Critical properties of disordered XY model on sparse random graphs Scuola di dottorato Vito Volterra Dottorato di Ricerca in Fisica – XXIX Ciclo Candidate Cosimo Lupo ID number 1266587 Thesis Advisor External Referees Prof. Federico Ricci - Tersenghi Prof. Florent Krzakala Prof. Juan Jesús Ruiz - Lorenzo arXiv:1706.08899v1 [cond-mat.dis-nn] 27 Jun 2017 A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics February 2017 Thesis defended on 16 February 2017 in front of a Board of Examiners composed by: Prof. Mauro Carfora (chairman) Prof. Sergio Caracciolo Dr. Antonello Scardicchio Critical properties of disordered XY model on sparse random graphs Ph.D. thesis. Sapienza – University of Rome © 2017 Cosimo Lupo. All rights reserved The research leading to these results has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation pro- gramme, grant agreement No. 694925 (LoTGlasSy). This thesis has been typeset by LATEX and the Sapthesis class. Version: June 28, 2017 Author’s email: [email protected] A Serena, per quanto cercassi il senso delle cose, la risposta sei sempre stata tu. “Il caos è un ordine da decifrare.” José Saramago, L’uomo duplicato “Tutte le isole, anche quelle conosciute, sono sconosciute finché non vi si sbarca.” José Saramago, Il racconto dell’isola sconosciuta “Chaos is order yet undeciphered.” José Saramago, The double “Even known islands remain unknown until we set foot on them.” José Saramago, The tale of the unknown island Abstract This thesis focuses on the XY model, the simplest vector spin model, used for describing numerous physical systems, from random lasers to superconductors, from synchronization problems to superfluids. It is studied for different sources of quenched disorder: random couplings, random fields, or both them. The belief propagation algorithm and the cavity method are exploited to solve the model on the sparse topology provided by Bethe lattices. It is found that the discretized version of the XY model, the so-called Q-state clock model, provides a reliable and efficient proxy for the continuous model with an error going to zero exponentially in Q, so implying a remarkable speedup in numerical simulations. Interesting results regard the low-temperature solution of the spin glass XY model, which is by far more unstable toward the replica symmetry broken phase with respect to what happens in discrete models. Moreover, even the random field XY model with ferromagnetic couplings exhibits a replica symmetry broken phase, at variance with both the fully connected version of the same model and the diluted random field Ising model, as a further evidence of a more pronounced glassiness of the diluted XY model. Then, the instabilities of the spin glass XY model in an external field are characterized, recognizing different critical lines according to the different symmetries of the external field. Finally, the inherent structures in the energy landscape of the spin glass XY model in a random field are described, exploiting the capability of the zero-temperature belief propagation algorithm to actually reach the ground state of the system. Remarkably, the density of soft modes in the Hessian matrix shows a non-mean-field behaviour, typical of glasses in finite dimension, while the critical point of replica symmetry instability predicted by the belief propagation algorithm seems to correspond to a delocalization of such soft modes. Sommario Questa tesi si concentra sul modello XY, il più semplice modello con spin vettoriali, usato per descrivere diversi sistemi fisici, dai random laser ai superconduttori, dal problema della sincronizzazione ai superfluidi. Viene studiato per diverse sorgenti di disordine quenched: accoppiamenti random, campi random, o entrambi. Il mo- dello XY viene risolto su grafi di Bethe grazie all’algoritmo di belief propagation e al metodo della cavità. Si trova che la versione discreta del modello XY, il cosid- detto clock model a Q stati, fornisce un’approssimazione affidabile ed efficiente del modello continuo con un errore che va a zero esponenzialmente in Q, fornendo così un notevole guadagno nelle simulazioni numeriche. La soluzione di bassa temperatura riserva risultati interessanti e inaspettati, essendo di gran lunga più instabile verso la rottura di simmetria delle repliche rispetto a quanto accade nei modelli discreti. Inoltre, persino il modello XY ferromagnetico in campo random mostra una fase con rottura di simmetria delle repliche, a differenza di quanto accade nell’analogo modello fully connected e nel modello di Ising ferromagnetico in campo random su grafi diluiti, ad ulteriore conferma di una maggiore vetrosità del modello XY diluito. Poi, vengono caratterizzate le instabilità del modello XY spin glass in campo magnetico esterno, trovando così diverse linee critiche a se- conda delle simmetrie del campo esterno. Infine, vengono studiate le strutture inerenti del panorama energetico del modello XY spin glass in campo random, sfruttando la capacità dell’algoritmo di belief propagation a temperatura nulla di raggiungere esattamente il ground state del sistema. La densità dei modi a più bassa energia nella matrice Hessiana mostra un comportamento non di campo medio, tipico dei sistemi vetrosi in dimensione finita, mentre il punto critico della rottura di simmetria delle repliche dato dall’algoritmo di belief propagation sembra corrispondere ad una delocalizzazione di tali modi soffici. Acknowledgments This thesis is the final step of a long path started even before the PhD. Hence, there are a lot of people I should say thank you to. First of all, I would like to thank Federico. For me, he has been by far more than just a thesis advisor. I owe him a lot, starting from having taught me how to actually do research. Thanks to him, I learned that there is always a smarter way to look at things, it is just enough to find it. Then, I would like to thank Giorgio Parisi. His suggestions have always been unvaluable, providing each time a unique point of view about things. He has been for me an infinite source of new ideas and solutions. I should then thank all the large family of “Chimera” group at Sapienza, especially Andrea Maiorano and Luca Leuzzi, both always available for any question; I am very grateful to them. The two external referees, Florent Krzakala and Juan Jesús Ruiz-Lorenzo, deserve a special mention, for a sincere interest exhibited for my research and for useful suggestions about it. Last, but not the least, I would like to thank all the people with whom I had several stimulating discussions: Ada Altieri, Fabrizio Antenucci, Marco Baity-Jesi, Francesco Concetti, Carlo Lucibello, Enrico Malatesta, Alessia Marruzzo, Matteo Mori, Gabriele Perugini, Jacopo Rocchi, Francesco Zamponi, as well as all the other people I had the pleasure to met at Sapienza in Rome and during all the conferences, workshops and schools I took place in. Contents Introduction xix I Preliminaries 1 1 Statistical mechanics and the mean field 3 1.1 Statistical mechanics and thermodynamics . 3 1.2 The Ising model . 5 1.3 Variational approaches on the Gibbs free energy . 7 1.4 The naïve mean field . 9 1.5 The Bethe - Peierls approximation . 11 1.6 Belief Propagation . 14 1.7 Factor graph formalism . 18 1.8 Sparse random graphs and applicability of Belief Propagation . 21 2 Spin glasses: the replica approach 25 2.1 A new kind of magnetism . 25 2.2 The Ising spin glass model . 27 2.2.1 The Edwards - Anderson model . 27 2.2.2 The Sherrington - Kirkpatrick model . 30 2.2.3 The breaking of replica symmetry . 32 2.2.4 The nature of the spin glass phase . 38 2.3 Vector spin glasses . 40 2.3.1 The isotropic case . 40 2.3.2 The anisotropic case . 41 II The XY model and the clock model 47 3 The XY model in absence of a field 49 3.1 The model . 49 3.2 The BP equations . 50 3.2.1 Paramagnetic solution . 51 3.2.2 Expanding around the paramagnetic solution . 52 3.2.3 Scaling of the Fourier coefficients below Tc ........... 57 3.3 The RS cavity method . 59 3.3.1 Population Dynamics Algorithm . 59 xvi Contents 3.3.2 Numerical analysis of the Fourier expansion . 63 3.3.3 Exploring the low-temperature region . 63 3.3.4 Susceptibility Propagation . 66 3.4 The limit of zero temperature . 69 3.5 Phase diagram for the bimodal distribution . 74 3.6 The gauge glass model . 75 4 Discretizing the XY model: the clock model 79 4.1 From the XY model to the clock model . 79 4.2 RS solution of the BP equations in the bimodal case . 80 4.2.1 Paramagnetic solution and expansion around it . 81 4.2.2 Convergence of the discretized Bessel functions . 83 4.2.3 Low- and zero-temperature solutions . 86 4.3 Phase diagrams of the clock model . 90 4.4 Convergence of physical observables . 93 4.4.1 The finite-temperature regime . 93 4.4.2 The zero-temperature regime . 98 4.5 Beyond the RS solution . 102 4.5.1 The 1RSB picture . 103 4.5.2 The 1RSB BP equations . 105 4.5.3 1RSB solution of the Q-state clock model . 109 III The XY model in a field 113 5 The random field XY model 115 5.1 From random couplings to random fields . 115 5.2 A brief overview of the random field Ising model . 116 5.3 Back to the XY model . 118 5.3.1 The paramagnetic phase . 118 5.3.2 The low-temperature – low-field region . 121 6 The spin glass XY model in a field 127 6.1 Instabilities in the fully connected case .