically and conceptually highly nontrivial, and its correct treatment requires a powerful tool,
known as replica symmetry breaking (RSB). While researchers have gained some insight into the physics of
RSB, meanwhile, the question of its relevance in iteimensional spin glasses is still hotly debated and no
satisfactory theory which extends the meaneld model to ite dimensionality has been found, yet.
RSB as a fundamental issue in replica theory, however, has proved important in lots of other lds of
physical and even of interdisciplinary research. Thus, it is important to gain a deeper understanding of
RSB, to