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Response Functions of Correlated Systems in the Linear Regime and Beyond

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Response Functions of Correlated Systems in the Linear Regime and Beyond Response functions of correlated systems in the linear regime and beyond Dissertation der Mathematisch-Naturwissenschaftlichen Fakultät der Eberhard Karls Universität Tübingen zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) vorgelegt von M.Sc. Agnese Tagliavini aus Ferrara, Italien Tübingen, 2018 Gedruckt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakultät der Eber- hard Karls Universität Tübingen. Tag der mündlichen Qualifikation: 9 November 2018 Dekan: Prof. Dr. Wolfgang Rosenstiel 1. Berichterstatter: Prof. Dr. Sabine Andergassen 2. Berichterstatter: Prof. Dr. Alessandro Toschi Abstract The technological progress in material science has paved the way to engineer new con- densed matter systems. Among those, fascinating properties are found in presence of strong correlations among the electrons. The maze of physical phenomena they exhibit is coun- tered by the difficulty in devising accurate theoretical descriptions. This explains the plethora of approximated theories aiming at the closest reproduction of their properties. In this respect, the response of the system to an external perturbation bridges the experi- mental evidence with the theoretical description, representing a testing ground to prove or disprove the validity of the latter. In this thesis we develop a number of theoretical and numerical strategies to improve the computation of linear response functions in several of the forefront many-body techniques. In particular, the development of computationally efficient schemes, driven by physical arguments, has allowed (i) improvements in treating the local two-particle correlations and scattering functions, which represent essential building blocks for established non- perturbative theories, such as dynamical mean field theory (DMFT) and its diagrammatic extensions and (ii) the implementation of the groundbreaking multiloop functional renor- malization group (mfRG) technique and its application to a two-dimensional Hubbard model. Together with (i), the multiloop scheme has promoted the functional RG to pro- vide quantitative predictions. As the mfRG is able to build up a complete subset of Feyn- man diagrams, such as those of the parquet approximation, its results are independent on the choice of the cutoff scheme as well as on the way (direct or through a post-processing treatment) response functions are calculated. We demonstrate by hand of precise numerical calculations that both properties are fulfilled by a satisfactory degree of accuracy. We also elaborate how these properties could be exploited in the future for improving the numeri- cal solution of parquet-based algorithms. Finally, our study reveals that an important piece of physical information can be accessed by looking at the nonlinear response of the system to an external field. In particular, a DMFT study of the pairing response function to a superconducting probe beyond the lin- ear regime, has pinpointed a simple criterion to identify the presence of a preformed pair phase. This result provides a complementary information to cutting-edge theoretical ap- proaches and, possibly, to non-equilibrium experiments, to shed some light on the nature of the pseudogap in high-Tc superconductors. iii Zusammenfassung Der technologische Fortschritt in der Materialwissenschaft hat die Entwicklung neuar- tiger Systeme ermöglicht, deren starke elektronische Korrelationen faszinierende Eigen- schaften aufweisen. Der Vielzahl an physikalischen Phänomenen steht die Schwierigkeit einer genauen theoretischen Beschreibung gegenüber. Das hat Anlass zur Entwicklung ver- schiedener theoretischer Näherungsmethoden gegeben. In diesem Zusammenhang stellt die Antwort des Systems auf eine externe Störung eine Möglichkeit dar, eine theoretische Beschreibung durch experimentelle Nachweise zu untermauern oder zu widerlegen. In dieser Doktorarbeit entwicklen wir theoretische und numerische Ansätze, um die Berech- nung der linearen Antwortfunktionen in verschiedenen modernen Quantenvielteilchen- methoden zu verbessern. Die effiziente numerische Handhabung, die durch physikalis- che Überlegungen inspiriert ist, gestattete einerseits (i) Verbesserungen in der Behand- lung von zweiteilchen Korrelationen sowie Streufunktionen, die wiederum essentielle Be- standteile nicht-perturbativer Theorien sind, wie z.B. der dynamischen Molekularfeldthe- orie (DMFT) und ihrer diagrammatischen Erweiterungen, und ermöglichte andererseits (ii) die Implementierung der wegweisenden multiloop-funktionalen Renormierungsgrup- pentheorie (mfRG) und deren Anwendung auf das zweidimensionale Hubbard Modell. Durch die Errungenschaften aus (i) und der Verwendung des Multiloopschemas ermöglicht die funktionale RG quantitative Vorhersagen. Die inhärente Eigenschaft der mfRG, eine vollständige Teilmenge der Feynmandiagramme zu erzeugen, hier insbesondere die Dia- gramme der parquet Näherung, fuehrt ausserdem dazu, dass ihre Resultate “cutoff” un- abhängig sind und die Art und Weise der Berechnung der Antwortfunktionen (entweder mittels des Renormierungsgruppenflusses oder als “post-processing” der fRG Rechnung) keine Rolle spielen. Wir zeigen anhand numerischer Berechnungen, dass beide oben genan- nten analytischen Eigenschaften der mfRG erfüllt sind. Außerdem diskutieren wir, dass diese Eigenschaften einen bedeutenden Beitrag in der numerischen Konvergens parque- basierter Algorithmen liefern können. Schließlich zeigt unsere Untersuchung, dass die nichtlineare Antwort eines Materials auf ein externes Störungsfeld wichtige physikalische Information enthält. Insbesondere hat eine Betrachtung der Antwortfunktion der Paarbildung auf ein Supraleitungsfeld im nicht- linearen Regime im Rahmen der DMFT, ein einfaches Kriterium für das Vorhandensein einer “preformed pair phase” aufgezeigt. Dieses Resultat liefert neue Einblicke in aktuelle theoretische und experimentelle Methoden der Vielteilchenphysik, um die Hintergründe der Entsehung einer Pseudogap in Hochtemperatursupraleitern zu beleuchten. iv List of diagrammatic symbols Self-energy: Σ 1P Green’s function: G Used in both non-interacng and interact- ing (“dressed”) cases. The arrow specifies the direcon of propagaon. 2P vertex functions: Filled square ! Full 2P vertex γ4. Empty square ! 2P reducible vertex ϕr: (a) Red framed for r = pp; (b) Green framed for r = ph; (c) Blue framed for r = ph. Diagonally striped square ! 2PI vertex Ir: (a) Green-blue stripes for r = pp; (b) Red-blue stripes for r = ph; (c) Green-red stripes for r = ph. 3P vertex function: γ6 nonlocal interaction: Filled wiggle line: nonlocal (possibly dynamic) effecve interacon (Veff) Simple wiggle line: nonlocal (bare) Coulomb interacon (V (q)) Physical susceptibility: χη=sc,d,m Filled circe: Full vertex-corrected suscepbility Empty circle: Non-vertex corrected suscepbil- η ity (bare bubble) χ0 v η=sc,d,m 3-point vertex: γ3 Filled triangle: Full screened 3-point fermion- boson vertex Small empty circle: Bare 3-point fermion-boson η vertex γ0;3 η=sc,d,m 4-point vertex: γ~4 4-point fermion-boson vertex η=sc,d,m 5-point vertex: γ5 5-point fermion-boson vertex vi Contents List of Diagrammatic Symbols v 1 Introduction 1 1.1 Introduction and Motivation ........................ 1 1.2 Structure of the work ............................ 7 2 Theory and formalism 9 2.1 Response function theory: linear regime and beyond ............ 9 2.1.1 Microscopic Response Theory: formalism and analytical properties 9 2.1.2 Kramers-Kronig relations ...................... 11 2.1.3 Thermodynamic susceptibility in functional integral formalism .. 12 2.1.4 Correlation function of fermionic bilinears: from asymptotic solutions to diagrammatic approaches .................... 15 2.1.5 Critical phenomena: fluctuations and response functions ..... 25 2.2 Investigating response functions in many-body systems: an overview .... 27 2.2.1 Lindhard response theory ..................... 28 2.2.2 Linear response function in dynamical mean-field theory ..... 29 2.2.3 Beyond DMFT: nonlocal effects at the two-particle level in DΓA .. 31 2.2.4 Linear response function in the functional renormalization group . 33 2.2.5 Beyond DMFT: nonlocal effects at two-particle level in DMF2RG . 39 3 Full many-body treatment of the response functions: from vertex corrections to higher order contributions 41 3.1 The high-frequency regime of the two-particle vertex: an AIM study .... 43 3.2 From DMFT to its diagrammatic extensions ................ 49 3.3 nonlocal response functions in multiloop functional RG .......... 54 3.4 Beyond the linear response regime: A DMFT study of the pseudogap physics 62 4 Conclusions and Outlook 67 Appendix A Lehmann representation of the linear response function and the fermion- boson vertex 73 vii Appendix B SU(2)P symmetry: degeneracy of nonlocal susceptibilities 77 Supplements 83 Personal contribution to publications ....................... 83 I High-frequency asymptotics of the vertex function: diagrammatic parametriza- tion and algorithmic implementation .................... 87 II Efficient Bethe-Salpeter equations treatment in dynamical mean-field theory . 113 III Multiloop functional renormalization group for the two-dimensional Hub- bard model: Loop convergence of the response functions ......... 131 IV Detecting a preformed pair phase: Response to a pairing forcing field .... 179 References 210 viii Tomy grandfather Franco and his unstoppable sense of humor. ix Acknowledgments The last years have been a challenging adventure both form a scientific and, much more important, form
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