Theses

Dynamic response functions of strongly correlated electrons - Results from real-frequency quantum field theory

Ph.D. thesis, Ludwig-Maximilians-University (LMU) Munich, April 2025

Supervisor: Prof. Jan von Delft

Summary: 

The quantitative description of correlated electron systems remains one of the central challenges in theoretical condensed matter physics. Many fascinating phenomena, such as high-temperature superconductivity or the formation of a pseudogap in cuprates and other correlated materials, are driven by strong electron-electron interactions, rendering perturbative techniques insufficient in many regimes. Of particular importance are dynamical correlation functions, which characterize how such systems respond to time-dependent external perturbations and are key to interpreting a wide range of experimental observations.

This thesis is dedicated to accurately computing dynamic response functions in interacting fermionic systems. The introductory chapters revisit the physical significance of these functions and outline experimental techniques to probe them. Special emphasis is placed on four-point functions, whose contributions to dynamic response functions are essential for both qualitative and, in particular, quantitative accuracy.

The central theme of the thesis is the direct computation of these quantities in real frequencies, as opposed to the more common imaginary-frequency approach. This choice circumvents the notoriously ill-conditioned problem of analytic continuation plaguing the accuracy and reliability of numerical studies. To this end, we adopt the real-frequency Keldysh formalism applied here in thermal equilibrium. Within this framework, two main quantum field-theoretical approaches are presented: the parquet equations, which describe self-consistent relations for the two-point self-energy and four-point vertex at the two-particle level, and the functional renormalization group (fRG), which offers a renormalization-group perspective at the level of correlation functions.

We demonstrate that the full three-dimensional real-frequency structure of the four-point vertex can be accurately resolved in this formalism. As a concrete example, we solve the parquet and fRG equations for the single-impurity Anderson model (SIAM) and provide a detailed account of the numerous technical challenges encountered and overcome.

In the final part of the thesis, we discuss pathways toward extending these real-frequency methods to spatially extended, correlated lattice systems. We argue that combining the above diagrammatic approaches with dynamical mean-field theory (DMFT) offers a promising strategy, using DMFT as a non-perturbative local starting point. However, this requires a DMFT impurity solver capable of computing two- and four-point functions. The newly developed multipoint extension to the numerical renormalization group (mpNRG) is a method for this purpose, though some numerical limitations persist at the four-point level. We, therefore, perform extensive consistency checks of mpNRG results, verifying the parquet equations and a Ward identity that we derive in full generality within the Keldysh formalism for the first time. With only a few exceptions, these relations are found to hold with high accuracy, validating the use of mpNRG in future diagrammatic extensions of DMFT. Finally, we highlight the potential of the quantics tensor cross interpolation (QTCI) method to find compressed representations of dynamic response functions efficiently. This technique shows considerable promise in managing the computational demands of future large-scale calculations.

In summary, this thesis establishes a robust framework for computing real-frequency response functions in strongly correlated electron systems. Through proof-of-principle quantum field theory calculations on the SIAM and thorough consistency checks of the mpNRG method, it lays a solid foundation for future extensions to correlated lattice systems, which compression techniques such as the QTCI promise to make computationally feasible.

Quantum Field Theory of Periodically Driven Bosonic Many-Particle Systems in Two Spatial Dimensions

Master's thesis, Technical University Munich (TUM), January 2020

Supervisor: Prof. Michael Knap

Summary: 

What happens to interacting bosonic systems, like the weakly interacting Bose gas or the Bose-Hubbard model, once they are coupled to a periodic drive? As interacting systems, they are believed to heat up to the infinite-temperature state eventually. The question is whether these models show some interesting behavior on the way, like a prethermal plateau in the energy, as recently discovered for the O(N)-model [1]. The consequent small heating rates would pave the way for studying newly discovered exotic topological phases of matter using bosonic systems of ultracold atoms. In this thesis we use the Schwinger-Keldysh approach to quantum field theory to answer this question. The formalism provides the time-evolution of one- and two-point correlation functions which let us compute physical observables like the particle number occupations in momentum space, condensate fractions and energy densities. We consider different approximation methods to treat the effects of the interactions. Focusing on the periodically driven Bose-Hubbard model we start from the Bogoliubov-approximation for the ground state in the superfluid phase and drive the system out of equilibrium. Using different driving parameters we extend the results of [2] by taking interaction effects into account systematically. We find that in the experimentally relevant regimes it is not justified to neglect scattering effects between quasiparticle excitations when approximating the bosonic self-energy. Nevertheless, our results for the short-time dynamics indeed indicate the existence of a prethermal steady-state for specific driving parameters.

Characterization of Many-Body Localization in One-Dimensional Spin Systems

Bachelor's thesis, Technical University Munich (TUM), September 2017

Supervisor: Prof. Michael Knap

Summary: 

Disorderd many-body quantum systems can fail to evolve into thermal equilibrium even in the presence of weak interactions. This phenomenon is known as many-body localization (MBL) and fundamentally different from the usual case of thermalization. In this thesis we introduce these two phenomena as well as Anderson localization for which disorder but no interactions have to be present. Afterwards we review a possibility to distinguish MBL from Anderson localization using spin-echo type protocols. In the end we support our analytical considerations by numerical simulations.